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Absolute Value Function

Let A= x:2<|x...

Question

Let $A = {x : 2 < | x | \leq 5 and x \in Z}$ and $B$ be the set of values of $a$ for which the equation $∣ ∣ | x - 1 | + a ∣ ∣ = 4$ can have real solutions. Then $n (A \cap B)$ is

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Solution

$A = {- 5, - 4, - 3, 3, 4, 5}$

$∣ ∣ | x - 1 | + a ∣ ∣ = 4$
$\Rightarrow | x - 1 | + a = \pm 4$
$\Rightarrow | x - 1 | = (- a \pm 4)$
For real solutions, we must have
$4 - a \geq 0$ or $- 4 - a \geq 0$
$\Rightarrow a \leq 4$ or $a \leq - 4$
$\Rightarrow a \in (- \infty, 4]$

$∴ A \cap B = {- 5, - 4, - 3, 3, 4}$
$\Rightarrow n (A \cap B) = 5$

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