Inequalities

More than one answer

An equation like x + 3 = 7 has one solution. An inequality like x + 3 > 7 has many: any x greater than 4 works.

Inequality symbols

Real-life inequalities

• You must be at least 16 to drive: a ≥ 16
• Carry-on bags must be under 10 kg: m < 10
• The elevator holds at most 12 people: p ≤ 12

Graphing on a number line

Mark the boundary value and shade the solution set.

Open means "not included"; closed means "included".

Solving inequalities

Solve just like equations, using inverse operations.

Example: 2x + 5 > 13

• 2x > 8
• x > 4

The one special rule

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

Example: −3x ≤ 12

• Divide by −3 and flip: x ≥ −4

Why? Start with 2 < 5 (true). Multiply both by −1: −2 and −5. Now −2 > −5. The order flipped!

Checking

Pick a number in your solution set and test it. For x > 4, try x = 5: 2(5) + 5 = 15 > 13 ✓. Try one outside: x = 3: 11 > 13 ✗ (good, it shouldn’t work).

Word problem

Problem: You have $50. Movie tickets cost $12. How many can you buy?

• 12t ≤ 50
• t ≤ 4.17
• Since t is a whole number: at most 4 tickets.

Worked examples

Example 1: x − 7 < 2 → x < 9

Example 2: 4 − x ≥ 1 → −x ≥ −3 → x ≤ 3 (flipped)

Example 3: n/2 > −5 → n > −10

Common mistakes

• Forgetting to reverse the sign when dividing by a negative.
• Reversing it when you don’t need to (dividing by a positive).
• Using the wrong circle on the number line.

Try it yourself

1. x + 4 ≥ 10
2. 5x < 35
3. −2x > 8
4. 3x − 1 ≤ 11
5. Describe the graph of x < −1.

Answers: 1) x ≥ 6 2) x < 7 3) x < −4 4) x ≤ 4 5) open circle at −1, shaded to the left

Quick recap

• An inequality has a solution set, not just one answer.
• Use an open circle for < and >, a closed circle for ≤ and ≥.
• Reverse the sign when multiplying or dividing by a negative.