Find $x$ from $1 < | x | < 2$ and represent it on number line

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Solution

$1 < | x | \Rightarrow | x | > 1 \Rightarrow x > 1 o r < - 1$
$∴ x \in (- \infty, - 1) \cup (1, \infty)$ .....................(i)
also |x| < 2 $\Rightarrow$ $x < 2 o r x > - 2$
$∴$ x lies between $- 2 & 2$
$\Rightarrow x \in (- 2, 2)$ ...........(ii)
Combining the two results we get
$1 < | x | < 2 \Rightarrow - 2 < x < - 1 \cup 1 < x < 2$ i.e. $x \in (- 2, - 1) \cup (1, 2)$

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