https://file.aiquickdraw.com/mathbot/file/1726576414491qma270om.jpg

The inequality shown in the image is:

\[
\frac{x - 5}{x + 5} < \frac{x - 9}{x + 6}
\]

To solve this inequality, we can follow these steps:

### Step 1: Clear the denominators by multiplying both sides by \((x + 5)(x + 6)\). 
However, we must be cautious about the values of \(x\) that make the denominators zero, as the inequality is undefined for those values. So:

- \(x + 5 \neq 0 \)  ⟶ \(x \neq -5\)
- \(x + 6 \neq 0 \)  ⟶ \(x \neq -6\)

### Step 2: Multiply both sides by \((x + 5)(x + 6)\), keeping track of the signs of the factors. This gives:

\[
(x - 5)(x + 6) < (x - 9)(x + 5)
\]

### Step 3: Expand both sides.

For the left-hand side:
\[
(x - 5)(x + 6) = x^2 + 6x - 5x - 30 = x^2 + x - 30
\]

For the right-hand side:
\[
(x - 9)(x + 5) = x^2 + 5x - 9x - 45 = x^2 - 4x - 45
\]

### Step 4: Subtract the terms from the right-hand side to bring everything to one side of the inequality:

\[
x^2 + x - 30 - (x^2 - 4x - 45) < 0
\]

Simplify:
\[
x^2 + x - 30 - x^2 + 4x + 45 < 0
\]
\[
5x + 15 < 0
\]

### Step 5: Solve the inequality:

\[
5x < -15
\]
\[
x < -3
\]

### Step 6: Consider the restrictions \(x \neq -5\) and \(x \neq -6\).

Thus, the solution is:

\[
x \in (-\infty, -6) \cup (-6, -5) \cup (-5, -3)
\]

This is the solution to the inequality.

Do you have any questions or would you like more details on this solution?

Here are 5 related questions:
1. What happens if we multiply or divide an inequality by a negative number?
2. How can we check if our solution to a rational inequality is correct?
3. What are the conditions for solving rational inequalities involving multiple fractions?
4. How do we handle cases where the inequality involves absolute values?
5. How can we graph the solution to a rational inequality?

**Tip:** When solving inequalities with rational expressions, always pay attention to the values that make the denominator zero, as these create restrictions on the solution.

Solve the inequality: (x - 5)/(x + 5) < (x - 9)/(x + 6)

Learn how to solve the rational inequality (x - 5)/(x + 5) < (x - 9)/(x + 6) with a clear step-by-step guide. This solution involves multiplying rational expressions, expanding binomials, and considering restrictions on the variable. Ideal for students in Grades 9-12.

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