823 lines
92 KiB
TypeScript
823 lines
92 KiB
TypeScript
import { type PracticeQuestion } from "../../types/lesson";
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export const LINEAR_INEQ_EASY: PracticeQuestion[] = [
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{
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id: "2c121b25",
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type: "mcq",
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questionHtml:
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"Valentina bought two containers of beads. In the first container 30% of the beads are red, and in the second container 70% of the beads are red. Together, the containers have at least 400 red beads. Which inequality shows this relationship, where x is the total number of beads in the first container and y is the total number of beads in the second container?",
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choices: [
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{ label: "A", text: "<strong>0 . 3 x + 0 . 7 y ≥ 400</strong>" },
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{ label: "B", text: "<strong>0 . 7 x + 0 . 3 y ≤ 400</strong>" },
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{
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label: "C",
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text: "<strong>the fraction x over 3, end fraction + the fraction y over 7, end fraction ≤ 400</strong>",
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},
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{ label: "D", text: "<strong>30 x + 70 y ≥ 400</strong>" },
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],
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correctAnswer: "A",
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explanation:
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"Choice A is correct. It is given that x is the total number of beads in the first container and that 30% of those beads are red; therefore, the expression 0.3x represents the number of red beads in the first container. It is given that y is the total number of beads in the second container and that 70% of those beads are red; therefore, the expression 0.7y represents the number of red beads in the second container. It is also given that, together, the containers have at least 400 red beads, so the inequality that shows this relationship is 0.3x + 0.7y ≥ 400.Choice B is incorrect because it represents the containers having a total of at most, rather than at least, 400 red beads. Choice C is incorrect and may be the result of misunderstanding how to represent a percentage of beads in each container. Also, the inequality shows the containers having a combined total of at most, rather than at least, 400 red beads. Choice D is incorrect because the percentages were not converted to decimals.",
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hasFigure: false,
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},
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{
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id: "563407e5",
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type: "mcq",
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questionHtml:
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"A bakery sells trays of cookies. Each tray contains at least 50 cookies but no more than 60. Which of the following could be the total number of cookies on 4 trays of cookies?",
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choices: [
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{ label: "A", text: "165" },
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{ label: "B", text: "205" },
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{ label: "C", text: "245" },
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{ label: "D", text: "285" },
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],
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correctAnswer: "B",
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explanation:
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"Choice B is correct. If each tray contains the least number of cookies possible, 50 cookies, then the least number of cookies possible on 4 trays is 50 × 4 = 200 cookies. If each tray contains the greatest number of cookies possible, 60 cookies, then the greatest number of cookies possible on 4 trays is 60 × 4 = 240 cookies. If the least number of cookies on 4 trays is 200 and the greatest number of cookies is 240, then 205 could be the total number of cookies on these 4 trays of cookies because <strong>200 ≤ 205, which ≤ 240.</strong>.Choices A, C, and D are incorrect. The least number of cookies on 4 trays is 200 cookies, and the greatest number of cookies on 4 trays is 240 cookies. The choices 165, 245, and 285 are each either less than 200 or greater than 240; therefore, they cannot represent the total number of cookies on 4 trays.",
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hasFigure: false,
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},
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{
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id: "59a49431",
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type: "mcq",
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questionHtml:
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"The boundary of the inequality is a solid line.<br><br>The line slants sharply down from left to right.<br>The line passes through the following points:<br><br>(1.0 comma 6.5)<br>(3.0 comma negative 4.5)<br><br>The area above and to the right of the boundary is shaded.<br><br>The shaded region shown represents solutions to an inequality. Which ordered pair <strong>(x, y)</strong> is a solution to this inequality?",
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choices: [
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{ label: "A", text: "<strong>(0 −4)</strong>" },
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{ label: "B", text: "<strong>(0, 4)</strong>" },
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{ label: "C", text: "<strong>(−4, 0)</strong>" },
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{ label: "D", text: "<strong>(4, 0)</strong>" },
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],
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correctAnswer: "D",
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explanation:
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"Choice D is correct. Since the shaded region shown represents solutions to an inequality, an ordered pair <strong>(x, y)</strong> is a solution to the inequality if it's represented by a point in the shaded region. Of the given choices, only <strong>(4, 0)</strong> is represented by a point in the shaded region. Therefore, <strong>(4, 0)</strong> is a solution to the inequality.<br>Choice A is incorrect and may result from conceptual errors.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.",
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hasFigure: true,
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figureUrl: "/practice-images/59a49431_svg1.svg",
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},
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{
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id: "68f2cbaf",
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type: "mcq",
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questionHtml:
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"Ty set a goal to walk at least <strong>24</strong> kilometers every day to prepare for a multiday hike. On a certain day, Ty plans to walk at an average speed of <strong>4</strong> kilometers per hour. What is the minimum number of hours Ty must walk on that day to fulfill the daily goal?",
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choices: [
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{ label: "A", text: "<strong>4</strong>" },
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{ label: "B", text: "<strong>6</strong>" },
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{ label: "C", text: "<strong>20</strong>" },
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{ label: "D", text: "<strong>24</strong>" },
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],
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correctAnswer: "B",
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explanation:
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"Choice B is correct. It's given that Ty plans to walk at an average speed of <strong>4</strong> kilometers per hour. The number of kilometers Ty will walk is determined by the expression <strong>4 s</strong>, where <strong>s</strong> is the number of hours Ty walks. The given goal of at least <strong>24</strong> kilometers means that the inequality <strong>4 s > or = 24</strong> represents the situation. Dividing both sides of this inequality by <strong>4</strong> gives <strong>s > or = 6</strong> , which corresponds to a minimum of <strong>6</strong> hours Ty must walk. <br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
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hasFigure: false,
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},
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{
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id: "7d6928bd",
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type: "mcq",
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questionHtml:
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"A cleaning service that cleans both offices and homes can clean at most <strong>14</strong> places per day. Which inequality represents this situation, where <strong>f</strong> is the number of offices and <strong>h</strong> is the number of homes?",
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choices: [
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{ label: "A", text: "<strong>f + h < or = 14</strong>" },
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{ label: "B", text: "<strong>f + h > or = 14</strong>" },
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{ label: "C", text: "<strong>f − h < or = 14</strong>" },
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{ label: "D", text: "<strong>f − h > or = 14</strong>" },
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],
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correctAnswer: "A",
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explanation:
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"Choice A is correct. It's given that the cleaning service cleans both offices and homes, where <strong>f</strong> is the number of offices and <strong>h</strong> is the number of homes the cleaning service can clean per day. Therefore, the expression <strong>f + h</strong> represents the number of places the cleaning service can clean per day. It's also given that the cleaning service can clean at most <strong>14</strong> places per day. Since <strong>f + h</strong> represents the number of places the cleaning service can clean per day and the service can clean at most <strong>14</strong> places per day, it follows that the inequality <strong>f + h < or = 14</strong> represents this situation.<br>Choice B is incorrect. This inequality represents a cleaning service that cleans at least <strong>14</strong> places per day.<br>Choice C is incorrect. This inequality represents a cleaning service that cleans at most <strong>14</strong> more offices than homes per day.<br>Choice D is incorrect. This inequality represents a cleaning service that cleans at least <strong>14</strong> more offices than homes per day.",
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hasFigure: false,
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},
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{
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id: "84d0d07e",
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type: "mcq",
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questionHtml:
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"A clothing store is having a sale on shirts and pants. During the sale, the cost of each shirt is $15 and the cost of each pair of pants is $25. Geoff can spend at most $120 at the store. If Geoff buys s shirts and p pairs of pants, which of the following must be true?",
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choices: [
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{ label: "A", text: "<strong>15 s + 25 p ≤ 120</strong>" },
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{ label: "B", text: "<strong>15 s + 25 p ≥ 120</strong>" },
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{ label: "C", text: "<strong>25 s + 15 p ≤ 120</strong>" },
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{ label: "D", text: "<strong>25 s + 15 p ≥ 120</strong>" },
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],
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correctAnswer: "A",
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explanation:
