Answer:
Explanation:
To solve the rational inequality (x + 5) / (x + 8) > 0, we can follow these steps:
1. Find the critical points of the rational expression, which are the values of x that make the numerator or denominator equal to zero. In this case, the critical point is when x + 8 = 0, so x = -8.
2. Test the intervals defined by these critical points. We will choose test values in each interval and determine whether the rational expression is positive or negative in each interval.
a. Choose a test value less than -8, say x = -10:
(x + 5) / (x + 8) = (-10 + 5) / (-10 + 8) = (-5) / (-2) = 5/2, which is positive.
b. Choose a test value between -8 and 0, say x = -4:
(x + 5) / (x + 8) = (-4 + 5) / (-4 + 8) = (1) / (4) = 1/4, which is positive.
c. Choose a test value greater than 0, say x = 2:
(x + 5) / (x + 8) = (2 + 5) / (2 + 8) = (7) / (10) = 7/10, which is positive.
3. Based on the test values, we can see that the rational expression is positive for all values of x in the intervals:
a. (-∞, -8)
b. (-8, 0)
c. (0, ∞)
Now, we can express the solution set in interval notation:
Solution set: (-∞, -8) U (-8, 0) U (0, ∞)
This is the set of values of x for which the rational expression (x + 5) / (x + 8) is greater than 0.