Which graph shows the solution set of (image)

Choice C



Explanation:

Note how all of the answer choices have the same boundaries:
x = -11, x = -4, x = 2.

The only difference between the choices is the difference in shading and the use of open or closed circles.

Another thing to note: the x+4 in the denominator. If x = -4, then x+4 = -4+4 = 0 which leads to a division by zero error. If we plug in x = -4, then we get trouble. Therefore, x = -4 is not part of the domain. We use an open circle at x = -4 to represent this. 

We can plug in x = -11 and x = 2 just fine, so we use closed circles at these endpoints.



The fact that x = -4 has an open circle, while x = -11 and x = 2 have closed circles, means we know the answer is A or C. 

Let's try plugging in some value smaller than -11. Plug in x = -12
f(x) = (x^2+9x-22)/(x+4)
f(-12) = ((-12)^2+9(-12)-22)/(-12+4)
f(-12) = -1.75
The result -1.75 is negative, so f(x) >= 0 is false when x < -11

We can eliminate choice A because of the shaded region for x < -11, which is false. 

The only thing left is choice C. Pick any point you want from the red shaded region. Whatever point you pick from graph C can be plugged into f(x) to make f(x) >= 0 true.

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