First slide

Trigonometric Identities

Question

If $\log_{1 / 2} (4 - x) \geq \log_{1 / 2} 2 - \log_{1 / 2} (x - 1),$ then x belongs to

Moderate

A

$(1, 2]$
B

$[3, 4)$
C

$(1, 3]$
D

$[1, 4)$

Solution

$\begin{array}{ll} \log_{1 / 2} (4 - x) \geq \log_{1 / 2} 2 - \log_{1 / 2} (x - 1) \\ or \log_{1 / 2} (4 - x) (x - 1) \geq \log_{1 / 2} 2 \\ or (4 - x) (x - 1) \leq 2 \\ or x^{2} - 5 x + 6 \geq 0 \\ or (x - 3) (x - 2) \geq 0 \\ or x \geq 3 or x \leq 2 \\ But x \in (1, 4) \\ \Rightarrow x \in (1, 2] \cup [3, 4) \end{array}$

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