📊 Inequalities

Grade 8 · Cambridge Lower Secondary Stage 8

x < 3 x ≤ 3 x > 3 x ≥ 3

Symbols

< less than > greater than
≤ less than or equal to ≥ greater than or equal to
Open circle on number line = strict inequality (< or >)
Filled circle = includes the endpoint (≤ or ≥)

What you'll learn

Represent inequalities on a number line
Solve linear inequalities
Solve double inequalities
List integer solutions
Solve inequality word problems

📖 Learn

1. Number Line Representation

Draw an arrow from the boundary value in the correct direction.

x > 2: open circle at 2, arrow to the right → ○→

x ≤ −1: filled circle at −1, arrow to the left ←●

−2 < x ≤ 4: open circle at −2, filled circle at 4, line between

2. Solving Inequalities

Solve exactly like equations, with one important exception:

⚠️ When you multiply or divide by a negative number, the inequality sign flips!

3x + 5 > 14 → 3x > 9 → x > 3

2 − x < 5 → −x < 3 → x > −3 (flip when dividing by −1)

4x − 3 ≥ 2x + 7 → 2x ≥ 10 → x ≥ 5

3. Double Inequalities

Perform the same operation on all three parts simultaneously.

−1 < 2x + 3 ≤ 9 → subtract 3: −4 < 2x ≤ 6 → divide by 2: −2 < x ≤ 3

4. Integer Solutions

After solving, list all whole numbers (integers) satisfying the inequality.

−2 < x ≤ 3 → integers: −1, 0, 1, 2, 3 (−2 excluded, 3 included)

x² < 16 → −4 < x < 4 → integers: −3, −2, −1, 0, 1, 2, 3

5. Inequality in Context

"A lift can carry at most 500 kg. Each person weighs 80 kg on average. Maximum n people?" → 80n ≤ 500 → n ≤ 6.25 → n ≤ 6

"Budget > £200 after buying n items at £35 each from £500." → 500 − 35n > 200 → 35n < 300 → n < 8.57 → n ≤ 8

✏️ Worked Examples

Example 1 — Solve and Represent

Solve 5x − 3 < 2x + 9 and show on a number line.

Step 1: 5x − 2x < 9 + 3 → 3x < 12

Step 2: x < 4

Number line: open circle at 4, arrow pointing left

Example 2 — Negative Multiplication

Solve 3 − 2x ≥ 7

Step 1: −2x ≥ 4

Step 2: Divide by −2, flip sign: x ≤ −2

Example 3 — Double Inequality

Solve 1 ≤ 3x − 2 < 10

Step 1: Add 2: 3 ≤ 3x < 12

Step 2: Divide by 3: 1 ≤ x < 4

Integer solutions: 1, 2, 3 (4 excluded)

Example 4 — Integer Solutions

Find integers n such that n² ≤ 20.

n² ≤ 20 → −√20 ≤ n ≤ √20 → −4.47 ≤ n ≤ 4.47

Integer solutions: −4, −3, −2, −1, 0, 1, 2, 3, 4

🎨 Visualizer

📏 Number Line Builder

Enter an inequality to draw it on the number line.

⚖️ Inequality Solver

Enter a linear inequality ax + b < c to solve step by step.

x

🔢 Integer Finder

Enter a solved inequality range and find all integers in it.

x

🎯 Quick Fire — True or False?

Is the value a solution to the inequality?

Exercise 1 — Solving Simple Inequalities

Solve and enter the boundary value (the number x is compared to).

Exercise 2 — Two-step Inequalities

Exercise 3 — Negative Coefficients

Remember to flip the sign when dividing by a negative. Enter the boundary value.