## Solving Inequalities

In text formatting, > means greater than, < means less than, >= means greater than or equal to, and <= means less than or equal to. You can treat these signs just like equal signs when solving inequalities, with one exception. Before we look at the exception, let's look at solving a basic inequality.

2x - 5 > 19
2x – 5 + 5 > 19 + 5
2x > 24
2x/2 > 24/2
x > 12

This means that any value of x so that x is greater than 12 will make 2x – 5 greater than 19. We’ll leave it to the student to check this answer.

Solving an inequality is very much like solving an equation, except there’s one new rule to remember. If you ever multiply or divide both sides by a negative number, you have to switch the direction of the inequality’s sign. For example:

-2x < 14

Here we'll divide both sides by -2, so we'll also have to change the < to a >.

-2x/(-2) > 14/(-2)
x > -7

To check this, let’s pick a number slightly greater than -7 for x. Let’s choose -6.9.

-2x < 14
-2(-6.9) < 14
13.8 < 14

And our inequality checks out just fine.

This rule of having to switch the sign comes from how the integers are ordered. While 3 < 4, -3 > -4. So if we multiply or divide both sides by a negative number, we have to switch the sign. This means < will become >, > will become <, <= will become >=, and >= will become <=.

The single most common mistake that people make when solving inequalities is that they forget to switch the direction of the sign when they multiply or divide by a negative number.