The set of real values of x for which log0.2x+2x≤1 is

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The set of real values of x for which

l o g_{0.2} \frac{x + 2}{x} \leq 1

is

Options:

A . (−∞,−52]∪(0,+∞)

B . [52,+∞)

C . (−∞,−2)∪(0,+∞)

D . None of these

Answer: Option A
:
A

l o g_{0.2} \frac{x + 2}{x} \leq 1

....(i)
For log to be defined,

\frac{x + 2}{x} > 0 \Rightarrow x > 0

or
x < -2
Now from(i),

l o g_{0.2} \frac{x + 2}{x} \leq l o g_{0.2} 0.2

\Rightarrow \frac{x + 2}{x} \geq 0.2

.....(ii)
Case (i) x > 0
From (ii),

x + 2 \geq 0.2 x

\Rightarrow 0.8 x \geq - 2

\Rightarrow x \geq - \frac{5}{2} .

Case (ii) x < -2
From(ii),

x + 2 \leq 0.2 x \Rightarrow 0.8 x \leq - 2 \Rightarrow x \leq - \frac{5}{2}

\Rightarrow x ϵ (- \infty, - \frac{5}{2}] \cup (0, \infty)

.

The Set Of Real Values Of X For Which Log0.2x+2x≤1 Is

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