Not all inequailities are simple. Some have a third part to the equation, or have a separate equation altogether. Regardless of what it looks like, it’s still solved the same way as a simple inequality.
Let’s look at this first example:
3 < x – 4 < 8
Although there are three parts to this inequality, we’re still going to get x by itself. In this case, we’ll add 4 to all three sides of the equation. We add it to the middle to cancel out the -4, and then we add to both the front and back terms to keep our equation balanced. The new equation looks like:
7 < x < 12
This means that x could only be a number between, but not including, 7 and 12.
Even if the compound inequality has you multiplying or dividing by a negative number, you’ll still perform the operation on all three terms. For example:
4 < x/-2 < 6
resolves to:
-8 > x > -12
when you multiply each term by -2. If you think about the number line, you’ll realize it makes more sense for -8 to be greater than -12.
Not all compound inequalities are in a line like this, though. Some compound inequalities describe two different lines. For example:
x > 3 or x < -3
If you have to solve for a variable in this type of compound inequality, all the inequality rules apply.
