Study Guide

Solving Inequalities

Solving Inequalities

Solving inequalities isn't that much different than solving equations. Instead of having an equal sign divide the two sides, there's an inequality sign.

However, there's one really important rule:

If we multiply or divide by a negative number, we need to flip the inequality sign.

For example, let's look at -2x + 3 > 5.

-2x + 3 > 5solve like you would -2x + 3 = 5
-2x + 3 - 3 > 5 - 3subtract 3 from each side
-2x > 2simplify
-2x/-2 > 2/-2divide each side by -2
x < -1switch the sign from > to <

With this last example, if we had divided by positive 2 instead of -2, we would've found that -x > 1. So, x < -1 and -x > 1 are really the same thing! That's why we need to switch the sign when we divide or multiply by a negative.

Since we divided by -2, we switched the sign from > to <. Now, just like with equations, we can check our answers. Since x < -1, pick any number less than -1 and plug it into the original inequality (we picked -2).

-2(-2) + 3 > 5

4 + 3 > 5

7 > 5

Yup, 7 is greater than 5, so we can be pretty confident that we solved this correctly. However, unlike equations, we can't be completely sure. If we want to double check our work, that wouldn't be a horrible idea.

Look Out: we only switch the inequality sign if we multiply or divide by a negative number. We don't switch it if we add or subtract a negative number.