Linear Inequalities

Understanding ranges, rules, and representations

CAPS Grade 10 Mathematics

Linear inequalities are a fundamental aspect of Grade 10 algebra, providing a framework to describe a range of possible solutions rather than a single fixed value. This document outlines the key rules for solving linear inequalities, methods for representing solutions, and graphical interpretations.

Learning Outcomes

  • Understand and apply the negative rule when multiplying/dividing by negative numbers
  • Solve linear inequalities using addition, subtraction, multiplication, and division
  • Represent solutions using inequality notation, number lines, and interval notation
  • Interpret graphical solutions for inequalities with two variables
  • Use test points to determine shading regions

Key Rules for Solving Linear Inequalities

!

The Negative Rule

When you multiply or divide both sides by a negative number, you must reverse (flip) the inequality sign.

Example: -2x < 6 → divide by -2 → x > -3
+/-

Addition/Subtraction

Adding or subtracting the same value on both sides does not change the inequality sign.

x - 3 < 5 → x < 8

Quiz 1 - The Negative Rule

Solve: -3x > 12

A) x > -4
B) x < -4
C) x > 4
D) x < 4

Representing Solutions

1. Inequality Notation

Simple expressions to represent the solution.

x < 5
x ≥ 3

2. Number Lines

Open Circle (○): > or < (excluded)

Closed Circle (●): ≥ or ≤ (included)

3. Interval Notation

(3,5) → 3 and 5 not included
[3,5] → 3 and 5 included

Open Circle Example: x < 3

3

Closed Circle Example: x ≥ -2

-2

Quiz 2 - Representations

What does a closed circle on a number line indicate?

A) The value is excluded
B) The value is included
C) The value is undefined
D) The value is infinite

Graphical Solutions (Two Variables)

Dashed Line

Used for strict inequalities (< or >). Points on the line are not part of the solution.

Solid Line

Used for inclusive inequalities (≤ or ≥). Points on the line are included.

Shading

Use a test point (often the origin) to determine which side to shade.

Graphical Example

Problem

Graph y ≤ 2x + 1

Solution
  • Draw line y = 2x + 1 (solid because ≤)
  • Test point (0,0): 0 ≤ 1 ✓ true
  • Shade region containing (0,0) (below the line)

Quiz 3 - Graphical Solutions

When graphing y > 2x - 1, what kind of line is used?

A) Solid line
B) Dashed line
C) Dotted line
D) Thick line

Practice & Assess

Match - Inequality Symbols

<
Less than
>
Greater than
≤
Less than or equal
≥
Greater than or equal

Fill - Negative Rule

When dividing by a negative number, the inequality sign _____.

Practice Questions

Q1

Solve: 2x - 5 < 7

Q2

Solve: -4x ≥ 12

Q3

Write in interval notation: x > 3

Answers

Q1: x < 6 | Q2: x ≤ -3 | Q3: (3, ∞)

Summary of Key Concepts

Negative Rule: Flip the sign when multiplying/dividing by a negative.
Representations: Inequality notation, number lines (open/closed circles), interval notation.
Graphical: Dashed line for < or >, solid line for ≤ or ≥, shade using test point.

Key Terms

Inequality Less than Greater than Less than or equal Greater than or equal Negative rule Number line Open circle Closed circle Interval notation Test point Shading
Exponents and Surds Linear Equations