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A&G-Inequalities
📊 E2.6 Inequalities - Learner Notes
Master the fundamentals of representing, solving, and graphing inequalities
1. Representing Inequalities on a Number Line
Open circles (○) = strict inequalities (< or >)
→ The number is NOT included in the solution
→ The number is NOT included in the solution
Closed circles (●) = inclusive inequalities (⩽ or ⩾)
→ The number IS included in the solution
→ The number IS included in the solution
Example: −3 ⩽ x < 1
- Closed circle at −3 (because x can equal −3)
- Open circle at 1 (because x cannot equal 1)
- Shade the region between them
💡 Reading tip: The inequality points to the smaller number!
2. Solving Linear Inequalities
Examples to practice:
- 3x < 2x + 4
- −3 ⩽ 3x − 2 < 7
Important rules:
- Treat inequalities like equations EXCEPT...
- You can add/subtract from all parts of a compound inequality
- Keep track of whether your inequality is strict (<, >) or inclusive (⩽, ⩾)
⚠️ CRITICAL RULE: FLIP the inequality sign when multiplying/dividing by a negative number
Example: −2x < 6 becomes x > −3 (sign flips!)
Example: −2x < 6 becomes x > −3 (sign flips!)
3. Graphing Linear Inequalities (Two Variables)
Visual conventions:
- Broken/dashed lines = strict inequalities (<, >)
→ Points ON the line are NOT included - Solid lines = inclusive inequalities (⩽, ⩾)
→ Points ON the line ARE included - Shading = the region of solutions (unwanted regions can be shaded if specified)
Steps to graph:
- Rearrange inequality to y = mx + c form if needed
- Draw the line (dashed or solid)
- Test a point (often (0,0)) to see which side to shade
- Shade the correct region
Examples shown:
- x < 1 (vertical dashed line, shade left)
- y ⩾ 1 (horizontal solid line, shade above)
4. Defining Regions with Inequalities
What you need to do:
- Look at a shaded region on a graph
- Write down ALL the inequalities that define its boundaries
- Check each boundary: Is it solid or dashed? Which side is shaded?
Note: Linear programming problems are NOT included in this section (they come later!)
✓ Study Tips:
- Practice drawing number lines with different combinations of open/closed circles
- Always check your inequality direction after each step
- Remember: the shaded region contains all the solutions
- When in doubt, test a point to verify your answer!
import React, { useState } from 'react';
import { ChevronLeft, ChevronRight } from 'lucide-react';
const InequalityGraphs = () => {
const [currentQuestion, setCurrentQuestion] = useState(0);
const questions = [
{
number: 1,
description: "Plot and shade the unwanted region for:",
inequalities: ["x >= 2", "y <= 4", "y >= x + 1"],
drawLines: (g) => {
// Shaded unwanted regions
g.push(x = 2 );
g.push(y = 4 );
g.push(y = x + 1 );
}
},
{
number: 2,
description: "Plot and shade the unwanted region for:",
inequalities: ["y < 2x + 3", "y > -x + 1", "x <= 5"],
drawLines: (g) => {
// Shaded unwanted regions
g.push(y = 2x + 3 );
g.push(y = -x + 1 );
g.push(x = 5 );
}
},
{
number: 3,
description: "Plot and shade the unwanted region for:",
inequalities: ["y >= 0", "x >= 0", "y <= -2x + 8", "y <= x + 2"],
drawLines: (g) => {
// Shaded unwanted regions
g.push(y = -2x + 8 );
g.push(y = x + 2 );
}
},
{
number: 4,
description: "Plot and shade the unwanted region for:",
inequalities: ["2x + y <= 10", "x + 2y >= 8", "x >= 0", "y >= 0"],
drawLines: (g) => {
// Shaded unwanted regions
g.push(2x + y = 10 );
g.push(x + 2y = 8 );
}
},
{
number: 5,
description: "Plot and shade the unwanted region for:",
inequalities: ["y < 3x - 2", "y > x - 4", "x < 4", "y > -2"],
drawLines: (g) => {
// Shaded unwanted regions
g.push(y = 3x - 2 );
g.push(y = x - 4 );
g.push(x = 4 );
g.push(y = -2 );
}
},
{
number: 6,
description: "Plot and shade the unwanted region for:",
inequalities: ["3x + 2y <= 12", "x - y >= -2", "x >= 1", "y >= 0"],
drawLines: (g) => {
// Shaded unwanted regions
g.push(3x + 2y = 12 );
g.push(x - y = -2 );
g.push(x = 1 );
}
},
{
number: 7,
description: "Plot and shade the unwanted region for:",
inequalities: ["y <= 6 - x", "y >= 2x - 3", "x >= 0", "y >= 0"],
drawLines: (g) => {
// Shaded unwanted regions
g.push(y = 6 - x );
g.push(y = 2x - 3 );
}
},
{
number: 8,
description: "Plot and shade the unwanted region for:",
inequalities: ["x + y < 7", "2x - y > -4", "x > 0", "y < 5"],
drawLines: (g) => {
// Shaded unwanted regions
g.push(x + y = 7 );
g.push(2x - y = -4 );
g.push(y = 5 );
}
},
{
number: 9,
description: "Plot and shade the unwanted region for:",
inequalities: ["y >= 0.5x + 1", "y <= -x + 6", "y >= 1", "x <= 6"],
drawLines: (g) => {
// Shaded unwanted regions
g.push(y = 0.5x + 1 );
g.push(y = -x + 6 );
g.push(y = 1 );
g.push(x = 6 );
}
},
{
number: 10,
description: "Plot and shade the unwanted region for:",
inequalities: ["4x + 3y <= 24", "x + y >= 3", "x >= 0", "y >= 0"],
drawLines: (g) => {
// Shaded unwanted regions
g.push(4x + 3y = 24 );
g.push(x + y = 3 );
}
}
];
const currentQ = questions[currentQuestion];
const graphElements = [];
currentQ.drawLines(graphElements);
const nextQuestion = () => {
if (currentQuestion < questions.length - 1) {
setCurrentQuestion(currentQuestion + 1);
}
};
const prevQuestion = () => {
if (currentQuestion > 0) {
setCurrentQuestion(currentQuestion - 1);
}
};
return (
);
};
export default InequalityGraphs;
Plotting and Shading Unwanted Regions
Question {currentQ.number}
{currentQ.description}
-
{currentQ.inequalities.map((ineq, idx) => (
- • {ineq} ))}
Blue diagonal lines = UNWANTED region (shaded)
|
White area = WANTED region (unshaded)
|
Solid line = <= or >= | Dashed line = < or >
Question {currentQuestion + 1} of {questions.length}