Solving inequalities

Solve it like an equation

Inequalities are solved with the same inverse operations as equations. Adding or subtracting on both sides shifts both numbers the same way, so their order never changes.

Multiplying or dividing by a positive number is safe too.

Something goes wrong

Solve $-2x < 8$ the same way: divide both sides by $-2$ and you get $x < -4$. Before trusting it, test it. If $x < -4$ is right, numbers like $-5$ and $-6$ should work, and $0$ shouldn't.

$-2x < 8$

Tap a number on the line to test it.

Test every pointShow the solution set

It's backwards. $-5$ fails and $0$ works. The real answer is $x > -4$. Something about dividing by a negative turned the inequality around.

Why negatives flip it

Start with something true: $2 < 5$. Multiply both numbers by $2$, then try $-1$, and watch the dots.

$2 < 5$$2 < 5$

Multiply both by 2Multiply both by −1Multiply both by −2

Multiplying by a positive stretches the line, and the order stays. Multiplying by a negative mirrors every number across zero: $2$ lands on $-2$ and $5$ lands on $-5$. The number that was further right is now further left, so the inequality between them reverses.

Now $-2x < 8$ comes out right:

Multi-step inequalities

Simplify, collect, then solve, flipping only when you multiply or divide by a negative.

Subtracting $5$ didn't flip anything. Dividing by $-3$ did.

Key ideas

• Solve inequalities with the same steps as equations.
• Multiplying or dividing by a negative mirrors the number line, so the inequality sign flips.
• Adding, subtracting, or multiplying and dividing by a positive never flips it.
• Test a value from your answer in the original inequality to check.