Solving inequalities

An equation usually has one answer. An inequality has a whole range of them. $x > 3$ means 3.1 works, 4 works, 1000 works, and 3 itself doesn't.

On a number line,

x > 3

is everything to the right of 3. The circle at 3 is left open on a graph you draw by hand, because 3 isn't included.

Adding, subtracting and multiplying by a positive all work exactly as they do with equations. Only negatives change things. Start with $2 < 5$ and multiply both sides by $- 1$ . You get $- 2$ and $- 5$ , and $- 2$ is the bigger one, so the sign has to turn around: $- 2 > - 5$ .

Multiply or divide by a negative number, and the inequality sign flips.

Work through it

Solve $3 x - 4 \geq 11$ .

Add 4 to both sides. $3 x \geq 15$

Divide by 3. It's positive, so the sign stays. $x \geq 5$

Solve $5 - 2 x < 11$ .

Subtract 5. $- 2 x < 6$

Divide by $- 2$ and flip the sign. $x > - 3$

Test $x = 0$ , which should work. $5 - 0 < 11$

Watch out for this

Solving $- 2 x < 6$ and writing $x < - 3$ .

Dividing by $- 2$ flips the sign, so it's $x > - 3$ . Testing one value catches this every time.

The short version
Solve it like an equation. If you multiply or divide by a negative, flip the sign.

Your turn

Prefer paper? Print the Inequalities and absolute value worksheet, with an answer key.

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Work through it

Solve 3x−4≥11.

Solve 5−2x<11.

Watch out for this

Your turn

Solve $3 x - 4 \geq 11$ .

Solve $5 - 2 x < 11$ .