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!