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The inequality to solve is:

\[
\frac{8}{x - 1} < \frac{3}{x - 8}
\]

We need to solve this inequality and express the solution in the form as requested, such as \( x = a \) or \( (x \leq a) \lor (x \geq b) \), etc.

### Steps to Solve the Inequality:

1. **Cross-multiply**, ensuring that we account for the change in the inequality direction when multiplying by negative numbers. This gives us:

\[
8(x - 8) < 3(x - 1)
\]

2. **Expand both sides**:

\[
8x - 64 < 3x - 3
\]

3. **Move all terms involving \(x\) to one side and constants to the other side**:

\[
8x - 3x < -3 + 64
\]

\[
5x < 61
\]

4. **Solve for \(x\)**:

\[
x < \frac{61}{5}
\]

This gives us a critical value, but we also need to account for any restrictions or undefined points.

### Undefined Points:

- \( x = 1 \) makes \( \frac{8}{x - 1} \) undefined.
- \( x = 8 \) makes \( \frac{3}{x - 8} \) undefined.

### Final Solution:

Considering the undefined points and the inequality \( x < \frac{61}{5} \), we can express the solution as follows:

\[
x \in (-\infty, 1) \cup (1, 8) \cup (8, \frac{61}{5})
\]

This means that \(x\) cannot be equal to 1 or 8, but it can take values in the intervals \( (-\infty, 1) \), \( (1, 8) \), and \( (8, \frac{61}{5}) \).

Would you like more details on this solution or have any other questions?

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**5 Related Questions:**
1. How do you determine when to reverse the inequality sign in a problem like this?
2. What happens if we multiply or divide both sides by a negative number?
3. Could we express the solution using set notation or interval notation?
4. How do we know when a fraction becomes undefined in an inequality?
5. What other types of inequalities involve similar steps?

**Tip:** Always remember to check for undefined points in rational inequalities by setting the denominator equal to zero and solving for the variable!

Find all x with 8/(x - 1) < 3/(x - 8). Provide your answer in the form x = a or (x <= a) ∨ (x >= b), etc.

Learn how to solve the rational inequality 8/(x - 1) < 3/(x - 8) with step-by-step instructions. This algebra problem involves rational expressions and requires careful handling of undefined points. Suitable for high school students in Grades 10-12.

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