https://file.aiquickdraw.com/mathbot/file/1726975273802721iz4dx.jpg

The inequality you uploaded is:

\[
\frac{8}{x - 3} < \frac{2}{x - 8}
\]

### Step-by-step solution:

1. **Identify Restrictions**: The denominators cannot be zero, so we exclude \(x = 3\) and \(x = 8\).

2. **Cross Multiply**: Since both sides of the inequality are fractions, cross-multiplication is valid as long as we account for the signs of the denominators.

   \[
   8(x - 8) < 2(x - 3)
   \]

3. **Simplify**:
   - Expand both sides:
   
     \[
     8x - 64 < 2x - 6
     \]

4. **Move terms to one side**:
   \[
   8x - 2x < -6 + 64
   \]
   Simplifying this gives:
   \[
   6x < 58
   \]

5. **Solve for \(x\)**:
   \[
   x < \frac{58}{6} = \frac{29}{3}
   \]

6. **Consider the domain restrictions**:
   - \(x \neq 3\)
   - \(x \neq 8\)

Since we cross-multiplied, we need to check the sign changes. This inequality will be valid for \(x\) in the following intervals:

- \(x < 3\) 
- \(3 < x < 8\)
- \(x > 8\)

Given the solution \(x < \frac{29}{3}\), we must verify within the intervals that exclude the points where the denominators are undefined.

### Final Solution:

- \(x < 3\) and \(3 < x < \frac{29}{3}\)

Thus, the solution set is:

\[
x \in (-\infty, 3) \cup (3, \frac{29}{3})
\]

Would you like further details or clarifications on this?

### Related Questions:
1. How do we handle inequalities when multiplying both sides by variables?
2. Why is it necessary to exclude certain values for \(x\)?
3. What changes when cross-multiplying terms in an inequality with variables?
4. How would the solution change if this were an equality instead of an inequality?
5. What is the difference between solving a rational equation and a rational inequality?

### Tip:
Always check for extraneous solutions when dealing with inequalities involving fractions—some solutions might cause the denominators to be zero!

Find all x with (8 / (x - 3)) < (2 / (x - 8))

This tutorial walks through solving the rational inequality 8 / (x - 3) < 2 / (x - 8). Learn how to cross-multiply and simplify inequalities involving fractions while considering domain restrictions for x. Suitable for Grades 9-12 students.

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