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"Choice A is correct. Since the cost of each shirt is $15 and Geoff buys s shirts, the expression <strong>15 s</strong> represents the amount Geoff spends on shirts. Since the cost of each pair of pants is $25 and Geoff buys p pairs of pants, the expression <strong>25 p</strong> represents the amount Geoff spends on pants. Therefore, the sum <strong>15 s + 25 p</strong> represents the total amount Geoff spends at the store. Since Geoff can spend at most $120 at the store, the total amount he spends must be less than or equal to 120. Thus, <strong>15 s + 25 p ≤ 120</strong>.Choice B is incorrect. This represents the situation in which Geoff spends at least, rather than at most, $120 at the store. Choice C is incorrect and may result from reversing the cost of a shirt and that of a pair of paints. Choice D is incorrect and may result from both reversing the cost of a shirt and that of a pair of pants and from representing a situation in which Geoff spends at least, rather than at most, $120 at the store.",
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hasFigure: false,
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},
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{
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id: "89541f9b",
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type: "mcq",
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questionHtml:
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"Which of the following ordered pairs <strong>x, y</strong> satisfies the inequality <strong>5 x − 3 y < 4</strong> ?<br> <br> <br> <strong>Statement 1, 1, 1</strong><br> <br> <br> <br> <br> <br> <strong>Statement 2, 2, 5</strong><br> <br> <br> <br> <br> <br> <strong>Statement 3, 3, 2</strong>",
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choices: [
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{ label: "A", text: "I only" },
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{ label: "B", text: "II only" },
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{ label: "C", text: "I and II only" },
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{ label: "D", text: "I and III only" },
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],
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correctAnswer: "C",
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explanation:
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"Choice C is correct. Substituting <strong>the ordered pair 1, 1</strong> into the inequality gives <strong>5 · 1 − 3 · 1 < 4</strong>, or <strong>2 < 4</strong>, which is a true statement. Substituting <strong>the ordered pair 2, 5</strong> into the inequality gives <strong>5 · 2 − 3 · 5 < 4</strong>, or <strong>−5 < 4</strong>, which is a true statement. Substituting <strong>the ordered pair 3, 2</strong> into the inequality gives <strong>5 · 3 − 3 · 2 < 4</strong>, or <strong>9 < 4</strong>, which is not a true statement. Therefore, <strong>the ordered pair 1, 1</strong> and <strong>the ordered pair 2, 5</strong> are the only ordered pairs shown that satisfy the given inequality.Choice A is incorrect because the ordered pair <strong>2, 5</strong> also satisfies the inequality. Choice B is incorrect because the ordered pair <strong>1, 1</strong> also satisfies the inequality. Choice D is incorrect because the ordered pair <strong>3, 2</strong> does not satisfy the inequality.",
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hasFigure: false,
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},
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{
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id: "915463e0",
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type: "mcq",
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questionHtml:
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"Normal body temperature for an adult is between <strong>97 . 8 ° Fahrenheit</strong> and <strong>99 ° Fahrenheit</strong>, inclusive. If Kevin, an adult male, has a body temperature that is considered to be normal, which of the following could be his body temperature?",
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choices: [
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{ label: "A", text: "<strong>96 . 7 ° Fahrenheit</strong>" },
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{ label: "B", text: "<strong>97 . 6 ° Fahrenheit</strong>" },
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{ label: "C", text: "<strong>97 . 9 ° Fahrenheit</strong>" },
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{ label: "D", text: "<strong>99 . 7 ° Fahrenheit</strong>" },
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],
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correctAnswer: "C",
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explanation:
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"Choice C is correct. Normal body temperature must be greater than or equal to 97.8°F but less than or equal to 99°F. Of the given choices, 97.9°F is the only temperature that fits these restrictions.Choices A and B are incorrect. These temperatures are less than 97.8°F, so they don’t fit the given restrictions. Choice D is incorrect. This temperature is greater than 99°F, so it doesn’t fit the given restrictions.",
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hasFigure: false,
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},
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{
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id: "b64e2c7f",
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type: "mcq",
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questionHtml:
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"Monarch butterflies can fly only with a body temperature of at least <strong>55.0 ° Fahrenheit (° F)</strong>. If a monarch butterfly's body temperature is <strong>51.3 ° F</strong>, what is the minimum increase needed in its body temperature, in <strong>° F</strong>, so that it can fly?",
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choices: [
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{ label: "A", text: "<strong>1.3</strong>" },
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{ label: "B", text: "<strong>3.7</strong>" },
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{ label: "C", text: "<strong>5.0</strong>" },
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{ label: "D", text: "<strong>6.3</strong>" },
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],
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correctAnswer: "B",
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explanation:
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"Choice B is correct. It's given that monarch butterflies can fly only with a body temperature of at least <strong>55.0 ° Fahrenheit (° F)</strong>. Let <strong>x</strong> represent the minimum increase needed in the monarch butterfly's body temperature to fly. If the monarch butterfly's body temperature is <strong>51.3 ° F</strong>, the inequality <strong>51.3 + x > or = 55.0</strong> represents this situation. Subtracting <strong>51.3</strong> from both sides of this inequality yields <strong>x > or = 3.7</strong>. Therefore, if the monarch butterfly's body temperature is <strong>51.3 ° F</strong>, the minimum increase needed in its body temperature, in <strong>° F</strong>, so that it can fly is <strong>3.7</strong>.<br>Choice A is incorrect. This is the minimum increase needed in body temperature if the monarch butterfly's body temperature is <strong>53.7 ° F</strong>, not <strong>51.3 ° F</strong>.<br>Choice C is incorrect. This is the minimum increase needed in body temperature if the monarch butterfly's body temperature is <strong>50.0 ° F</strong>, not <strong>51.3 ° F</strong>.<br>Choice D is incorrect. This is the minimum increase needed in body temperature if the monarch butterfly's body temperature is <strong>48.7 ° F</strong>, not <strong>51.3 ° F</strong>.",
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hasFigure: false,
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},
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{
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id: "b75f7812",
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type: "spr",
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questionHtml:
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"Maria plans to rent a boat. The boat rental costs $60 per hour, and she will also have to pay for a water safety course that costs $10. Maria wants to spend no more than $280 for the rental and the course. If the boat rental is available only for a whole number of hours, what is the maximum number of hours for which Maria can rent the boat?",
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choices: [],
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correctAnswer: "",
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explanation:
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"The correct answer is 4. The equation <strong>60 h + 10 ≤ 280</strong>, where h is the number of hours the boat has been rented, can be written to represent the situation. Subtracting 10 from both sides and then dividing by 60 yields <strong>h ≤ 4 . 5</strong>. Since the boat can be rented only for whole numbers of hours, the maximum number of hours for which Maria can rent the boat is 4.",
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hasFigure: false,
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},
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{
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id: "c50ede6d",
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type: "mcq",
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questionHtml:
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"The total cost, in dollars, to rent a surfboard consists of a <strong>dollar sign 25</strong> service fee and a <strong>dollar sign 10</strong> per hour rental fee. A person rents a surfboard for <strong>t</strong> hours and intends to spend a maximum of <strong>dollar sign 75</strong> to rent the surfboard. Which inequality represents this situation?",
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choices: [
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{ label: "A", text: "<strong>10 t < or = 75</strong>" },
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{ label: "B", text: "<strong>10 + 25 t < or = 75</strong>" },
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{ label: "C", text: "<strong>25 t < or = 75</strong>" },
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{ label: "D", text: "<strong>25 + 10 t < or = 75</strong>" },
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],
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correctAnswer: "D",
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explanation:
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"Choice D is correct. The cost of the rental fee depends on the number of hours the surfboard is rented. Multiplying <strong>t</strong> hours by <strong>10</strong> dollars per hour yields a rental fee of <strong>10 t</strong> dollars. The total cost of the rental consists of the rental fee plus the <strong>25</strong> dollar service fee, which yields a total cost of <strong>25 + 10 t</strong> dollars. Since the person intends to spend a maximum of <strong>75</strong> dollars to rent the surfboard, the total cost must be at most <strong>75</strong> dollars. Therefore, the inequality <strong>25 + 10 t < or = 75</strong> represents this situation.<br>Choice A is incorrect. This represents a situation where the rental fee, not the total cost, is at most <strong>75</strong> dollars.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
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hasFigure: false,
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},
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{
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id: "cfe67646",
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type: "mcq",
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questionHtml:
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"The point <strong>(8, 2)</strong> in the xy-plane is a solution to which of the following systems of inequalities?",
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choices: [
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{ label: "A", text: "<strong>x > 0</strong><br><strong>y > 0</strong>" },
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{ label: "B", text: "<strong>x > 0</strong><br><strong>y < 0</strong>" },
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{ label: "C", text: "<strong>x < 0</strong><br><strong>y > 0</strong>" },
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{ label: "D", text: "<strong>x < 0</strong><br><strong>y < 0</strong>" },
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],
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correctAnswer: "A",
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explanation:
