1.5 Solving Inequalities

In order to express the solutions to inequalities, we will need some notation.

Here’s a worksheet that covers some aspects of expressing inequalities.

For linear inequalities, we solve these mostly like linear equations, taking care to flip the inequality symbol if we multiply or divide by a negative number.

Here is a worksheet where you can solve linear inequalities and graph their solutions. Note you will be required to express the solutions to inequalities in all the ways shown above, even though this worksheet only has you express them as a graph.

We now solve absolute value inequalities. Taking note that your approach will differ depending on whether the absolute value is greater or less than a number.

Here is a worksheet with extra problems on absolute value inequalities.

Now we move on to general polynomial inequalities. Our approach for these will be to get a polynomial on one side which is greater than or less than 0. Then we factor and find the critical points where the factors can change signs. Those points divide the number line into regions, which we test for validity.

Here is a worksheet with polynomial inequalities.

Finally, if you have a rational inequality, you can use a similar technique to solve them.

Here is a worksheet on rational inequalities.