To solve the inequality [tex]\(\frac{(x+5)(x+1)}{x-4} \geq 0\)[/tex], we need to find the values of [tex]\(x\)[/tex] for which the expression is non-negative. We'll do this by following these steps:
1. Identify Critical Points: These are the values of [tex]\(x\)[/tex] where the numerator or denominator equals zero.
2. Determine Sign Changes: We need to examine the sign of the expression in each interval created by these critical points.
3. Construct the Solution: Based on the signs, we can determine the intervals where the expression is non-negative.
### 1. Identify Critical Points
Numerator:
[tex]\[
(x + 5)(x + 1) = 0
\][/tex]
This gives us two solutions:
[tex]\[
x + 5 = 0 \quad \Rightarrow \quad x = -5
\][/tex]
[tex]\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\][/tex]
Denominator:
[tex]\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\][/tex]
### 2. Determine Sign Changes
Now, we have the critical points: [tex]\(x = -5\)[/tex], [tex]\(x = -1\)[/tex], and [tex]\(x = 4\)[/tex]. These points divide the number line into four intervals:
- [tex]\( (-\infty, -5) \)[/tex]
- [tex]\( (-5, -1) \)[/tex]
- [tex]\( (-1, 4) \)[/tex]
- [tex]\( (4, \infty) \)[/tex]
We will test a point within each interval to determine whether the expression is positive or negative in that interval.
- For [tex]\(x < -5\)[/tex], pick [tex]\(x = -6\)[/tex]:
[tex]\[
\frac{(-6 + 5)(-6 + 1)}{-6 - 4} = \frac{(-1)(-5)}{-10} = \frac{5}{-10} = -0.5 \quad \text{(negative)}
\][/tex]
- For [tex]\(-5 < x < -1\)[/tex], pick [tex]\(x = -3\)[/tex]:
[tex]\[
\frac{(-3 + 5)(-3 + 1)}{-3 - 4} = \frac{(2)(-2)}{-7} = \frac{-4}{-7} = \frac{4}{7} \quad \text{(positive)}
\][/tex]
- For [tex]\(-1 < x < 4\)[/tex], pick [tex]\(x = 0\)[/tex]:
[tex]\[
\frac{(0 + 5)(0 + 1)}{0 - 4} = \frac{(5)(1)}{-4} = \frac{5}{-4} = -1.25 \quad \text{(negative)}
\][/tex]
- For [tex]\(x > 4\)[/tex], pick [tex]\(x = 5\)[/tex]:
[tex]\[
\frac{(5 + 5)(5 + 1)}{5 - 4} = \frac{(10)(6)}{1} = 60 \quad \text{(positive)}
\][/tex]
### 3. Construct the Solution
From our test points, we have determined that the expression is:
- Negative for [tex]\(x < -5\)[/tex]
- Positive for [tex]\(-5 < x < -1\)[/tex]
- Negative for [tex]\(-1 < x < 4\)[/tex]
- Positive for [tex]\(x > 4\)[/tex]
Finally, since we need [tex]\(\frac{(x+5)(x+1)}{x-4} \geq 0\)[/tex], we include the intervals where the expression is positive and also check if the critical points themselves are included:
- [tex]\(x = -5\)[/tex] and [tex]\(x = -1\)[/tex] make the numerator zero, thus satisfying the inequality.
- [tex]\(x = 4\)[/tex] makes the denominator zero, which must be excluded.
Therefore, the solution to the inequality is:
[tex]\[
-5 \leq x \leq -1
\][/tex]