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"Choice A is correct. The given point, <strong>(8, 2)</strong>, is located in the first quadrant in the xy-plane. The system of inequalities in choice A represents all the points in the first quadrant in the xy-plane. Therefore, <strong>(8, 2)</strong> is a solution to the system of inequalities in choice A.<br>Alternate approach: Substituting <strong>8</strong> for <strong>x</strong> in the first inequality in choice A, <strong>x > 0</strong>, yields <strong>8 > 0</strong>, which is true. Substituting <strong>2</strong> for <strong>y</strong> in the second inequality in choice A, <strong>y > 0</strong>, yields <strong>2 > 0</strong>, which is true. Since the coordinates of the point <strong>(8, 2)</strong> make the inequalities <strong>x > 0</strong> and <strong>y > 0</strong> true, the point <strong>(8, 2)</strong> is a solution to the system of inequalities consisting of <strong>x > 0</strong> and <strong>y > 0</strong>.<br>Choice B is incorrect. This system of inequalities represents all the points in the fourth quadrant, not the first quadrant, in the xy-plane.<br>Choice C is incorrect. This system of inequalities represents all the points in the second quadrant, not the first quadrant, in the xy-plane.<br>Choice D is incorrect. This system of inequalities represents all the points in the third quadrant, not the first quadrant, in the xy-plane.",
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hasFigure: false,
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},
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{
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id: "df32b09c",
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type: "mcq",
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questionHtml:
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"Tom scored 85, 78, and 98 on his first three exams in history class. Solving which inequality gives the score, G, on Tom’s fourth exam that will result in a mean score on all four exams of at least 90 ?",
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choices: [
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{ label: "A", text: "<strong>90 − (85 + 78 + 98, ) ≤ 4 G</strong>" },
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{ label: "B", text: "<strong>4 G + 85 + 78 + 98 ≥ 360</strong>" },
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{
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label: "C",
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text: "<strong>the fraction with numerator, (G + 85 + 78 + 98, ), and denominator 4, end fraction ≥ 90</strong>",
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},
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{
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label: "D",
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text: "<strong>the fraction with numerator, (85 + 78 + 98, ), and denominator 4, end fraction ≥ 90 − 4 G</strong>",
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},
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],
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correctAnswer: "C",
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explanation:
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"Choice C is correct. The mean of the four scores (G, 85, 78, and 98) can be expressed as <strong>the fraction with numerator G + 85 + 78 + 98, and denominator 4</strong>. The inequality that expresses the condition that the mean score is at least 90 can therefore be written as <strong>the fraction with numerator G + 85 + 78 + 98, and denominator 4 ≥ 90</strong>.Choice A is incorrect. The sum of the scores (G, 85, 78, and 98) isn’t divided by 4 to express the mean. Choice B is incorrect and may be the result of an algebraic error when multiplying both sides of the inequality by 4. Choice D is incorrect because it doesn’t include G in the mean with the other three scores.",
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hasFigure: false,
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},
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{
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id: "e744499e",
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type: "mcq",
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questionHtml:
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"An elementary school teacher is ordering x workbooks and y sets of flash cards for a math class. The teacher must order at least 20 items, but the total cost of the order must not be over $80. If the workbooks cost $3 each and the flash cards cost $4 per set, which of the following systems of inequalities models this situation?",
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choices: [
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{ label: "A", text: "<strong>x + y ≥ 20, and, 3 x + 4 y ≤ 80</strong>" },
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{ label: "B", text: "<strong>x + y ≥ 20, and, 3 x + 4 y ≥ 80</strong>" },
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{ label: "C", text: "<strong>3 x + 4 y ≤ 20, and, x + y ≥ 80</strong>" },
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{ label: "D", text: "<strong>x + y ≤ 20, and, 3 x + 4 y ≥ 80</strong>" },
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],
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correctAnswer: "A",
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explanation:
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"Choice A is correct. The total number of workbooks and sets of flash cards ordered is represented by x + y. Since the teacher must order at least 20 items, it must be true that x + y ≥ 20. Each workbook costs $3; therefore, 3x represents the cost, in dollars, of x workbooks. Each set of flashcards costs $4; therefore, 4y represents the cost, in dollars, of y sets of flashcards. It follows that the total cost for x workbooks and y sets of flashcards is 3x + 4y. Since the total cost of the order must not be over $80, it must also be true that 3x + 4y ≤ 80. Of the choices given, these inequalities are shown only in choice A.<br> <br><br>Choice B is incorrect. The second inequality says that the total cost must be greater, not less than or equal to $80. Choice C incorrectly limits the cost by the minimum number of items and the number of items with the maximum cost. Choice D is incorrect. The first inequality incorrectly says that at most 20 items must be ordered, and the second inequality says that the total cost of the order must be at least, not at most, $80.",
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hasFigure: false,
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},
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{
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id: "ee439cff",
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||
type: "mcq",
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questionHtml:
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"On a car trip, Rhett and Jessica each drove for part of the trip, and the total distance they drove was under <strong>220</strong> miles. Rhett drove at an average speed of <strong>35 miles per hour (mph)</strong>, and Jessica drove at an average speed of <strong>40 mph</strong>. Which of the following inequalities represents this situation, where <strong>r</strong> is the number of hours Rhett drove and <strong>j</strong> is the number of hours Jessica drove?",
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choices: [
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{ label: "A", text: "<strong>35 r + 40 j > 220</strong>" },
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{ label: "B", text: "<strong>35 r + 40 j < 220</strong>" },
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{ label: "C", text: "<strong>40 r + 35 j > 220</strong>" },
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{ label: "D", text: "<strong>40 r + 35 j < 220</strong>" },
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],
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correctAnswer: "B",
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explanation:
|
||
"Choice B is correct. It’s given that Rhett drove at an average speed of <strong>35</strong> miles per hour and that he drove for <strong>r</strong> hours. Multiplying <strong>35</strong> miles per hour by <strong>r</strong> hours yields <strong>35 r</strong> miles, or the distance that Rhett drove. It’s also given that Jessica drove at an average speed of <strong>40</strong> miles per hour and that she drove for <strong>j</strong> hours. Multiplying <strong>40</strong> miles per hour by <strong>j</strong> hours yields <strong>40 j</strong> miles, or the distance that Jessica drove. The total distance, in miles, that Rhett and Jessica drove can be represented by the expression <strong>35 r + 40 j</strong>. It’s given that the total distance they drove was under <strong>220</strong> miles. Therefore, the inequality <strong>35 r + 40 j < 220</strong> represents this situation.<br>Choice A is incorrect. This inequality represents a situation in which the total distance Rhett and Jessica drove was over, rather than under, <strong>220</strong> miles.<br>Choice C is incorrect. This inequality represents a situation in which Rhett drove at an average speed of <strong>40</strong>, rather than <strong>35</strong>, miles per hour, Jessica drove at an average speed of <strong>35</strong>, rather than <strong>40</strong>, miles per hour, and the total distance they drove was over, rather than under, <strong>220</strong> miles.<br>Choice D is incorrect. This inequality represents a situation in which Rhett drove at an average speed of <strong>40</strong>, rather than <strong>35</strong>, miles per hour, and Jessica drove at an average speed of <strong>35</strong>, rather than <strong>40</strong>, miles per hour.",
|
||
hasFigure: false,
|
||
},
|
||
];
|
||
|
||
export const LINEAR_INEQ_MEDIUM: PracticeQuestion[] = [
|
||
{
|
||
id: "64c85440",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"In North America, the standard width of a parking space is at least 7.5 feet and no more than 9.0 feet. A restaurant owner recently resurfaced the restaurant’s parking lot and wants to determine the number of parking spaces, n, in the parking lot that could be placed perpendicular to a curb that is 135 feet long, based on the standard width of a parking space. Which of the following describes all the possible values of n ?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>18 ≤ n, which ≤ 135</strong>" },
|
||
{ label: "B", text: "<strong>7 . 5 ≤ n, which ≤ 9</strong>" },
|
||
{ label: "C", text: "<strong>15 ≤ n, which ≤ 135</strong>" },
|
||
{ label: "D", text: "<strong>15 ≤ n, which ≤ 18</strong>" },
|
||
],
|
||
correctAnswer: "D",
|
||
explanation:
|
||
"Choice D is correct. Placing the parking spaces with the minimum width of 7.5 feet gives the maximum possible number of parking spaces. Thus, the maximum number that can be placed perpendicular to a 135-foot-long curb is <strong>135 over 7 . 5 = 18</strong>. Placing the parking spaces with the maximum width of 9 feet gives the minimum number of parking spaces. Thus, the minimum number that can be placed perpendicular to a 135-foot-long curb is <strong>135 over 9 = 15</strong>. Therefore, if n is the number of parking spaces in the lot, the range of possible values for n is <strong>15 ≤ n, which ≤ 18</strong>.Choices A and C are incorrect. These choices equate the length of the curb with the maximum possible number of parking spaces. Choice B is incorrect. This is the range of possible values for the width of a parking space instead of the range of possible values for the number of parking spaces.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "74c98c82",
|
||
type: "spr",
|
||
questionHtml:
|
||
"An event planner is planning a party. It costs the event planner a onetime fee of <strong>dollar sign 35</strong> to rent the venue and <strong>dollar sign 10.25</strong> per attendee. The event planner has a budget of <strong>dollar sign 200</strong>. What is the greatest number of attendees possible without exceeding the budget?",
|
||
choices: [],
|
||
correctAnswer: "16",
|
||
explanation:
|
||
"The correct answer is <strong>16</strong>. The total cost of the party is found by adding the onetime fee of the venue to the cost per attendee times the number of attendees. Let <strong>x</strong> be the number of attendees. The expression <strong>35 + 10.25 x</strong> thus represents the total cost of the party. It's given that the budget is <strong>dollar sign 200</strong>, so this situation can be represented by the inequality <strong>35 + 10.25 x < or = 200</strong>. The greatest number of attendees can be found by solving this inequality for <strong>x</strong>. Subtracting <strong>35</strong> from both sides of this inequality gives <strong>10.25 x < or = 165</strong>. Dividing both sides of this inequality by <strong>10.25</strong> results in approximately <strong>x < or = 16.098</strong>. Since the question is stated in terms of attendees, rounding <strong>x</strong> down to the nearest whole number, <strong>16</strong>, gives the greatest number of attendees possible.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "80da233d",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"A certain elephant weighs 200 pounds at birth and gains more than 2 but less than 3 pounds per day during its first year. Which of the following inequalities represents all possible weights w, in pounds, for the elephant 365 days after birth?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>400 < w, which < 600</strong>" },
