The Solution:
Given:
[tex]\frac{x^2+4x-21}{x-5}\leq0[/tex]Required:
Find the critical points of the given rational inequality.
Recall:
The critical values are simply the zeros of both the numerator and the denominator.
Thus, the critical points are:
[tex]\begin{gathered} x-5=0 \\ x=5 \\ \\ solve\text{ the quadratic equation:} \\ x^2+4x-21=0 \\ x^2+7x-3x-21=0 \end{gathered}[/tex][tex]\begin{gathered} x(x+7)-3(x+7)=0 \\ (x-3)(x+7)=0 \\ x-3=0 \\ x=3 \\ or \\ x+7=0 \\ x=-7 \\ \end{gathered}[/tex]Thus, the critical points are:
[tex]x=5,\text{ }x=3\text{ or }x=-7[/tex]Therefore, the correct answer is [option C]