To solve the inequality
[tex]\[
\frac{x-5}{-x-1} \geq 0,
\][/tex]
we will go through several steps to determine the solution set. Here's a detailed step-by-step solution:
### Step 1: Determine the critical points
Critical points occur where the numerator or the denominator is equal to zero, as these are points where the expression can change sign or become undefined.
1. Numerator [tex]\( x - 5 = 0 \)[/tex]:
[tex]\[
x = 5
\][/tex]
2. Denominator [tex]\( -x - 1 = 0 \)[/tex]:
[tex]\[
-x - 1 = 0 \Rightarrow -x = 1 \Rightarrow x = -1
\][/tex]
These critical points divide the number line into intervals that we must test.
### Step 2: Identify the intervals
The critical points [tex]\( x = 5 \)[/tex] and [tex]\( x = -1 \)[/tex] divide the real line into three intervals:
1. [tex]\((-\infty, -1)\)[/tex]
2. [tex]\((-1, 5)\)[/tex]
3. [tex]\((5, \infty)\)[/tex]
### Step 3: Test the intervals
We need to determine the sign of the expression [tex]\(\frac{x-5}{-x-1}\)[/tex] in each interval:
1. For [tex]\( x \in (-\infty, -1)\)[/tex]:
Choose [tex]\( x = -2 \)[/tex]:
[tex]\[
\frac{-2-5}{-(-2)-1} = \frac{-7}{2-1} = \frac{-7}{1} = -7
\][/tex]
The expression is negative.
2. For [tex]\( x \in (-1, 5)\)[/tex]:
Choose [tex]\( x = 0 \)[/tex]:
[tex]\[
\frac{0-5}{-0-1} = \frac{-5}{-1} = 5
\][/tex]
The expression is positive.
3. For [tex]\( x \in (5, \infty)\)[/tex]:
Choose [tex]\( x = 6 \)[/tex]:
[tex]\[
\frac{6-5}{-6-1} = \frac{1}{-7} = -\frac{1}{7}
\][/tex]
The expression is negative.
### Step 4: Include the boundaries
We must also consider the boundaries of each interval:
- At [tex]\( x = -1 \)[/tex], the denominator [tex]\(-x - 1\)[/tex] is zero, making the fraction undefined. Thus, [tex]\( x = -1 \)[/tex] cannot be included in the solution set.
- At [tex]\( x = 5 \)[/tex], the numerator [tex]\( x - 5 \)[/tex] is zero, and since zero divided by a non-zero number is zero,
[tex]\[
\frac{0}{-6} = 0,
\][/tex]
which satisfies [tex]\( \geq 0 \)[/tex]. Thus, [tex]\( x = 5 \)[/tex] can be included in the solution set.
### Step 5: Combine the results
From our interval testing, the expression is non-negative (i.e., [tex]\(\geq 0\)[/tex]) in the interval [tex]\((-1, 5]\)[/tex].
### Conclusion:
The interval where the inequality is satisfied is:
[tex]\[
(-1, 5]
\][/tex]
### Answer in interval notation:
[tex]\[
(-1, 5]
\][/tex]