|
||
{ label: "B", text: "<strong>565 < w, which < 930</strong>" },
|
||
{ label: "C", text: "<strong>730 < w, which < 1, 095</strong>" },
|
||
{ label: "D", text: "<strong>930 < w, which < 1, 295</strong>" },
|
||
],
|
||
correctAnswer: "D",
|
||
explanation:
|
||
"Choice D is correct. It’s given that the elephant weighs 200 pounds at birth and gains more than 2 pounds but less than 3 pounds per day during its first year. The inequality <strong>200 + 2 d < w, which < 200 + 3 d</strong> represents this situation, where d is the number of days after birth. Substituting 365 for d in the inequality gives <strong>200 + 2 · 365 < w, which < 200 + 3 · 365</strong>, or <strong>930 < w, which < 1, 295</strong>.Choice A is incorrect and may result from solving the inequality <strong>200 · 2 < w, which < 200 · 3</strong>. Choice B is incorrect and may result from solving the inequality for a weight range of more than 1 pound but less than 2 pounds: <strong>200 + 1 · 365 < w, which < 200 + 2 · 365</strong>. Choice C is incorrect and may result from calculating the possible weight gained by the elephant during the first year without adding the 200 pounds the elephant weighed at birth.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "8f0c82e2",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"The minimum value of <strong>x</strong> is <strong>12</strong> less than <strong>6</strong> times another number <strong>n</strong>. Which inequality shows the possible values of <strong>x</strong>?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>x < or = 6 n − 12</strong>" },
|
||
{ label: "B", text: "<strong>x > or = 6 n − 12</strong>" },
|
||
{ label: "C", text: "<strong>x < or = 12 − 6 n</strong>" },
|
||
{ label: "D", text: "<strong>x > or = 12 − 6 n</strong>" },
|
||
],
|
||
correctAnswer: "B",
|
||
explanation:
|
||
"Choice B is correct. It’s given that the minimum value of <strong>x</strong> is <strong>12</strong> less than <strong>6</strong> times another number <strong>n</strong>. Therefore, the possible values of <strong>x</strong> are all greater than or equal to the value of <strong>12</strong> less than <strong>6</strong> times <strong>n</strong>. The value of <strong>6</strong> times <strong>n</strong> is given by the expression <strong>6 n</strong>. The value of <strong>12</strong> less than <strong>6 n</strong> is given by the expression <strong>6 n − 12</strong>. Therefore, the possible values of <strong>x</strong> are all greater than or equal to <strong>6 n − 12</strong>. This can be shown by the inequality <strong>x > or = 6 n − 12</strong>.<br>Choice A is incorrect. This inequality shows the possible values of <strong>x</strong> if the maximum, not the minimum, value of <strong>x</strong> is <strong>12</strong> less than <strong>6</strong> times <strong>n</strong>.<br>Choice C is incorrect. This inequality shows the possible values of <strong>x</strong> if the maximum, not the minimum, value of <strong>x</strong> is <strong>6</strong> times <strong>n</strong> less than <strong>12</strong>, not <strong>12</strong> less than <strong>6</strong> times <strong>n</strong>.<br>Choice D is incorrect. This inequality shows the possible values of <strong>x</strong> if the minimum value of <strong>x</strong> is <strong>6</strong> times <strong>n</strong> less than <strong>12</strong>, not <strong>12</strong> less than <strong>6</strong> times <strong>n</strong>.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "90bd9ef8",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"The average annual energy cost for a certain home is $4,334. The homeowner plans to spend $25,000 to install a geothermal heating system. The homeowner estimates that the average annual energy cost will then be $2,712. Which of the following inequalities can be solved to find t, the number of years after installation at which the total amount of energy cost savings will exceed the installation cost?",
|
||
choices: [
|
||
{
|
||
label: "A",
|
||
text: "<strong>25, 000 > (4, 334 − 2, 712, ) · t</strong>",
|
||
},
|
||
{
|
||
label: "B",
|
||
text: "<strong>25, 000 < (4, 334 − 2, 712, ) · t</strong>",
|
||
},
|
||
{ label: "C", text: "<strong>25, 000 − 4, 334 > 2, 712 t</strong>" },
|
||
{
|
||
label: "D",
|
||
text: "<strong>25, 000 > the fraction with numerator 4, 332, and denominator 2, 712, end fraction, t</strong>",
|
||
},
|
||
],
|
||
correctAnswer: "B",
|
||
explanation:
|
||
"Choice B is correct. The savings each year from installing the geothermal heating system will be the average annual energy cost for the home before the geothermal heating system installation minus the average annual energy cost after the geothermal heating system installation, which is <strong>4, 334 − 2, 712</strong> dollars. In t years, the savings will be <strong>(4, 334 − 2, 712, ) · t</strong> dollars. Therefore, the inequality that can be solved to find the number of years after installation at which the total amount of energy cost savings will exceed (be greater than) the installation cost, $25,000, is <strong>25, 000 < (4, 334 − 2, 712, ) · t</strong>.Choice A is incorrect. It gives the number of years after installation at which the total amount of energy cost savings will be less than the installation cost. Choice C is incorrect and may result from subtracting the average annual energy cost for the home from the onetime cost<br><br>of the geothermal heating system installation. To find the predicted total savings, the predicted average cost should be subtracted from the average annual energy cost before the installation, and the result should be multiplied by the number of years, t. Choice D is incorrect and may result from misunderstanding the context. The ratio <strong>4, 332 over 2, 712</strong> compares the average energy cost before installation and the average energy cost after installation; it does not represent the savings.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "948087f2",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"Which of the following ordered pairs (x, y) satisfies the system of inequalities above?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>−2 −1</strong>" },
|
||
{ label: "B", text: "<strong>−1, 3</strong>" },
|
||
{ label: "C", text: "<strong>1, 5</strong>" },
|
||
{ label: "D", text: "<strong>2 −1</strong>" },
|
||
],
|
||
correctAnswer: "D",
|
||
explanation:
|
||
"Choice D is correct. Any point (x, y) that is a solution to the given system of inequalities must satisfy both inequalities in the system. The second inequality in the system can be rewritten as <strong>x > y + 1</strong>. Of the given answer choices, only choice D satisfies this inequality, because inequality <strong>2 > −1 + 1</strong> is a true statement. The point <strong>with coordinates 2 −1</strong> also satisfies the first inequality.Alternate approach: Substituting <strong>the ordered pair 2 −1</strong> into the first inequality gives <strong>−1 ≤ 3 · 2 + 1</strong>, or <strong>−1 ≤ 7</strong>, which is a true statement. Substituting <strong>the ordered pair 2 −1</strong> into the second inequality gives <strong>2 − −1 > 1</strong>, or <strong>3 > 1</strong>, which is a true statement. Therefore, since <strong>the ordered pair 2 −1</strong> satisfies both inequalities, it is a solution to the system.<br>Choice A is incorrect because substituting −2 for x and −1 for y in the first inequality gives <strong>−1 ≤ 3 · −2 + 1</strong>, or <strong>−1 ≤ −5</strong>, which is false. Choice B is incorrect because substituting −1 for x and 3 for y in the first inequality gives <strong>3 ≤ 3 · −1 + 1</strong>, or <strong>3 ≤ −2</strong>, which is false. Choice C is incorrect because substituting 1 for x and 5 for y in the first inequality gives <strong>5 ≤ 3 · 1 + 1</strong>, or <strong>5 ≤ 4</strong>, which is false.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "968e9e51",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"Which of the following ordered pairs <strong>x, y</strong> is a solution to the system of inequalities above?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>1, 0</strong>" },
|
||
{ label: "B", text: "<strong>−1, 0</strong>" },
|
||
{ label: "C", text: "<strong>0, 1</strong>" },
|
||
{ label: "D", text: "<strong>0 −1</strong>" },
|
||
],
|
||
correctAnswer: "D",
|
||
explanation:
|
||
"Choice D is correct. The solutions to the given system of inequalities is the set of all ordered pairs <strong>x, y</strong> that satisfy both inequalities in the system. For an ordered pair to satisfy the inequality <strong>y ≤ x</strong>, the value of the ordered pair’s y-coordinate must be less than or equal to the value of the ordered pair’s x-coordinate. This is true of the ordered pair <strong>0 −1</strong>, because <strong>−1 ≤ 0</strong>. To satisfy the inequality <strong>y ≤ −x</strong>, the value of the ordered pair’s y-coordinate must be less than or equal to the value of the additive inverse of the ordered pair’s x-coordinate. This is also true of the ordered pair <strong>0 −1</strong>. Because 0 is its own additive inverse, <strong>−1 ≤ the − of 0</strong> is the same as <strong>−1 ≤ 0</strong>. Therefore, the ordered pair <strong>0 −1</strong> is a solution to the given system of inequalities.Choice A is incorrect. This ordered pair satisfies only the inequality <strong>y ≤ x</strong> in the given system, not both inequalities. Choice B incorrect. This ordered pair satisfies only the inequality <strong>y ≤ −x</strong> in the system, but not both inequalities. Choice C is incorrect. This ordered pair satisfies neither inequality.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "b1228811",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"Marisa needs to hire at least 10 staff members for an upcoming project. The staff members will be made up of junior directors, who will be paid $640 per week, and senior directors, who will be paid $880 per week. Her budget for paying the staff members is no more than $9,700 per week. She must hire at least 3 junior directors and at least 1 senior director. Which of the following systems of inequalities represents the conditions described if x is the number of junior directors and y is the number of senior directors?",
|
||
choices: [
|
||
{
|
||
label: "A",
|
||
text: "<strong>640 x + 880 y ≥ 9, 700; x + y ≤ 10; x ≥ 3; y ≥ 1</strong>",
|
||
},
|
||
{
|
||
label: "B",
|
||
text: "<strong>640 x + 880 y ≤ 9, 700; x + y ≥ 10; x ≥ 3; y ≥ 1</strong>",
|
||
},
|
||
{
|
||
label: "C",
|
||
text: "<strong>640 x + 880 y ≥ 9, 700; x + y ≥ 10; x ≤ 3; y ≤ 1</strong>",
|
||
},
|
||
{
|
||
label: "D",
|
||
text: "<strong>640 x + 880 y ≤ 9, 700; x + y ≤ 10; x ≤ 3; y ≤ 1</strong>",
|
||
},
|
||
],
|
||
correctAnswer: "B",
|
||
explanation:
|
||
"Choice B is correct. Marisa will hire x junior directors and y senior directors. Since she needs to hire at least 10 staff members, <strong>x + y ≥ 10</strong>. Each junior director will be paid $640 per week, and each senior director will be paid $880 per week. Marisa’s budget for paying the new staff is no more than $9,700 per week; in terms of x and y, this condition is <strong>640 x + 880 y ≤ 9, 700</strong>. Since Marisa must hire at least 3 junior directors and at least 1 senior director, it follows that <strong>x ≥ 3</strong> and <strong>y ≥ 1</strong>. All four of these conditions are represented correctly in choice B.Choices A and C are incorrect. For example, the first condition, <strong>640 x + 880 y ≥ 9, 700</strong>, in each of these options implies that Marisa can pay the new staff members more than her budget of $9,700. Choice D is incorrect because Marisa needs to hire at least 10 staff members, not at most 10 staff members, as the inequality <strong>x + y ≤ 10</strong> implies.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "b31c3117",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"The Karvonen formula above shows the relationship between Alice’s target heart rate H, in beats per minute (bpm), and the intensity level p of different activities. When <strong>p = 0</strong>, Alice has a resting heart rate. When <strong>p = 1</strong>, Alice has her maximum heart rate. It is recommended that p be between 0.5 and 0.85 for Alice when she trains. Which of the following inequalities describes Alice’s target training heart rate?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>120 ≤ H, which ≤ 162</strong>" },
|
||
{ label: "B", text: "<strong>102 ≤ H, which ≤ 120</strong>" },
|
||
{ label: "C", text: "<strong>60 ≤ H, which ≤ 162</strong>" },
|
||
{ label: "D", text: "<strong>60 ≤ H, which ≤ 102</strong>" },
|
||
],
|
||
correctAnswer: "A",
|
||
explanation:
|
||
