Inequalities: reading a number line

You may see a question like this on the GREs

First, ignore the options. They are designed to confuse you! Focus on the number line and write down the inequality the number line represents. (If you need a refresher with number lines, just take a minute to read over the Number Lines post under Tools You Can Use.)

Starting from the left, we see that -3 is involved. Can x be worth -3? No, because there is a donut hole over it. So we know part of the inequality will be -3 < x…but we are not done yet.

Keep going down the number line, and we see that x can be worth up to 2. There is a big dot on the 2, so we also include 2, giving us -3 < x < 2. Great!

Now do we see that in our list of options?

We can immediately eliminate option b and c, since they only have one inequality and we know from the number line that we will need two.

Option e is so tempting, but it has a < when we need a <. We double-check to make sure it should be a < and since there is a donut hole over the 3, we confirm we do need it to be <. We eliminate Option e.

Option a we can eliminate because we know that x can be worth at least some negative values, and option a only gives positive values for x.

That leaves option d: -9 < 3x < 6. Is that the same as what we got: -3 < x < 2? We know we need to get x alone to find out.

Since in -9 < 3x < 6 the 3 is being multiplied by the x, we need to divide everywhere by 3.

That matches what we got: -3 < x < 2. The answer is option d. Great work!