Find all values of 𝑥 that satisfy negative 18 is less than or equal to negative 𝑥 plus four, which is less than or equal to negative one. Write your answer as an interval.
This is a compound linear inequality. It has more than one inequality symbol in the problem. This means that when we solve this, we’ll get a range of what 𝑥 can be.
Solving a compound inequality is basically the same as solving a simple inequality, except that whatever we do to one side, we have to do to all three pieces of our equation. Let’s take a look.
To get 𝑥 by itself, I’ll want to subtract four, but that means that I’ll need to subtract four from negative 18, negative one, and from the middle. Negative 18 minus four is negative 22.
We don’t change our inequality symbol; we just bring it down. We haven’t changed our negative 𝑥, so we bring that down as well. Positive four minus four equals zero; they cancel each other out; we have nothing to bring down here. We bring down our second inequality symbol, and then we say negative one minus four equals negative five.
Now it looks like 𝑥 is isolated, but our problem is asking to find all the values of 𝑥, and what we’re currently showing is the values of negative 𝑥. In order to solve this problem, we’ll need to multiply our entire compound inequality by negative one.
Starting here, we’ll say negative one times negative 22 equals 22. Now here’s something very important to remember. When we’re dealing with inequalities and we’re multiplying or dividing by a negative, we have to flip the sign. We’ve just turned this less than symbol into a greater than symbol.
We can then move on to multiply negative one times negative 𝑥, which would equal 𝑥. We have to distribute this negative one to our second inequality symbol and flip that sign as well. And finally we can distribute negative one to the negative five, which leaves us with five.
If I want to read this inequality, I would read it like this: 22 is greater than or equal to 𝑥, which is greater than or equal to five. On a graph, it would look something like this: 𝑥 has to be greater than or equal to five but less than or equal to 22. That in interval notation looks like this: bracket five comma 22 bracket.