"Choice A is correct. When Alice trains, it’s recommended that p be between 0.5 and 0.85. Therefore, her target training heart rate is represented by the values of H corresponding to <strong>0 . 5 ≤ p, which ≤ 0 . 8 5</strong>. When <strong>p = 0 . 5</strong>, <strong>H = 120 · 0 . 5 + 60</strong>, or <strong>H = 120</strong>. When <strong>p = 0 . 8 5</strong>, <strong>H = 120 · 0 . 8 5 + 60</strong>, or <strong>H = 162</strong>. Therefore, the inequality that describes Alice’s target training heart rate is <strong>120 ≤ H, which ≤ 162</strong>.Choice B is incorrect. This inequality describes Alice’s target heart rate for <strong>0 . 3 5 ≤ p, which ≤ 0 . 5</strong>. Choice C is incorrect. This inequality describes her target heart rate for <strong>0 ≤ p, which ≤ 0 . 8 5</strong>. Choice D is incorrect. This inequality describes her target heart rate for <strong>0 ≤ p, which ≤ 0 . 3 5</strong>.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "bf5f80c6",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"<em>(expression)</em><br>Which point (<em>(expression)</em>, <em>(expression)</em>) is a solution to the given inequality in the <em>(expression)</em>-plane?",
|
||
choices: [
|
||
{ label: "A", text: "(<em>(expression)</em>, <em>(expression)</em>)" },
|
||
{ label: "B", text: "(<em>(expression)</em>, <em>(expression)</em>)" },
|
||
{ label: "C", text: "(<em>(expression)</em>, <em>(expression)</em>)" },
|
||
{ label: "D", text: "(<em>(expression)</em>, <em>(expression)</em>)" },
|
||
],
|
||
correctAnswer: "A",
|
||
explanation:
|
||
"Choice D is correct. For a point <strong>(x, y)</strong> to be a solution to the given inequality in the xy-plane, the value of the point’s y-coordinate must be less than the value of <strong>− 4 x + 4</strong>, where <strong>x</strong> is the value of the x-coordinate of the point. This is true of the point <strong>(−4, 0)</strong> because <strong>0 < − 4 (−4) + 4</strong>, or <strong>0 < 20</strong>. Therefore, the point <strong>(−4, 0)</strong> is a solution to the given inequality.<br><br>Choices A, B, and C are incorrect. None of these points are a solution to the given inequality because each point’s y-coordinate is greater than the value of <strong>− 4 x + 4</strong> for the point’s x-coordinate.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "c17d9ba9",
|
||
type: "spr",
|
||
questionHtml:
|
||
"A number <strong>x</strong> is at most <strong>17</strong> less than <strong>5</strong> times the value of <strong>y</strong>. If the value of <strong>y</strong> is <strong>3</strong>, what is the greatest possible value of <strong>x</strong>?",
|
||
choices: [],
|
||
correctAnswer: "-2",
|
||
explanation:
|
||
"The correct answer is <strong>−2</strong>. It's given that a number <strong>x</strong> is at most <strong>17</strong> less than <strong>5</strong> times the value of <strong>y</strong>, or <strong>x < or = 5 y − 17</strong>. Substituting <strong>3</strong> for <strong>y</strong> in this inequality yields <strong>x < or = 5 (3) − 17</strong>, or <strong>x < or = −2</strong>. Thus, if the value of <strong>y</strong> is <strong>3</strong>, the greatest possible value of <strong>x</strong> is <strong>−2</strong>.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "d02193fb",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"The boundary of the inequality is a dashed line.<br><br>The line slants sharply down from left to right.<br>The line passes through the following points:<br><br>(negative 1 comma 5)<br>(0 comma 1)<br>(1 comma negative 3)<br><br>The area above and to the right of the boundary is shaded.<br><br>The shaded region shown represents the solutions to which inequality?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>y < 1 + 4 x</strong>" },
|
||
{ label: "B", text: "<strong>y < 1 − 4 x</strong>" },
|
||
{ label: "C", text: "<strong>y > 1 + 4 x</strong>" },
|
||
{ label: "D", text: "<strong>y > 1 − 4 x</strong>" },
|
||
],
|
||
correctAnswer: "D",
|
||
explanation:
|
||
"Choice D is correct. The equation for the line representing the boundary of the shaded region can be written in slope-intercept form <strong>y = b + m x</strong>, where <strong>m</strong> is the slope and <strong>(0, b)</strong> is the y-intercept of the line. For the graph shown, the boundary line passes through the points <strong>(0, 1)</strong> and <strong>(1 −3)</strong>. Given two points on a line, <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong>, the slope of the line can be calculated using the equation <strong>m = (y 2 − y 1) / (x 2 − x 1)</strong>. Substituting the points <strong>(0, 1)</strong> and <strong>(1 −3)</strong> for <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong> in this equation yields <strong>m = (−3 − 1) / (1 − 0)</strong>, which is equivalent to <strong>m = (−4) / (1)</strong>, or <strong>m = −4</strong>. Since the point <strong>(0, 1)</strong> represents the y-intercept, it follows that <strong>b = 1</strong>. Substituting <strong>−4</strong> for <strong>m</strong> and <strong>1</strong> for <strong>b</strong> in the equation <strong>y = b + m x</strong> yields <strong>y = 1 − 4 x</strong> as the equation of the boundary line. Since the shaded region represents all the points above this boundary line, it follows that the shaded region shown represents the solutions to the inequality <strong>y > 1 − 4 x</strong>.<br>Choice A is incorrect. This inequality represents a region below, not above, a boundary line with a slope of <strong>4</strong>, not <strong>−4</strong>.<br>Choice B is incorrect. This inequality represents a region below, not above, the boundary line shown.<br>Choice C is incorrect. This inequality represents a region whose boundary line has a slope of <strong>4</strong>, not <strong>−4</strong>.",
|
||
hasFigure: true,
|
||
figureUrl: "/practice-images/d02193fb_svg1.svg",
|
||
},
|
||
{
|
||
id: "e9ef0e6b",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"A model estimates that whales from the genus Eschrichtius travel <strong>72</strong> to <strong>77</strong> miles in the ocean each day during their migration. Based on this model, which inequality represents the estimated total number of miles, <strong>x</strong>, a whale from the genus Eschrichtius could travel in <strong>16</strong> days of its migration?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>72 + 16 < or = x < or = 77 + 16</strong>" },
|
||
{
|
||
label: "B",
|
||
text: "<strong>(72) (16) < or = x < or = (77) (16)</strong>",
|
||
},
|
||
{ label: "C", text: "<strong>72 < or = 16 + x < or = 77</strong>" },
|
||
{ label: "D", text: "<strong>72 < or = 16 x < or = 77</strong>" },
|
||
],
|
||
correctAnswer: "B",
|
||
explanation:
|
||
"Choice B is correct. It's given that the model estimates that whales from the genus Eschrichtius travel <strong>72</strong> to <strong>77</strong> miles in the ocean each day during their migration. If one of these whales travels <strong>72</strong> miles each day for <strong>16</strong> days, then the whale travels <strong>72 (16)</strong> miles total. If one of these whales travels <strong>77</strong> miles each day for <strong>16</strong> days, then the whale travels <strong>77 (16)</strong> miles total. Therefore, the model estimates that in <strong>16</strong> days of its migration, a whale from the genus Eschrichtius could travel at least <strong>72 (16)</strong> and at most <strong>77 (16)</strong> miles total. Thus, the inequality <strong>(72) (16) < or = x < or = (77) (16)</strong> represents the estimated total number of miles, <strong>x</strong>, a whale from the genus Eschrichtius could travel in <strong>16</strong> days of its migration.<br>Choice A is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect and may result from conceptual errors.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "f02b4509",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"A moving truck can tow a trailer if the combined weight of the trailer and the boxes it contains is no more than <strong>4, 600</strong> pounds. What is the maximum number of boxes this truck can tow in a trailer with a weight of <strong>500</strong> pounds if each box weighs <strong>120</strong> pounds?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>34</strong>" },
|
||
{ label: "B", text: "<strong>35</strong>" },
|
||
{ label: "C", text: "<strong>38</strong>" },
|
||
{ label: "D", text: "<strong>39</strong>" },
|
||
],
|
||
correctAnswer: "A",
|
||
explanation:
|
||
"Choice A is correct. It’s given that the truck can tow a trailer if the combined weight of the trailer and the boxes it contains is no more than <strong>4, 600</strong> pounds. If the trailer has a weight of <strong>500</strong> pounds and each box weighs <strong>120</strong> pounds, the expression <strong>500 + 120 b</strong>, where <strong>b</strong> is the number of boxes, gives the combined weight of the trailer and the boxes. Since the combined weight must be no more than <strong>4, 600</strong> pounds, the possible numbers of boxes the truck can tow are given by the inequality <strong>500 + 120 b < or = 4, 600</strong>. Subtracting <strong>500</strong> from both sides of this inequality yields <strong>120 b < or = 4, 100</strong>. Dividing both sides of this inequality by <strong>120</strong> yields <strong>b < or = (205) / (6)</strong>, or <strong>b</strong> is less than or equal to approximately <strong>34.17</strong>. Since the number of boxes, <strong>b</strong>, must be a whole number, the maximum number of boxes the truck can tow is the greatest whole number less than <strong>34.17</strong>, which is <strong>34</strong>.<br>Choice B is incorrect. Towing the trailer and <strong>35</strong> boxes would yield a combined weight of <strong>4, 700</strong> pounds, which is greater than <strong>4, 600</strong> pounds.<br>Choice C is incorrect. Towing the trailer and <strong>38</strong> boxes would yield a combined weight of <strong>5, 060</strong> pounds, which is greater than <strong>4, 600</strong> pounds.<br>Choice D is incorrect. Towing the trailer and <strong>39</strong> boxes would yield a combined weight of <strong>5, 180</strong> pounds, which is greater than <strong>4, 600</strong> pounds.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "f224df07",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"A cargo helicopter delivers only 100-pound packages and 120-pound packages. For each delivery trip, the helicopter must carry at least 10 packages, and the total weight of the packages can be at most 1,100 pounds. What is the maximum number of 120-pound packages that the helicopter can carry per trip?",
|
||
choices: [
|
||
{ label: "A", text: "2" },
|
||
{ label: "B", text: "4" },
|
||
{ label: "C", text: "5" },
|
||
{ label: "D", text: "6" },
|
||
],
|
||
correctAnswer: "C",
|
||
explanation:
|
||
"Choice C is correct. Let a equal the number of 120-pound packages, and let b equal the number of 100-pound packages. It’s given that the total weight of the packages can be at most 1,100 pounds: the inequality <strong>120 a + 100 b ≤ 1, 100</strong> represents this situation. It’s also given that the helicopter must carry at least 10 packages: the inequality <strong>a + b ≥ 10</strong> represents this situation. Values of a and b that satisfy these two inequalities represent the allowable numbers of 120-pound packages and 100-pound packages the helicopter can transport. To maximize the number of 120-pound packages, a, in the helicopter, the number of 100-pound packages, b, in the helicopter needs to be minimized. Expressing b in terms of a in the second inequality yields <strong>b ≥ 10 − a</strong>, so the minimum value of b is equal to <strong>10 − a</strong>. Substituting <strong>10 − a</strong> for b in the first inequality results in <strong>120 a + 100 · (10 − a, ) ≤ 1, 100</strong>. Using the distributive property to rewrite this inequality yields <strong>120 a + 1, 000 − 100 a ≤ 1, 100</strong>, or <strong>20 a + 1, 000 ≤ 1, 100</strong>. Subtracting 1,000 from both sides of this inequality yields <strong>20 a ≤ 100</strong>. Dividing both sides of this inequality by 20 results in <strong>a ≤ 5</strong>. This means that the maximum number of 120-pound packages that the helicopter can carry per trip is 5.Choices A, B, and D are incorrect and may result from incorrectly creating or solving the system of inequalities.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "f2bbd43d",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"<strong>y > 14</strong><br><strong>4 x + y < 18</strong><br>The point <strong>(x, 53)</strong> is a solution to the system of inequalities in the xy-plane. Which of the following could be the value of <strong>x</strong>?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>−9</strong>" },
|
||
{ label: "B", text: "<strong>−5</strong>" },
|
||
{ label: "C", text: "<strong>5</strong>" },
|
||
{ label: "D", text: "<strong>9</strong>" },
|
||
],
|
||
correctAnswer: "A",
|
||
explanation:
|
||
