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Mathematics 50 Online
OpenStudy (anonymous):

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

OpenStudy (shadowfiend):

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.

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