Question

The set of values of $x$ satisfying simultaneously the inequalities $\frac{\sqrt{(x-8)(2-x)}}{{\log }_{0.3}\left(\frac{10}{7}\left({\log }_{2}5-1\right)\right)}\ge 0$ 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 $N^r\ge 0$
i.e. $(x-8)(x-2)<0$ and $2^{x-3}>31$
i.e. $2\le x\le 8$ and $2^{x-3}\ge 31$
only $x=8$ satisfies both the inequality