Answer to Question #312169 in Real Analysis for Jyo

Question

Find all real numbers x such that

1<x^2<4

Answer 1

1 < x² < 4

=> x² > 1 and x² <4

=> x² - 1 > 0 and x² - 4 < 0

=> (x-1)(x+1) > 0 and (x-2)(x+2) < 0

For inequality (x-1)(x+1) > 0

By wavy curve rule

x < -1 or x > 1

=> x $\in (-∞, -1) \bigcup(1,∞)$

For inequality (x-2)(x+2) < 0

By wavy curve rule

-2 < x < 2 => x $\in (-2,2)$

So the solution is intersection of the sets $(-∞, -1) \bigcup(1,∞)$ and (-2,2)

So the real numbers x$\in$ (-2,-1) U (1,2) satisfy the inequality 1< x²<4