The number of integral solutions $x$ of $\log_{(x+7/2)}\!\left(\dfrac{x-7}{2x-3}\right)^2\geq 0$ is
Step-by-Step Solution
Key Concept: Domain: $x+\frac{7}{2}>0,\ x+\frac{7}{2}\neq 1,\ \frac{x-7}{2x-3}\neq 0$, giving domain $(-\frac{7}{2},\infty)\setminus\{-\frac{5}{2},\frac{3}{2},7\}$. Then split on whether base $>1$ or $0<\text{base}<1$.
Combining cases: $x\in(-\frac{5}{2},\frac{3}{2})\cup(\frac{3}{2},\frac{10}{3}]$. Integers: $-2,-1,0,1,2,3$. Count $=6$.
Correct Answer: 3