4x + 5 Over x + 4 is less than or equal to 9

We are given the inequality:

\[
\frac{4x + 5}{x + 4} \leq 9
\]

To solve this inequality, follow these steps:

### Step 1: Eliminate the denominator
Multiply both sides of the inequality by \(x + 4\), but keep in mind that this can change the direction of the inequality if \(x + 4\) is negative. We have two cases to consider:

#### Case 1: \(x + 4 > 0\) (i.e., \(x > -4\))

Multiplying by \(x + 4\), the inequality becomes:

\[
4x + 5 \leq 9(x + 4)
\]

Expanding the right side:

\[
4x + 5 \leq 9x + 36
\]

Now, subtract \(4x\) from both sides:

\[
5 \leq 5x + 36
\]

Subtract 36 from both sides:

\[
-31 \leq 5x
\]

Divide by 5:

\[
x \geq -\frac{31}{5}
\]

So for this case, we have \(x \geq -\frac{31}{5}\) and \(x > -4\). The solution for this case is:

\[
x \geq -\frac{31}{5} \quad \text{and} \quad x > -4
\]

Since \(-\frac{31}{5} = -6.2\), this simplifies to:

\[
x \geq -\frac{31}{5} \quad (\text{since} \, -6.2 > -4 \, \text{is false,})
\]

#### Case 2: \(x + 4 < 0\) (i.e., \(x < -4\))

Multiplying both sides by \(x + 4\) (which is negative), the inequality changes direction:

\[
4x + 5 \geq 9(x + 4)
\]

Expanding the right side:

\[
4x + 5 \geq 9x + 36
\]

Subtract \(4x\) from both sides:

\[
5 \geq 5x + 36
\]

Subtract 36 from both sides:

\[
-31 \geq 5x
\]

Divide by 5:

\[
x \leq -\frac{31}{5}
\]

For this case, we also need \(x < -4\), so the solution here is:

\[
x \leq -4
\]

### Final solution:
Combining both cases, the solution is:

\[
x \leq -4 \quad \text{or} \quad x \geq -\frac{31}{5}
\]

Would you like further details or clarifications?

Here are five related questions:
1. How do you solve rational inequalities in general?
2. What happens when the denominator is zero in inequalities?
3. Can inequalities always be solved using similar steps as equations?
4. How does the inequality direction change when multiplying by negative numbers?
5. What is the significance of boundary points in solving inequalities?

**Tip:** When solving inequalities involving fractions, always check for cases where the denominator is zero and consider how the sign of the denominator affects the inequality.

4x + 5 over x + 4 is less than or equal to 9

Learn how to solve the rational inequality (4x + 5)/(x + 4) <= 9 step by step. This problem covers key algebraic concepts, including rational inequalities, boundary points, and the impact of negative denominators. Suitable for students in Grades 9-12.

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