"Choice A is correct. It’s given that the point <strong>(x, 53)</strong> is a solution to the given system of inequalities in the xy-plane. This means that the coordinates of the point, when substituted for the variables <strong>x</strong> and <strong>y</strong>, make both of the inequalities in the system true. Substituting <strong>53</strong> for <strong>y</strong> in the inequality <strong>y > 14</strong> yields <strong>53 > 14</strong>, which is true. Substituting <strong>53</strong> for <strong>y</strong> in the inequality <strong>4 x + y < 18</strong> yields <strong>4 x + 53 < 18</strong>. Subtracting <strong>53</strong> from both sides of this inequality yields <strong>4 x < −35</strong>. Dividing both sides of this inequality by <strong>4</strong> yields <strong>x < −8.75</strong>. Therefore, <strong>x</strong> must be a value less than <strong>−8.75</strong>. Of the given choices, only <strong>−9</strong> is less than <strong>−8.75</strong>.<br>Choice B is incorrect. Substituting <strong>−5</strong> for <strong>x</strong> and <strong>53</strong> for <strong>y</strong> in the inequality <strong>4 x + y < 18</strong> yields <strong>4 (−5) + 53 < 18</strong>, or <strong>33 < 18</strong>, which is not true.<br>Choice C is incorrect. Substituting <strong>5</strong> for <strong>x</strong> and <strong>53</strong> for <strong>y</strong> in the inequality <strong>4 x + y < 18</strong> yields <strong>4 (5) + 53 < 18</strong>, or <strong>73 < 18</strong>, which is not true.<br>Choice D is incorrect. Substituting <strong>9</strong> for <strong>x</strong> and <strong>53</strong> for <strong>y</strong> in the inequality <strong>4 x + y < 18</strong> yields <strong>4 (9) + 53 < 18</strong>, or <strong>89 < 18</strong>, which is not true.",
|
||
hasFigure: false,
|
||
},
|
||
];
|
||
|
||
export const LINEAR_INEQ_HARD: PracticeQuestion[] = [
|
||
{
|
||
id: "03503d49",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"A business owner plans to purchase the same model of chair for each of the <strong>81</strong> employees. The total budget to spend on these chairs is <strong>dollar sign 14, 000</strong>, which includes a <strong>7 % sign</strong> sales tax. Which of the following is closest to the maximum possible price per chair, before sales tax, the business owner could pay based on this budget?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>dollar sign 148.15</strong>" },
|
||
{ label: "B", text: "<strong>dollar sign 161.53</strong>" },
|
||
{ label: "C", text: "<strong>dollar sign 172.84</strong>" },
|
||
{ label: "D", text: "<strong>dollar sign 184.94</strong>" },
|
||
],
|
||
correctAnswer: "B",
|
||
explanation:
|
||
"Choice B is correct. It’s given that a business owner plans to purchase <strong>81</strong> chairs. If <strong>p</strong> is the price per chair, the total price of purchasing <strong>81</strong> chairs is <strong>81 p</strong>. It’s also given that <strong>7 % sign</strong> sales tax is included, which is equivalent to <strong>81 p</strong> multiplied by <strong>1.07</strong>, or <strong>81 (1.07) p</strong>. Since the total budget is <strong>dollar sign 14, 000</strong>, the inequality representing the situation is given by <strong>81 (1.07) p < or = 14, 000</strong>. Dividing both sides of this inequality by <strong>81 (1.07)</strong> and rounding the result to two decimal places gives <strong>p < or = 161.53</strong>. To not exceed the budget, the maximum possible price per chair is <strong>dollar sign 161.53</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect. This is the maximum possible price per chair including sales tax, not the maximum possible price per chair before sales tax.<br>Choice D is incorrect. This is the maximum possible price if the sales tax is added to the total budget, not the maximum possible price per chair before sales tax.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "1035faea",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"A psychologist set up an experiment to study the tendency of a person to select the first item when presented with a series of items. In the experiment, 300 people were presented with a set of five pictures arranged in random order. Each person was asked to choose the most appealing picture. Of the first 150 participants, 36 chose the first picture in the set. Among the remaining 150 participants, p people chose the first picture in the set. If more than 20% of all participants chose the first picture in the set, which of the following inequalities best describes the possible values of p ?",
|
||
choices: [
|
||
{
|
||
label: "A",
|
||
text: "<strong>p > 0 . 20 · (300 − 36, )</strong>, where <strong>p ≤ 150</strong>",
|
||
},
|
||
{
|
||
label: "B",
|
||
text: "<strong>p > 0 . 20 · (300 + 36, )</strong>, where <strong>p ≤ 150</strong>",
|
||
},
|
||
{
|
||
label: "C",
|
||
text: "<strong>p − 36 > 0 . 20 · 300</strong>, where <strong>p ≤ 150</strong>",
|
||
},
|
||
{
|
||
label: "D",
|
||
text: "<strong>p + 36 > 0 . 20 · 300</strong>, where <strong>p ≤ 150</strong>",
|
||
},
|
||
],
|
||
correctAnswer: "D",
|
||
explanation:
|
||
"Choice D is correct. Of the first 150 participants, 36 chose the first picture in the set, and of the 150 remaining participants, p chose the first picture in the set. Hence, the proportion of the participants who chose the first picture in the set is <strong>the fraction with numerator 36 + p, and denominator 300</strong>. Since more than 20% of all the participants chose the first picture, it follows that <strong>the fraction with numerator 36 + p, and denominator 300 > 0 . 2 0</strong>.This inequality can be rewritten as <strong>p + 36 > 0 . 2 0 · 300</strong>. Since p is a number of people among the remaining 150 participants, <strong>p ≤ 150</strong>.<br>Choices A, B, and C are incorrect and may be the result of some incorrect interpretations of the given information or of computational errors.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "1a621af4",
|
||
type: "spr",
|
||
questionHtml:
|
||
"A number <strong>x</strong> is at most <strong>2</strong> less than <strong>3</strong> times the value of <strong>y</strong>. If the value of <strong>y</strong> is <strong>−4</strong>, what is the greatest possible value of <strong>x</strong>?",
|
||
choices: [],
|
||
correctAnswer: "-14",
|
||
explanation:
|
||
"The correct answer is <strong>−14</strong>. It's given that a number <strong>x</strong> is at most <strong>2</strong> less than <strong>3</strong> times the value of <strong>y</strong>. Therefore, <strong>x</strong> is less than or equal to <strong>2</strong> less than <strong>3</strong> times the value of <strong>y</strong>. The expression <strong>3 y</strong> represents <strong>3</strong> times the value of <strong>y</strong>. The expression <strong>3 y − 2</strong> represents <strong>2</strong> less than <strong>3</strong> times the value of <strong>y</strong>. Therefore, <strong>x</strong> is less than or equal to <strong>3 y − 2</strong>. This can be shown by the inequality <strong>x < or = 3 y − 2</strong>. Substituting <strong>−4</strong> for <strong>y</strong> in this inequality yields <strong>x < or = 3 (−4) − 2</strong> or, <strong>x < or = −14</strong>. Therefore, if the value of <strong>y</strong> is <strong>−4</strong>, the greatest possible value of <strong>x</strong> is <strong>−14</strong>.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "45cfb9de",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"Adam’s school is a 20-minute walk or a 5-minute bus ride away from his house. The bus runs once every 30 minutes, and the number of minutes, w, that Adam waits for the bus varies between 0 and 30. Which of the following inequalities gives the values of w for which it would be faster for Adam to walk to school?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>w − 5 < 20</strong>" },
|
||
{ label: "B", text: "<strong>w − 5 > 20</strong>" },
|
||
{ label: "C", text: "<strong>w + 5 < 20</strong>" },
|
||
{ label: "D", text: "<strong>w + 5 > 20</strong>" },
|
||
],
|
||
correctAnswer: "D",
|
||
explanation:
|
||
"Choice D is correct. It is given that w is the number of minutes that Adam waits for the bus. The total time it takes Adam to get to school on a day he takes the bus is the sum of the minutes, w, he waits for the bus and the 5 minutes the bus ride takes; thus, this time, in minutes, is w + 5. It is also given that the total amount of time it takes Adam to get to school on a day that he walks is 20 minutes. Therefore, w + 5 > 20 gives the values of w for which it would be faster for Adam to walk to school.Choices A and B are incorrect because w – 5 is not the total length of time for Adam to wait for and then take the bus to school. Choice C is incorrect because the inequality should be true when walking 20 minutes is faster than the time it takes Adam to wait for and ride the bus, not less.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "48fb34c8",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"<strong>y > 13 x − 18</strong><br>For which of the following tables are all the values of <strong>x</strong> and their corresponding values of <strong>y</strong> solutions to the given inequality?",
|
||
choices: [
|
||
{
|
||
label: "A",
|
||
text: "<strong>x</strong><br><strong>y</strong><br><br><strong>3</strong><br><strong>21</strong><br><br><strong>5</strong><br><strong>47</strong><br><br><strong>8</strong><br><strong>86</strong>",
|
||
},
|
||
{
|
||
label: "B",
|
||
text: "<strong>x</strong><br><strong>y</strong><br><br><strong>3</strong><br><strong>26</strong><br><br><strong>5</strong><br><strong>42</strong><br><br><strong>8</strong><br><strong>86</strong>",
|
||
},
|
||
{
|
||
label: "C",
|
||
text: "<strong>x</strong><br><strong>y</strong><br><br><strong>3</strong><br><strong>16</strong><br><br><strong>5</strong><br><strong>42</strong><br><br><strong>8</strong><br><strong>81</strong>",
|
||
},
|
||
{
|
||
label: "D",
|
||
text: "<strong>x</strong><br><strong>y</strong><br><br><strong>3</strong><br><strong>26</strong><br><br><strong>5</strong><br><strong>52</strong><br><br><strong>8</strong><br><strong>91</strong>",
|
||
},
|
||
],
|
||
correctAnswer: "D",
|
||
explanation:
|
||
"Choice D is correct. All the tables in the choices have the same three values of <strong>x</strong>, so each of the three values of <strong>x</strong> can be substituted in the given inequality to compare the corresponding values of <strong>y</strong> in each of the tables. Substituting <strong>3</strong> for <strong>x</strong> in the given inequality yields <strong>y > 13 (3) − 18</strong>, or <strong>y > 21</strong>. Therefore, when <strong>x = 3</strong>, the corresponding value of <strong>y</strong> is greater than <strong>21</strong>. Substituting <strong>5</strong> for <strong>x</strong> in the given inequality yields <strong>y > 13 (5) − 18</strong>, or <strong>y > 47</strong>. Therefore, when <strong>x = 5</strong>, the corresponding value of <strong>y</strong> is greater than <strong>47</strong>. Substituting <strong>8</strong> for <strong>x</strong> in the given inequality yields <strong>y > 13 (8) − 18</strong>, or <strong>y > 86</strong>. Therefore, when <strong>x = 8</strong>, the corresponding value of <strong>y</strong> is greater than <strong>86</strong>. For the table in choice D, when <strong>x = 3</strong>, the corresponding value of <strong>y</strong> is <strong>26</strong>, which is greater than <strong>21</strong>; when <strong>x = 5</strong>, the corresponding value of <strong>y</strong> is <strong>52</strong>, which is greater than <strong>47</strong>; when <strong>x = 8</strong>, the corresponding value of <strong>y</strong> is <strong>91</strong>, which is greater than <strong>86</strong>. Therefore, the table in choice D gives values of <strong>x</strong> and their corresponding values of <strong>y</strong> that are all solutions to the given inequality.<br>Choice A is incorrect. In the table for choice A, when <strong>x = 3</strong>, the corresponding value of <strong>y</strong> is <strong>21</strong>, which is not greater than <strong>21</strong>; when <strong>x = 5</strong>, the corresponding value of <strong>y</strong> is <strong>47</strong>, which is not greater than <strong>47</strong>; when <strong>x = 8</strong>, the corresponding value of <strong>y</strong> is <strong>86</strong>, which is not greater than <strong>86</strong>.<br>Choice B is incorrect. In the table for choice B, when <strong>x = 5</strong>, the corresponding value of <strong>y</strong> is <strong>42</strong>, which is not greater than <strong>47</strong>; when <strong>x = 8</strong>, the corresponding value of <strong>y</strong> is <strong>86</strong>, which is not greater than <strong>86</strong>.<br>Choice C is incorrect. In the table for choice C, when <strong>x = 3</strong>, the corresponding value of <strong>y</strong> is <strong>16</strong>, which is not greater than <strong>21</strong>; when <strong>x = 5</strong>, the corresponding value of <strong>y</strong> is <strong>42</strong>, which is not greater than <strong>47</strong>; when <strong>x = 8</strong>, the corresponding value of <strong>y</strong> is <strong>81</strong>, which is not greater than <strong>86</strong>.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "541bef2f",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"<strong>y < or = x + 7</strong><br><strong>y > or = −2 x − 1</strong><br>Which point <strong>(x, y)</strong> is a solution to the given system of inequalities in the xy-plane?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>(−14, 0)</strong>" },
