This section discusses how to compute and/or verify inequalities

Text and Supplemental Content

In most cases, verifying inequalities is as simple as just computing both sides of the inequality down to a value, and making sure the inequality is true. As simple as that is to explain, however, sometimes it can be challenging in practice.

Consider the following example: Suppose we want to “verify” this inequality for when . To do this, we start by computing both sides independently.

Notice how, in the above computation, we didn’t continue to write the sign the whole way down. It’s helpful to record it to make sure we remember which way the sign goes, but we don’t know if the inequality is valid once the number is input, so we definitely shouldn’t include the sign once the number is put in for the variable.

Nonetheless, once we have gotten to the end of the computations on both sides, now we can ask if the inequality is true - that is, whether , which it is. Thus this is inequality is indeed true at .

This may seem silly and straight forward, and to some extent it is, but there are lots of times in the following sections and chapters (especially when we discuss graphing inequalities and a bunch of content in the absolute value chapter) where this kind of check or verification will be a necessary step in determining solution sets and/or whether a solution is extraneous.

To determine if an inequality is valid at a specific -value you should...
Check your work to make sure you didn’t make any errors when generating the inequality. Move everything to one side and make sure it is positive. Calculate both sides of the inequality independently, then compare using the original inequality to see if it is valid. Ask someone else, this stuff is weird.