Answer:
B. -2 < x < 4
Explanation:
The function is (x +2)/(x-4) and we are asked to determine the x interval when the inequality (x +2)/(x-4) < 0 is satisfied
1. Find the critical points
The critical points are
x = - 2 when the numerator becomes -2 + 2 = 0, and f(x) = 0
and
x = 4 when the denominator becomes 4 - 4 = 0 and (x + 2)/0 is undefined
So critical points are -2 and 4
2. Use these critical points to divide the number line into intervals
The intervals are -∞ < x < -2, -2 < x < 4 and 4 < x < ∞
these can be re-written as x < -2, -2 < x < -4 and x > 4
3. For each of these intervals take a test point and see if it satisfies that inequality
(a) -∞ < x < -2 Take x = -3 which is less than -2
We get f(-3) = (-3 + 2)/(-3 -4) = -1/-7 which is + 1/7 and does not satisfy the (x+2)/(x-4) < 0 inequality.
So we can discard this interval
(b) For interval -2 < x < 4, Let's take x = 0
(0+2)/(0-4) = 2/(-4) = -1/2 = -0.5 which does satisfy the inequality
(x+2)/(x-4) <0, so this is a valid interval
(c) For interval x > 4, take x = 5
(x+2)/(x-4 ) = (5 + 2)/(5-4) = 7/1 = 7 and does not satisfy (x+2)/(x-4 ) < 0
So the interval which satisfies the inequality is -2 < x < 4