# Inequality problem

Hey everyone. I'm taking Calculus at UofT and I got a question in a problem set that kind of got me thinking, and well, I'm not sure if I'm doing it correctly. This isn't the exact question, but, how would you go about solving this inequality:
||(x^2)-4|-3|< 1

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Ray Vickson
Homework Helper
Dearly Missed
Hey everyone. I'm taking Calculus at UofT and I got a question in a problem set that kind of got me thinking, and well, I'm not sure if I'm doing it correctly. This isn't the exact question, but, how would you go about solving this inequality:
||(x^2)-4|-3|< 1
The easiest way is to first gain some insight into the problem by plotting a graph of the function f(x) =||x^2 - 4|-3|.

A systematic way is to split up x into cases: (1) x^2 ≤ 4; (2) x^2 > 4. In case (1) we have f(x) = |4-x^2-3| = |1-x^2| = |x^2-1|. So, for x^2 ≤ 4 we also want |x^2-1| < 1. What do these requirements say about x? In case (2) we have f(x) = |x^2 - 4 - 3| = |x^2-7|, so for x^2 > 4 we also need |x^2-7| < 1. What do these tell you about x?

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berkeman
Mentor
Hey everyone. I'm taking Calculus at UofT and I got a question in a problem set that kind of got me thinking, and well, I'm not sure if I'm doing it correctly. This isn't the exact question, but, how would you go about solving this inequality:
||(x^2)-4|-3|< 1