Number of integers in the solution set of | x - 1 | log 2 ( 4 - x ) | x - 1 | log 2 ( 1 + x ) are

Question

Number of integers in the solution set of $| x - 1 |^{\log_{2} (4 - x)}$ $<$ $| x - 1 |^{\log_{2} (1 + x)}$ are ________.

Smart Achievers · Accepted Answer

(2)

$4 - x > 0, 1 + x > 0 \Rightarrow x \in (- 1, 4)$ $\dots (i)$

Let $| x - 1 |$ $< 1 \Rightarrow x \in (0, 2)$ $\dots (i i)$

The inequality implies $\log_{2} (4 - x) > \log_{2} (1 + x)$

$\Rightarrow$ $4 - x > 1 + x$

$\Rightarrow$ $x < \frac{3}{2}$ $\dots (i i i)$

From (i), (ii) and (iii) $\Rightarrow$ $x \in (0, \frac{3}{2})$

Let $| x - 1 |$ $> 1 \Rightarrow x \in (- \infty, 0) \cup (2, \infty)$ $\dots (i v)$

The inequality implies $\log_{2} (4 - x) < \log_{2} (1 + x)$

$\Rightarrow$ $4 - x < 1 + x \Rightarrow x > \frac{3}{2}$ $\dots (v)$

(i), (iv), (v) $\Rightarrow x \in (2, 4)$

Finally, we have $x \in (0, \frac{3}{2}) \cup (2, 4)$

$∴$ Number of integers = 2

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