Algebra · Lesson 11 of 16

Inequalities and Number Lines

An inequality compares two values and has many solutions. Solve it like an equation, but flip the sign if you multiply or divide by a negative.

Secondary
Updated 14 November 2026

What you will learn

Read inequality symbols.
Show solutions on a number line.
Solve simple inequalities.

Before you start: you should be able to solve equations and use negative numbers. See Solving Equations.

The idea

An inequality says one value is bigger or smaller than another. The symbols are $<$ (less than), $>$ (greater than), $\le$ (less than or equal to) and $\ge$ (greater than or equal to). Unlike an equation, it usually has many solutions.

The four inequality symbols. The small line under the symbol means the boundary value is included.

On a number line

The boundary is where the solutions start. Mark the boundary with a dot. On paper use an open circle when the boundary is not included and a filled circle when it is. Then draw an arrow in the direction of the solutions.

x > 3 starts just after 3 and continues to the right. The boundary 3 itself is not a solution.

Solving

Use the same steps as an equation. The one exception: multiplying or dividing both sides by a negative number reverses the sign. $-2x < 6$ becomes $x > -3$ .

The solution of -2x < 6 is x > -3. Test a value: x = 0 gives 0 < 6, which is true.

Worked examples

Example 1: One step

Solve $x + 3 > 7$ .

Subtract $3$ from both sides.
$x > 4$ .

Answer: $x > 4$

Example 2: Two steps

Solve $3x - 2 < 7$ .

Add $2$ : $3x < 9$ .
Divide by $3$ : $x < 3$ .

Answer: $x < 3$

Example 3: Negative

Solve $-2x < 6$ .

Divide both sides by $-2$ .
Flip the sign: $x > -3$ .

Answer: $x > -3$

Tip: test one value from your answer in the original inequality. It should be true.

Why it matters

Inequalities describe limits: minimum ages, speed limits, budgets and safe ranges. They also describe regions on graphs.

Practice

Your turn. Type or choose an answer, then press Check. Getting one wrong is fine, that is what practice is for. You can try as many times as you like.