First slide

Operations on Sets

Question

The number of integers satisfying $\log_{\frac{1}{x}} (\frac{2 (x - 2)}{(x + 1) (x - 5)}) \geq 1 is$

Moderate

A

0
B

1
C

2
D

3

Solution

$\begin{array}{l} Case I: 1 / x > 1 or 0 < x < 1 \\ ∴ \log_{\frac{1}{x}} (\frac{2 (x - 2)}{(x + 1) (x - 5)}) \geq 1 \\ \Rightarrow \frac{2 (x - 2)}{(x + 1) (x - 5)} \geq \frac{1}{x} \\ \Rightarrow \frac{2 (x - 2)}{(x + 1) (x - 5)} - \frac{1}{x} \geq 0 \\ \Rightarrow \frac{2 x (x - 2) - (x + 1) (x - 5)}{x (x + 1) (x - 5)} \geq 0 \\ \Rightarrow \frac{x^{2} + 5}{x (x + 1) (x - 5)} \geq 0 \Rightarrow x (x + 1) (x - 5) > 0 \sin c e x^{2} + 5 > 0 f o r a l l x \in R \\ \Rightarrow x \in (- 1, 0) \cup (5, \infty) \end{array}$
Hence, no solution in this case.
$\begin{array}{l} Case II: 0 < \frac{1}{x} < 1 or x > 1 \overset{}{\to (1)} \\ ∴ 0 < x < 5 \overset{}{\to (2)} \\ Also \frac{2 (x - 2)}{(x + 1) (x - 5)} > 0 \Rightarrow x \in (- 1, 2) \cup (5, \infty) \overset{}{\to} (3) \\ f r o m (1), (2), (3) w e c a n t a k e \\ x \in (1, 2) \end{array}$

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