Inequalities

An inequality like $x > 3$ has many solutions, not one: 4, 5, 3.5, and every number above 3. We show that on a number line as a ray.

x > 3

means every number to the right of 3, not including 3.

You solve them the same way as equations, with one catch. Start with the true statement $2 < 5$ , and multiply by $- 1$ : $- 2$ and $- 5$ . On the number line $- 2$ is now greater, so the sign has to flip.

Multiply or divide by a negative, and

<

becomes

>

Work through it

$2 x + 1 > 9$

Subtract 1. $2 x > 8$

Divide by 2, a positive, so nothing flips. $x > 4$

$5 - 3 x \leq 17$

Subtract 5. $- 3 x \leq 12$

Divide by $- 3$ , so flip. $x \geq - 4$

Watch out for this

Flipping the sign whenever there's a negative number around.

Only flip when you multiply or divide by a negative. Adding or subtracting never flips it.

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

Your turn

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