The number of real numbers $x$ such that $\frac{x}{x+4} = \frac{5[x]-7}{7[x]-5}$ is ____. $[x]$ denotes greatest integer $\leq x$.
Step-by-Step Solution
Key Concept: The floor function constraint $m \le x < m+1$ combined with the functional equation determines the possible integer values of the floor.
Let $[x] = m$, then $\frac{x}{x+4} = \frac{5m-7}{7m-5}$. Rearranging gives $x = \frac{10m-14}{m+1}$. Since $m \le x < m+1$, we have $m \le \frac{10m-14}{m+1} < m+1$. Solving these inequalities yields possible integral values of $m$ as $2, 6, 7$.
Correct Answer: 3