import React from "react"; import { Scale, ArrowRight, GitBranch, BarChart, Layers, BookOpen, } from "lucide-react"; import LessonShell, { ConceptCard, FormulaBox, ExampleCard, TipCard, PracticeFromDataset, } from "../../../components/lessons/LessonShell"; import InequalityRegionWidget from "../../../components/lessons/InequalityRegionWidget"; import { LINEAR_INEQ_EASY, LINEAR_INEQ_MEDIUM, } from "../../../data/math/linear-inequalities"; interface LessonProps { onFinish?: () => void; } const SECTIONS = [ { title: "Solving Inequalities", icon: Scale }, { title: "Compound Inequalities", icon: GitBranch }, { title: "Graphing on Number Lines", icon: ArrowRight }, { title: "Coordinate Plane", icon: BarChart }, { title: "Systems of Inequalities", icon: Layers }, { title: "Practice & Quiz", icon: BookOpen }, ]; export default function LinearInequalitiesLesson({ onFinish }: LessonProps) { return ( {/* Section 1: Solving Inequalities */}

Solving Inequalities

Inequalities work just like equations — apply the same operation to both sides. The one critical difference: when you multiply or divide by a negative number, you must{" "} flip the inequality sign.

<

Less than

>

Greater than

Less than or equal

Greater than or equal

Solve: −3x + 7 > 16

−3x > 9

Divide by −3 → FLIP → x < −3

Solve: 2x − 5 ≤ 3x + 2

−x ≤ 7

Multiply by −1 → x ≥ −7

The #1 inequality mistake: forgetting to flip the sign when multiplying or dividing by a negative. The SAT specifically designs trap answers for this.

{/* Section 2: Compound Inequalities */}

Compound Inequalities

A compound inequality combines two inequalities.{" "} AND means both must be true (intersection).{" "} OR means at least one must be true (union).

AND (Intersection)

−2 < x ≤ 5 means x is between −2 and 5

Both conditions satisfied simultaneously

OR (Union)

x < −3 OR x > 4 means outside the interval

At least one condition satisfied

Solve: −2 < 3x + 1 ≤ 10

Subtract 1: −3 < 3x ≤ 9

Divide by 3:{" "} −1 < x ≤ 3

Solve: x + 1 < −2 OR x + 1 > 4

x < −3 OR x > 3

Solution: (−∞, −3) ∪ (3, ∞)

{/* Section 3: Graphing on Number Lines */}

Graphing on Number Lines

When graphing inequalities on a number line, the circle type and shading direction matter.

Symbol Circle Meaning
< or > Open ○ Value NOT included
≤ or ≥ Closed ● Value IS included

The SAT frequently asks which number line graph matches a given inequality. Just check: open vs closed circle, and which direction the arrow points.

{/* Section 4: Coordinate Plane */}

Coordinate Plane Inequalities

When graphing a linear inequality in two variables:

1. Graph the boundary line (y = mx + b)

2. Use a dashed line for < or > (not included)

3. Use a solid line for ≤ or ≥ (included)

4. Shade above for y > or y ≥ and{" "} below for y < or y ≤

Boundary line: y = 2x − 1 (slope 2, y-intercept −1)

Solid line (≥ means included)

Shade above the line

{/* Section 5: Systems of Inequalities */}

Systems of Inequalities

The solution to a system of inequalities is the{" "} overlapping region where ALL inequalities are satisfied. Any point in this region satisfies every inequality in the system.

y ≤ x + 3 AND y > −x + 1

Graph both: shade below y = x + 3 (solid), shade above y = −x + 1 (dashed)

Solution is the overlapping region

To check if a point is in the solution region, plug it into BOTH inequalities. It must satisfy all of them.

{/* Section 6: Practice & Quiz */}

Practice & Quiz

{LINEAR_INEQ_EASY.slice(0, 2).map((q) => ( ))} {LINEAR_INEQ_MEDIUM.slice(0, 1).map((q) => ( ))}
); }