Question

The set of values of $x$ satisfying simultaneously the inequalities $\frac{\sqrt{(x-8)(2-x)}}{\log _{0.3}\left(\frac{10}{7}\left(\log _2 5-1\right)\right)} \geq 0$ and $2^{x-3}-31>0$ is

• A. a unit set
• B. an empty set
• C. an infinite set
• D. a set consisting of exactly two elements.

Answer 1

Answer: (A)

Note that $ D ^{ r }<0$, hence given inequality (1) is true only if $N ^{ r } \geq 0$
i.e. $(x-8)(x-2)<0$ and $2^{x-3}>31$
i.e. $ 2 \leq x \leq 8$ and $2^{x-3} \geq 31$
only $x=8$ satisfies both the inequality