https://file.aiquickdraw.com/mathbot/file/17274324082213s7kxa9b.jpg

The inequality to solve is:

\[
\frac{4x + 5}{x + 2} - 3 \leq 0
\]

Let's go step by step:

### Step 1: Simplify the inequality
First, bring the terms to a common denominator:
\[
\frac{4x + 5}{x + 2} - \frac{3(x+2)}{x+2} \leq 0
\]
This simplifies to:
\[
\frac{4x + 5 - 3(x + 2)}{x + 2} \leq 0
\]
Simplify the numerator:
\[
4x + 5 - 3x - 6 = x - 1
\]
So the inequality becomes:
\[
\frac{x - 1}{x + 2} \leq 0
\]

### Step 2: Find the critical points
The critical points are found by setting the numerator and denominator equal to zero:
- \( x - 1 = 0 \) gives \( x = 1 \)
- \( x + 2 = 0 \) gives \( x = -2 \) (this value is undefined for the rational expression).

### Step 3: Test intervals
Now we test the sign of the expression \( \frac{x - 1}{x + 2} \) in the intervals defined by these points: \( (-\infty, -2) \), \( (-2, 1) \), and \( (1, \infty) \).

- For \( x \in (-\infty, -2) \), choose \( x = -3 \):
  \[
  \frac{-3 - 1}{-3 + 2} = \frac{-4}{-1} = 4 > 0
  \]
  So the expression is positive here.

- For \( x \in (-2, 1) \), choose \( x = 0 \):
  \[
  \frac{0 - 1}{0 + 2} = \frac{-1}{2} = -\frac{1}{2} < 0
  \]
  So the expression is negative here.

- For \( x \in (1, \infty) \), choose \( x = 2 \):
  \[
  \frac{2 - 1}{2 + 2} = \frac{1}{4} > 0
  \]
  So the expression is positive here.

### Step 4: Solution
We want the expression to be less than or equal to zero, so we take the intervals where the expression is negative or zero.

- The expression is negative in \( (-2, 1) \).
- At \( x = 1 \), the expression is zero (since the numerator becomes zero), so we include \( x = 1 \).
- At \( x = -2 \), the expression is undefined (so we exclude this point).

Thus, the solution is:
\[
(-2, 1]
\]

### Conclusion:
The correct answer is the first option: \( (-2, 1] \).

Would you like any more details or have further questions?

Here are 5 related questions to explore further:
1. How do you solve rational inequalities with more complex numerators and denominators?
2. What happens if a rational inequality has no real solutions?
3. How do vertical asymptotes affect the solution of inequalities?
4. How can you solve inequalities involving absolute values?
5. What methods can be used to graph rational inequalities?

**Tip:** Always test intervals around critical points to ensure you capture the correct sign behavior.

Solve the inequality: \(\frac{4x + 5}{x + 2} - 3 \leq 0\)

Detailed solution for the rational inequality \(\frac{4x + 5}{x + 2} - 3 \leq 0\). This problem involves simplifying the rational expression, identifying critical points, and testing intervals for sign changes. The solution is suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation