How do you solve |3x-2|

Question

How do you solve |3x-2|<2 ? PLEASE HELP!

shadowfiend · Answer

So in an absolute value situation like this, what we really mean is that these must be true:

$$-2 < 3x-2 < 2$$

This is because anything between -2 and 0, when you take the absolute value of it, will be < 2. Obviously anything between 0 and 2, when you take the absolute value of it, will also be < 2.

So, we can solve this by solving the two sides simultaneously. First we add two to both sides:

$$0 < 3x < 4$$

Then we divide both sides by 3:

$$0 < x < \frac{4}{3}$$

Let's check the end points. 0 - 2 is -2, and its absolute value is 2. Note that our interval doesn't *include* 2, but neither does the solution *include* 0, it's just the closest endpoint.

Then we check $\frac{4}{3}$ -- $3 \cdot \frac{4}{3} - 2 = 4 - 2 = 2$. Again, we are at a proper endpoint. So the solution looks to be correct.