Solving Inequalities

An inequality is solved almost exactly like an equation: do the same to both sides.

One thing is different

Multiplying or dividing both sides by a negative reverses the symbol.

Why it reverses

3 < 5 is true. Multiply both by −1 and you get −3 and −5. But −3 > −5, so the symbol had to turn.

Test your answer

Pick a number from your solution and put it in the original. It should make it true.

Solved the same way, with one exception

Inequalities are solved by doing the same to both sides, exactly like equations — except that multiplying or dividing both sides by a negative number reverses the direction of the symbol.

Why the direction reverses

It is true that 3 < 5. Multiply both by −1 and you get −3 and −5, but −3 > −5. Multiplying by a negative reflects both numbers across zero, which reverses their order. The flip is a consequence of that reflection.

Adding and subtracting never flip it

Adding or subtracting the same number shifts both sides equally along the line and leaves their order intact — even if the number is negative. Only multiplying or dividing by a negative reverses the symbol.

Check with a value from your solution

If you solve −2x > 6 to get x < −3, test x = −4: −2(−4) = 8, and 8 > 6 is true. Testing one value from inside the solution range catches a forgotten flip immediately, which is the whole reason to do it.