|
||
{ label: "B", text: "<strong>(0 −14)</strong>" },
|
||
{ label: "C", text: "<strong>(0, 14)</strong>" },
|
||
{ label: "D", text: "<strong>(14, 0)</strong>" },
|
||
],
|
||
correctAnswer: "D",
|
||
explanation:
|
||
"Choice D is correct. A point <strong>(x, y)</strong> is a solution to a system of inequalities in the xy-plane if substituting the x-coordinate and the y-coordinate of the point for <strong>x</strong> and <strong>y</strong>, respectively, in each inequality makes both of the inequalities true. Substituting the x-coordinate and the y-coordinate of choice D, <strong>14</strong> and <strong>0</strong>, for <strong>x</strong> and <strong>y</strong>, respectively, in the first inequality in the given system, <strong>y < or = x + 7</strong>, yields <strong>0 < or = 14 + 7</strong>, or <strong>0 < or = 21</strong>, which is true. Substituting <strong>14</strong> for <strong>x</strong> and <strong>0</strong> for <strong>y</strong> in the second inequality in the given system, <strong>y > or = − 2 x − 1</strong>, yields <strong>0 > or = − 2 (14) − 1</strong>, or <strong>0 > or = −29</strong>, which is true. Therefore, the point <strong>(14, 0)</strong> is a solution to the given system of inequalities in the xy-plane.<br>Choice A is incorrect. Substituting <strong>−14</strong> for <strong>x</strong> and <strong>0</strong> for <strong>y</strong> in the inequality <strong>y < or = x + 7</strong> yields <strong>0 < or = −14 + 7</strong>, or <strong>0 < or = −7</strong>, which is not true.<br>Choice B is incorrect. Substituting <strong>0</strong> for <strong>x</strong> and <strong>−14</strong> for <strong>y</strong> in the inequality <strong>y > or = − 2 x − 1</strong> yields <strong>−14 > or = − 2 (0) − 1</strong>, or <strong>−14 > or = −1</strong>, which is not true.<br>Choice C is incorrect. Substituting <strong>0</strong> for <strong>x</strong> and <strong>14</strong> for <strong>y</strong> in the inequality <strong>y < or = x + 7</strong> yields <strong>14 < or = 0 + 7</strong>, or <strong>14 < or = 7</strong>, which is not true.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "5bf5136d",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of <strong>6</strong> and <strong>12</strong>, which inequality represents the possible lengths, <strong>x</strong>, of the third side of the triangle?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>x < 18</strong>" },
|
||
{ label: "B", text: "<strong>x > 18</strong>" },
|
||
{ label: "C", text: "<strong>6 < x < 18</strong>" },
|
||
{ label: "D", text: "<strong>x < 6</strong> or <strong>x > 18</strong>" },
|
||
],
|
||
correctAnswer: "C",
|
||
explanation:
|
||
"Choice C is correct. It’s given that a triangle has side lengths of <strong>6</strong> and <strong>12</strong>, and <strong>x</strong> represents the length of the third side of the triangle. It’s also given that the triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. Therefore, the inequalities <strong>6 + x > 12</strong>, <strong>6 + 12 > x</strong>, and <strong>12 + x > 6</strong> represent all possible values of <strong>x</strong>. Subtracting <strong>6</strong> from both sides of the inequality <strong>6 + x > 12</strong> yields <strong>x > 12 − 6</strong>, or <strong>x > 6</strong>. Adding <strong>6</strong> and <strong>12</strong> in the inequality <strong>6 + 12 > x</strong> yields <strong>18 > x</strong>, or <strong>x < 18</strong>. Subtracting <strong>12</strong> from both sides of the inequality <strong>12 + x > 6</strong> yields <strong>x > 6 − 12</strong>, or <strong>x > −6</strong>. Since all x-values that satisfy the inequality <strong>x > 6</strong> also satisfy the inequality <strong>x > −6</strong>, it follows that the inequalities <strong>x > 6</strong> and <strong>x < 18</strong> represent the possible values of <strong>x</strong>. Therefore, the inequality <strong>6 < x < 18</strong> represents the possible lengths, <strong>x</strong>, of the third side of the triangle.<br>Choice A is incorrect. This inequality gives the upper bound for <strong>x</strong> but does not include its lower bound.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
|
||
hasFigure: false,
|
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},
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{
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id: "6c71f3ec",
|
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type: "mcq",
|
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questionHtml:
|
||
"A salesperson’s total earnings consist of a base salary of <strong>x</strong> dollars per year, plus commission earnings of <strong>11 % sign</strong> of the total sales the salesperson makes during the year. This year, the salesperson has a goal for the total earnings to be at least <strong>3</strong> times and at most <strong>4</strong> times the base salary. Which of the following inequalities represents all possible values of total sales <strong>s</strong>, in dollars, the salesperson can make this year in order to meet that goal?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>2 x < or = s < or = 3 x</strong>" },
|
||
{
|
||
label: "B",
|
||
text: "<strong>(2) / (0.11) x < or = s < or = (3) / (0.11) x</strong>",
|
||
},
|
||
{ label: "C", text: "<strong>3 x < or = s < or = 4 x</strong>" },
|
||
{
|
||
label: "D",
|
||
text: "<strong>(3) / (0.11) x < or = s < or = (4) / (0.11) x</strong>",
|
||
},
|
||
],
|
||
correctAnswer: "B",
|
||
explanation:
|
||
"Choice B is correct. It’s given that a salesperson's total earnings consist of a base salary of <strong>x</strong> dollars per year plus commission earnings of <strong>11 % sign</strong> of the total sales the salesperson makes during the year. If the salesperson makes <strong>s</strong> dollars in total sales this year, the salesperson’s total earnings can be represented by the expression <strong>x + 0.11 s</strong>. It’s also given that the salesperson has a goal for the total earnings to be at least <strong>3</strong> times and at most <strong>4</strong> times the base salary, which can be represented by the expressions <strong>3 x</strong> and <strong>4 x</strong>, respectively. Therefore, this situation can be represented by the inequality <strong>3 x < or = x + 0.11 s < or = 4 x</strong>. Subtracting <strong>x</strong> from each part of this inequality yields <strong>2 x < or = 0.11 s < or = 3 x</strong>. Dividing each part of this inequality by <strong>0.11</strong> yields <strong>(2) / (0.11) x < or = s < or = (3) / (0.11) x</strong>. Therefore, the inequality <strong>(2) / (0.11) x < or = s < or = (3) / (0.11) x</strong> represents all possible values of total sales <strong>s</strong>, in dollars, the salesperson can make this year in order to meet their goal.<br>Choice A is incorrect. This inequality represents a situation in which the total sales, rather than the total earnings, are at least <strong>2</strong> times and at most <strong>3</strong> times, rather than at least <strong>3</strong> times and at most <strong>4</strong> times, the base salary.<br>Choice C is incorrect. This inequality represents a situation in which the total sales, rather than the total earnings, are at least <strong>3</strong> times and at most <strong>4</strong> times the base salary.<br>Choice D is incorrect. This inequality represents a situation in which the total earnings are at least <strong>4</strong> times and at most <strong>5</strong> times, rather than at least <strong>3</strong> times and at most <strong>4</strong> times, the base salary.",
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||
hasFigure: false,
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},
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||
{
|
||
id: "830120b0",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"Which of the following consists of the y-coordinates of all the points that satisfy the system of inequalities above?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>y > 6</strong>" },
|
||
{ label: "B", text: "<strong>y > 4</strong>" },
|
||
{ label: "C", text: "<strong>y > five-halves</strong>" },
|
||
{ label: "D", text: "<strong>y > three-halves</strong>" },
|
||
],
|
||
correctAnswer: "B",
|
||
explanation:
|
||
"Choice B is correct. Subtracting the same number from each side of an inequality gives an equivalent inequality. Hence, subtracting 1 from each side of the inequality <strong>2 x > 5</strong> gives <strong>2 x − 1 > 4</strong>. So the given system of inequalities is equivalent to the system of inequalities <strong>y > 2 x − 1</strong> and <strong>2 x − 1 > 4</strong>, which can be rewritten as <strong>y > 2 x − 1, which > 4</strong>. Using the transitive property of inequalities, it follows that <strong>y > 4</strong>.Choice A is incorrect because there are points with a y-coordinate less than 6 that satisfy the given system of inequalities. For example, <strong>3, 5 . 5</strong> satisfies both inequalities. Choice C is incorrect. This may result from solving the inequality <strong>2 x > 5</strong> for x, then replacing x with y. Choice D is incorrect because this inequality allows y-values that are not the y-coordinate of any point that satisfies both inequalities. For example, <strong>y = 2</strong> is contained in the set <strong>y > three halves</strong>; however, if 2 is substituted into the first inequality for y, the result is <strong>x < three halves</strong>. This cannot be true because the second inequality gives <strong>x > five halves</strong>.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "95cad55f",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"A laundry service is buying detergent and fabric softener from its supplier. The supplier will deliver no more than 300 pounds in a shipment. Each container of detergent weighs 7.35 pounds, and each container of fabric softener weighs 6.2 pounds. The service wants to buy at least twice as many containers of detergent as containers of fabric softener. Let d represent the number of containers of detergent, and let s represent the number of containers of fabric softener, where d and s are nonnegative integers. Which of the following systems of inequalities best represents this situation?",
|
||
choices: [
|
||
{
|
||
label: "A",
|
||
text: "<strong>7 . three five d + 6 . 2 s ≤ 300, and, d ≥ 2 s</strong>",
|
||
},
|
||
{
|
||
label: "B",
|
||
text: "<strong>7 . three five d + 6 . 2 s ≤ 300, and, 2 d ≥ s</strong>",
|
||
},
|
||
{
|
||
label: "C",
|
||
text: "<strong>14 . 7 d + 6 . 2 s ≤ 300, and, d ≥ 2 s</strong>",
|
||
},
|
||
{
|
||
label: "D",
|
||
text: "<strong>14 . 7 d + 6 . 2 s ≤ 300, and, 2 d ≥ s</strong>",
|
||
},
|
||
],
|
||
correctAnswer: "A",
|
||
explanation:
|
||
"Choice A is correct. The number of containers in a shipment must have a weight less than or equal to 300 pounds. The total weight, in pounds, of detergent and fabric softener that the supplier delivers can be expressed as the weight of each container multiplied by the number of each type of container, which is 7.35d for detergent and 6.2s for fabric softener. Since this total cannot exceed 300 pounds, it follows that <strong>7 . 3 5 d + 6 . 2 s ≤ 300</strong>. Also, since the laundry service wants to buy at least twice as many containers of detergent as containers of fabric softener, the number of containers of detergent should be greater than or equal to two times the number of containers of fabric softener. This can be expressed by the inequality <strong>d ≥ 2 s</strong>.<br>Choice B is incorrect because it misrepresents the relationship between the numbers of each container that the laundry service wants to buy. Choice C is incorrect because the first inequality of the system incorrectly doubles the weight per container of detergent. The weight of each container of detergent is 7.35, not 14.7 pounds. Choice D is incorrect because it doubles the weight per container of detergent and transposes the relationship between the numbers of containers.",
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||
hasFigure: false,
|
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},
|
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{
|
||
id: "963da34c",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"A shipping service restricts the dimensions of the boxes it will ship for a certain type of service. The restriction states that for boxes shaped like rectangular prisms, the sum of the perimeter of the base of the box and the height of the box cannot exceed 130 inches. The perimeter of the base is determined using the width and length of the box. If a box has a height of 60 inches and its length is 2.5 times the width, which inequality shows the allowable width x, in inches, of the box?",
|
||
choices: [
|
||
{ label: "A", text: "<strong>zero < x, which ≤ 10</strong>" },
|
||
{
|
||
label: "B",
|
||
text: "<strong>zero < x, which ≤ 11 and two-thirds</strong>",
|
||
},
|
||
{
|
||
label: "C",
|
||
text: "<strong>zero < x, which ≤ 17 and one-half</strong>",
|
||
},
|
||
{ label: "D", text: "<strong>zero < x, which ≤ 20</strong>" },
|
||
],
|
||
correctAnswer: "A",
|
||
explanation:
|
||
