The number of real solutions to the equation $3x - 7 = [x^2 - 3x + 2]$ is ____. $[x]$ denotes greatest integer $\leq x$.
Step-by-Step Solution
Key Concept: Apply the fundamental property of the greatest integer function $[t] ≤ t < [t] + 1$ to the equation $3x - 7 = [x^2 - 3x + 2]$, yielding the compound inequality $3x - 7 ≤ x^2 - 3x + 2 < 3x - 6$, which must be solved simultaneously to find integer or rational values of $x$ that satisfy both constraints.
Use the property $[t] \le t < [t] + 1$ for real $t$. Apply this to the given inequality $3x - 7 \le x^2 - 3x + 2 < (3x-7) + 1$ and solve both inequalities simultaneously to find $x \in \{\frac{7}{3}, \frac{8}{3}, \frac{9}{3}, \frac{10}{3}, \frac{11}{3}\}$.
Correct Answer: 5