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Mathematics 95 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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