To solve the inequality [tex]\(\frac{x-6}{x+5} \geq \frac{x+7}{x+3}\)[/tex], let's follow a step-by-step approach.
1. Start with the given inequality:
[tex]\[
\frac{x-6}{x+5} \geq \frac{x+7}{x+3}
\][/tex]
2. Combine the fractions onto one side:
[tex]\[
\frac{x-6}{x+5} - \frac{x+7}{x+3} \geq 0
\][/tex]
3. Subtract the fractions to create a single rational expression:
The easiest way to combine these fractions is to bring both terms to a common denominator:
[tex]\[
\frac{(x-6)(x+3) - (x+7)(x+5)}{(x+5)(x+3)} \geq 0
\][/tex]
4. Simplify the numerator:
[tex]\[
(x-6)(x+3) - (x+7)(x+5) = x^2 + 3x - 6x - 18 - (x^2 + 5x + 7x + 35)
\][/tex]
[tex]\[
= x^2 - 3x - 18 - x^2 - 12x - 35
\][/tex]
[tex]\[
= -15x - 53
\][/tex]
This reduces the combined fraction to:
[tex]\[
\frac{-15x - 53}{(x+5)(x+3)} \geq 0
\][/tex]
5. Analyze the key points:
[tex]\[
-15x - 53 \geq 0 \quad \Rightarrow \quad -15x \geq 53 \quad \Rightarrow \quad x \leq -\frac{53}{15}
\][/tex]
Also, we must consider the points where the denominator is zero since the rational function is undefined at these points:
[tex]\[
(x + 5)(x + 3) = 0 \quad \Rightarrow \quad x = -5 \quad \text{or} \quad x = -3
\][/tex]
6. Interval testing:
Evaluate the expression in the intervals determined by these critical points to determine where the expression [tex]\(\frac{-15x - 53}{(x+5)(x+3)}\)[/tex] is positive:
- When [tex]\(x < -5\)[/tex]
- When [tex]\(-5 < x < -3\)[/tex]
- When [tex]\(-3 < x \leq -\frac{53}{15}\)[/tex]
7. Constructing the solution:
Based on the detailed numerical calculation and critical points, the expression [tex]\(\frac{-15x - 53}{(x+5)(x+3)} \geq 0\)[/tex] holds true in the intervals:
[tex]\[
(-\infty, -5) \text{ and } (-3, -\frac{53}{15}]
\][/tex]
Therefore, the inequality is equivalent to:
[tex]\[
x \in (-\infty, -5) \cup (-3, -\frac{53}{15}]
\][/tex]
