Number of integers satisfying the inequality ${log}_{\frac{1}{2}} | x - 3 | > - 1$ is

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Solution

${log}_{\frac{1}{2}} | x - 3 |$ is meaningful when $x \neq 3$
${log}_{\frac{1}{2}} | x - 3 | > - 1$
$\Rightarrow - {log}_{2} | x - 3 | > - 1 [∵ {log}_{1 / a} b = - {log}_{a} b]$
$\Rightarrow {log}_{2} | x - 3 | < 1$ , multiplied both sides with $- 1$ , so inequality changes
$\Rightarrow | x - 3 | < 2$
$\Rightarrow - 2 < x - 3 < 2$
$\Rightarrow 1 < x < 5$
$\Rightarrow x \in (1, 5) - {3}$
Therefore, integral values are $2, 4$ that is $2$ integral values.
Ans: 2

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