Solve Inequality: x^2-4|-3|<1 | UofT Calculus

[AnthonyJohnAre]

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

 

[Ray Vickson]

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?

 

[berkeman]

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

Please check your PMs. You should show your attempt at solving the equation as part of your first post asking for help.