Year 9

Linear inequalities

Solve linear inequalities in one variable and represent solutions on a number line.

Back to Year 9 Previous lesson Next lessonProgress: not startedMastery: not started

Lesson overview

Algebra - inequalities

An inequality is like an equation but instead of an equals sign, it uses <, >, ≤ or ≥. You solve inequalities in almost the same way as equations — adding, subtracting, multiplying or dividing both sides — with one critical exception: when you multiply or divide both sides by a negative number, the inequality sign must flip. Solutions to inequalities are ranges of values, not single answers, and are best shown on a number line.

Solving a simple inequality

Solve 3x + 1 < 10.

Subtract 1 from both sides, then divide by 3. The inequality sign stays the same.

Critical rule: dividing by a negative flips the sign

Solve −2x > 8.

Why the sign flippedDividing both sides by −2 reverses the inequality

Always flip when ÷ or × by a negative

This is the most common mistake! Dividing by a negative number always flips the direction of the inequality.

Inequality notation

Understand the four symbols.

x < 3x is less than 3

Open circle on number line

x > 3x is greater than 3

Open circle on number line

x ≤ 3x is less than or equal to 3

Closed (filled) circle on number line

x ≥ 3x is greater than or equal to 3

Closed (filled) circle on number line

Open circle = value not included. Filled circle = value included.

Two-sided inequalities

Solve and show on a number line: −1 < 2x + 3 ≤ 9.

Do the same operation to all three parts. The answer is a range: x is greater than −2 and at most 3.

Worked example

Solve 4 − 3x ≥ 13. Show the solution on a number line.

Number lineFilled circle at −3, arrow pointing left

≤ means −3 is included

Subtract 4 from both sides: −3x ≥ 9.
Divide both sides by −3 — and FLIP the inequality sign: x ≤ −3.
The solution is all values of x that are −3 or less.
On the number line: filled circle at −3, shaded to the left.

Try it

Use the examples carefully, then choose the answer.

Question 1

Solve 2x + 5 > 13.

💡 Subtract 5, then divide by 2.

Question 2

Solve 5x − 3 ≤ 12.

💡 Add 3, then divide by 5.

Question 3

Solve −4x < 20.

💡 Divide by −4. Remember to flip the inequality sign!

Question 4

Solve 3 − 2x ≥ −7.

💡 Subtract 3 first, then divide by −2 and flip.

Question 5

Solve 1 < 3x − 2 ≤ 10 and write the solution.

💡 Add 2 to all three parts, then divide all three parts by 3.

Linear inequalities

Lesson overview

Worked example

Try it

Independent practice will plug in here