# math questions answered

**Solving Inequalities: An Overview** **(page 1 of 3)**

Solving linear inequalities is very similar to solving linear equations , except for one small but important detail: you flip the inequality sign whenever you multiply or divide the inequality by a negative. The easiest way to show this is with some examples:

Graphically, the solution is:

Note that the solution to a less than, but not equal to inequality is graphed with a parentheses (or else an open dot) at the endpoint, indicating that the endpoint is not included within the solution.

Graphically, the solution is:

Note that *x* in the solution does not have to be on the left. However, it is often easier to picture what the solution means with the variable on the left. Don’t be afraid to rearrange things to suit your taste.

Graphically, the solution is:

Note that the solution to a less than or equal to inequality is graphed with a square bracket (or else a closed dot) at the endpoint, indicating that the endpoint is included within the solution.

Graphically, the solution is:

Copyright Elizabeth Stapel 1999-2011 All Rights Reserved

Graphically, the solution is:

The rule for example 5 above often seems unreasonable to students the first time they see it. But think about inequalities with numbers in there, instead of variables. You know that the number four is larger than the number two: 4 2 . Multiplying through this inequality by 1 , we get 4 2 , which the number line shows is true:

If we hadn’t flipped the inequality, we would have ended up with 4 2 , which clearly isn’t true.

Cite this article as:

Stapel, Elizabeth. Solving Inequalities: An Overview. __Purplemath__. Available from