To solve the inequality \(\frac{3x-1}{x+2} \leq 5\), follow these steps:

### Step 1: Move everything to one side of the inequality
Subtract 5 from both sides to set the inequality to zero:
\[
\frac{3x - 1}{x + 2} - 5 \leq 0
\]
Rewrite the right-hand side as a single fraction:
\[
\frac{3x - 1}{x + 2} - 5 = \frac{3x - 1 - 5(x + 2)}{x + 2} = \frac{3x - 1 - 5x - 10}{x + 2}
\]
Simplify the numerator:
\[
= \frac{-2x - 11}{x + 2} \leq 0
\]

### Step 2: Solve the inequality
Now solve the inequality:
\[
\frac{-2x - 11}{x + 2} \leq 0
\]

The critical points occur where the numerator and denominator are zero:
- The numerator \(-2x - 11 = 0\) gives \(x = -\frac{11}{2}\).
- The denominator \(x + 2 = 0\) gives \(x = -2\).

### Step 3: Test intervals
These critical points divide the real line into intervals. We need to test each interval to determine where the expression is less than or equal to zero:

1. **Interval \( (-\infty, -\frac{11}{2}) \)**: Choose \(x = -6\), test in the inequality \(\frac{-2x - 11}{x + 2} \leq 0\).
   \[
   \frac{-2(-6) - 11}{-6 + 2} = \frac{12 - 11}{-4} = \frac{1}{-4} < 0
   \]
   This interval satisfies the inequality.

2. **Interval \((- \frac{11}{2}, -2)\)**: Choose \(x = -3\), test in the inequality:
   \[
   \frac{-2(-3) - 11}{-3 + 2} = \frac{6 - 11}{-1} = \frac{-5}{-1} > 0
   \]
   This interval does not satisfy the inequality.

3. **Interval \((-2, \infty)\)**: Choose \(x = 0\), test in the inequality:
   \[
   \frac{-2(0) - 11}{0 + 2} = \frac{-11}{2} < 0
   \]
   This interval satisfies the inequality.

### Step 4: Include or exclude critical points
- At \(x = -\frac{11}{2}\), \(\frac{-2x - 11}{x + 2} = 0\), so it satisfies the inequality.
- At \(x = -2\), the denominator is zero, making the expression undefined.

### Final Solution:
The solution is:
\[
x \in (-\infty, -2) \cup \left[-\frac{11}{2}, \infty\right)
\]

Would you like more details or explanations?

Here are 5 related questions:
1. How do you solve rational inequalities involving higher-degree polynomials?
2. What is the process for solving compound inequalities?
3. How do you graph the solution set for a rational inequality?
4. Can you explain why the expression is undefined at certain points?
5. What is the difference between solving equations and inequalities?

**Tip:** Always check for critical points where the denominator equals zero in rational inequalities!

Learn how to solve the rational inequality (3x-1)/(x+2) ≤ 5 with a detailed step-by-step solution. This algebra problem covers key concepts like critical points, interval testing, and solving rational inequalities. Suitable for students in Grades 9-11.

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