"Choice A is correct. If x is the width, in inches, of the box, then the length of the box is 2.5x inches. It follows that the perimeter of the base is <strong>2 · (2 . 5 x + x, )</strong>, or 7x inches. The height of the box is given to be 60 inches. According to the restriction, the sum of the perimeter of the base and the height of the box should not exceed 130 inches. Algebraically, this can be represented by <strong>7 x + 60 ≤ 130</strong>, or <strong>7 x ≤ 70</strong>. Dividing both sides of the inequality by 7 gives <strong>x ≤ 10</strong>. Since x represents the width of the box, x must also be a positive number. Therefore, the inequality <strong>0 < x, which ≤ 10</strong> represents all the allowable values of x that satisfy the given conditions.Choices B, C, and D are incorrect and may result from calculation errors or misreading the given information.",
|
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hasFigure: false,
|
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},
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{
|
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id: "b8e73b5b",
|
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type: "mcq",
|
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questionHtml:
|
||
"Ken is working this summer as part of a crew on a farm. He earned $8 per hour for the first 10 hours he worked this week. Because of his performance, his crew leader raised his salary to $10 per hour for the rest of the week. Ken saves 90% of his earnings from each week. What is the least number of hours he must work the rest of the week to save at least $270 for the week?",
|
||
choices: [
|
||
{ label: "A", text: "38" },
|
||
{ label: "B", text: "33" },
|
||
{ label: "C", text: "22" },
|
||
{ label: "D", text: "16" },
|
||
],
|
||
correctAnswer: "C",
|
||
explanation:
|
||
"Choice C is correct. Ken earned $8 per hour for the first 10 hours he worked, so he earned a total of $80 for the first 10 hours he worked. For the rest of the week, Ken was paid at the rate of $10 per hour. Let x be the number of hours he will work for the rest of the week. The total of Ken’s earnings, in dollars, for the week will be <strong>10 x + 80</strong>. He saves 90% of his earnings each week, so this week he will save <strong>0 . 9 · (10 x + 80, )</strong> dollars. The inequality <strong>0 . 9 · (10 x + 80, ) ≥ 270</strong> represents the condition that he will save at least $270 for the week. Factoring 10 out of the expression <strong>10 x + 80</strong> gives <strong>10 · (x + 8, )</strong>. The product of 10 and 0.9 is 9, so the inequality can be rewritten as <strong>9 · (x + 8, ) ≥ 270</strong>. Dividing both sides of this inequality by 9 yields <strong>x + 8 ≥ 30</strong>, so <strong>x ≥ 22</strong>. Therefore, the least number of hours Ken must work the rest of the week to save at least $270 for the week is 22.Choices A and B are incorrect because Ken can save $270 by working fewer hours than 38 or 33 for the rest of the week. Choice D is incorrect. If Ken worked 16 hours for the rest of the week, his total earnings for the week will be <strong>80 dollars + 160 dollars = 240 dollars</strong>, which is less than $270. Since he saves only 90% of his earnings each week, he would save even less than $240 for the week.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "d8539e09",
|
||
type: "mcq",
|
||
questionHtml:
|
||
"<strong>y < 6 x + 2</strong><br>For which of the following tables are all the values of <strong>x</strong> and their corresponding values of <strong>y</strong> solutions to the given inequality?",
|
||
choices: [
|
||
{
|
||
label: "A",
|
||
text: "<strong>x</strong><br><strong>y</strong><br><br><strong>3</strong><br><strong>20</strong><br><br><strong>5</strong><br><strong>32</strong><br><br><strong>7</strong><br><strong>44</strong>",
|
||
},
|
||
{
|
||
label: "B",
|
||
text: "<strong>x</strong><br><strong>y</strong><br><br><strong>3</strong><br><strong>16</strong><br><br><strong>5</strong><br><strong>36</strong><br><br><strong>7</strong><br><strong>40</strong>",
|
||
},
|
||
{
|
||
label: "C",
|
||
text: "<strong>x</strong><br><strong>y</strong><br><br><strong>3</strong><br><strong>16</strong><br><br><strong>5</strong><br><strong>28</strong><br><br><strong>7</strong><br><strong>40</strong>",
|
||
},
|
||
{
|
||
label: "D",
|
||
text: "<strong>x</strong><br><strong>y</strong><br><br><strong>3</strong><br><strong>24</strong><br><br><strong>5</strong><br><strong>36</strong><br><br><strong>7</strong><br><strong>48</strong>",
|
||
},
|
||
],
|
||
correctAnswer: "C",
|
||
explanation:
|
||
"Choice C is correct. All the tables in the choices have the same three values of <strong>x</strong>, so each of the three values of <strong>x</strong> can be substituted in the given inequality to compare the corresponding values of <strong>y</strong> in each of the tables. Substituting <strong>3</strong> for <strong>x</strong> in the given inequality yields <strong>y < 6 (3) + 2</strong>, or <strong>y < 20</strong>. Therefore, when <strong>x = 3</strong>, the corresponding value of <strong>y</strong> is less than <strong>20</strong>. Substituting <strong>5</strong> for <strong>x</strong> in the given inequality yields <strong>y < 6 (5) + 2</strong>, or <strong>y < 32</strong>. Therefore, when <strong>x = 5</strong>, the corresponding value of <strong>y</strong> is less than <strong>32</strong>. Substituting <strong>7</strong> for <strong>x</strong> in the given inequality yields <strong>y < 6 (7) + 2</strong>, or <strong>y < 44</strong>. Therefore, when <strong>x = 7</strong>, the corresponding value of <strong>y</strong> is less than <strong>44</strong>. For the table in choice C, when <strong>x = 3</strong>, the corresponding value of <strong>y</strong> is <strong>16</strong>, which is less than <strong>20</strong>; when <strong>x = 5</strong>, the corresponding value of <strong>y</strong> is <strong>28</strong>, which is less than <strong>32</strong>; when <strong>x = 7</strong>, the corresponding value of <strong>y</strong> is <strong>40</strong>, which is less than <strong>44</strong>. Therefore, the table in choice C gives values of <strong>x</strong> and their corresponding values of <strong>y</strong> that are all solutions to the given inequality.<br>Choice A is incorrect. In the table for choice A, when <strong>x = 3</strong>, the corresponding value of <strong>y</strong> is <strong>20</strong>, which is not less than <strong>20</strong>; when <strong>x = 5</strong>, the corresponding value of <strong>y</strong> is <strong>32</strong>, which is not less than <strong>32</strong>; when <strong>x = 7</strong>, the corresponding value of <strong>y</strong> is <strong>44</strong>, which is not less than <strong>44</strong>.<br>Choice B is incorrect. In the table for choice B, when <strong>x = 5</strong>, the corresponding value of <strong>y</strong> is <strong>36</strong>, which is not less than <strong>32</strong>.<br>Choice D is incorrect. In the table for choice D, when <strong>x = 3</strong>, the corresponding value of <strong>y</strong> is <strong>24</strong>, which is not less than <strong>20</strong>; when <strong>x = 5</strong>, the corresponding value of <strong>y</strong> is <strong>36</strong>, which is not less than <strong>32</strong>; when <strong>x = 7</strong>, the corresponding value of <strong>y</strong> is <strong>48</strong>, which is not less than <strong>44</strong>.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "e8f9e117",
|
||
type: "spr",
|
||
questionHtml:
|
||
"The formula above is Ohm’s law for an electric circuit with current I, in amperes, potential difference V, in volts, and resistance R, in ohms. A circuit has a resistance of 500 ohms, and its potential difference will be generated by n six-volt batteries that produce a total potential difference of <strong>6 n</strong> volts. If the circuit is to have a current of no more than 0.25 ampere, what is the greatest number, n, of six-volt batteries that can be used?",
|
||
choices: [],
|
||
correctAnswer: "",
|
||
explanation:
|
||
"The correct answer is 20. For the given circuit, the resistance R is 500 ohms, and the total potential difference V generated by n batteries is <strong>6 n</strong> volts. It’s also given that the circuit is to have a current of no more than 0.25 ampere, which can be expressed as <strong>I < 0 . 2 5</strong>. Since Ohm’s law says that <strong>I = V over R</strong>, the given values for V and R can be substituted for I in this inequality, which yields <strong>6 n over 500 < 0 . 2 5</strong>. Multiplying both sides of this inequality by 500 yields <strong>6 n < 125</strong>, and dividing both sides of this inequality by 6 yields <strong>n < 20 . 8 3 3</strong>. Since the number of batteries must be a whole number less than 20.833, the greatest number of batteries that can be used in this circuit is 20.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "ee2f611f",
|
||
type: "spr",
|
||
questionHtml:
|
||
"A local transit company sells a monthly pass for $95 that allows an unlimited number of trips of any length. Tickets for individual trips cost $1.50, $2.50, or $3.50, depending on the length of the trip. What is the minimum number of trips per month for which a monthly pass could cost less than purchasing individual tickets for trips?",
|
||
choices: [],
|
||
correctAnswer: "",
|
||
explanation:
|
||
"The correct answer is 28. The minimum number of individual trips for which the cost of the monthly pass is less than the cost of individual tickets can be found by assuming the maximum cost of the individual tickets, $3.50. If n tickets costing $3.50 each are purchased in one month, the inequality 95 n represents this situation. Dividing both sides of the inequality by 3.50 yields 27.14 n, which is equivalent to n > 27.14. Since only a whole number of tickets can be purchased, it follows that 28 is the minimum number of trips.",
|
||
hasFigure: false,
|
||
},
|
||
{
|
||
id: "ee7b1de1",
|
||
type: "spr",
|
||
questionHtml:
|
||
"A small business owner budgets <strong>dollar sign 2, 200</strong> to purchase candles. The owner must purchase a minimum of <strong>200</strong> candles to maintain the discounted pricing. If the owner pays <strong>dollar sign 4.90</strong> per candle to purchase small candles and <strong>dollar sign 11.60</strong> per candle to purchase large candles, what is the maximum number of large candles the owner can purchase to stay within the budget and maintain the discounted pricing?",
|
||
choices: [],
|
||
correctAnswer: "182",
|
||
explanation:
|
||
"The correct answer is <strong>182</strong>. Let <strong>s</strong> represent the number of small candles the owner can purchase, and let <strong>script l</strong> represent the number of large candles the owner can purchase. It’s given that the owner pays <strong>dollar sign 4.90</strong> per candle to purchase small candles and <strong>dollar sign 11.60</strong> per candle to purchase large candles. Therefore, the owner pays <strong>4.90 s</strong> dollars for <strong>s</strong> small candles and <strong>11.60 script l</strong> dollars for <strong>script l</strong> large candles, which means the owner pays a total of <strong>4.90 s + 11.60 script l</strong> dollars to purchase candles. It’s given that the owner budgets <strong>dollar sign 2, 200</strong> to purchase candles. Therefore, <strong>4.90 s + 11.60 script l < or = 2, 200</strong>. It’s also given that the owner must purchase a minimum of <strong>200</strong> candles. Therefore, <strong>s + script l > or = 200</strong>. The inequalities <strong>4.90 s + 11.60 script l < or = 2, 200</strong> and <strong>s + script l > or = 200</strong> can be combined into one compound inequality by rewriting the second inequality so that its left-hand side is equivalent to the left-hand side of the first inequality. Subtracting <strong>script l</strong> from both sides of the inequality <strong>s + script l > or = 200</strong> yields <strong>s > or = 200 − script l</strong>. Multiplying both sides of this inequality by <strong>4.90</strong> yields <strong>4.90 s > or = 4.90 (200 − script l)</strong>, or <strong>4.90 s > or = 980 − 4.90 script l</strong>. Adding <strong>11.60 script l</strong> to both sides of this inequality yields <strong>4.90 s + 11.60 script l > or = 980 − 4 . 90 script l + 11 . 60 script l</strong>, or <strong>4.90 s + 11.60 script l > or = 980 + 6.70 script l</strong>. This inequality can be combined with the inequality <strong>4.90 s + 11.60 script l < or = 2, 200</strong>, which yields the compound inequality <strong>980 + 6.70 script l < or = 4.90 s + 11.60 script l < or = 2, 200</strong>. It follows that <strong>980 + 6.70 script l < or = 2, 200</strong>. Subtracting <strong>980</strong> from both sides of this inequality yields <strong>6.70 script l < or = 2, 200</strong>. Dividing both sides of this inequality by <strong>6.70</strong> yields approximately <strong>script l < or = 182.09</strong>. Since the number of large candles the owner purchases must be a whole number, the maximum number of large candles the owner can purchase is the largest whole number less than <strong>182.09</strong>, which is <strong>182</strong>.",
|
||
hasFigure: false,
|
||
},
|
||
];
|