Solve the absolute-value inequality. Express the answer using interval notation.
9 − |2x + 5| ≤ 8

To solve the absolute-value inequality \( 9 - |2x + 5| \leq 8 \), follow these steps:

### Step 1: Isolate the absolute value expression
First, subtract 9 from both sides of the inequality to isolate the absolute value term:

\[
- |2x + 5| \leq -1
\]

### Step 2: Multiply both sides by -1
When you multiply both sides of an inequality by a negative number, the direction of the inequality sign reverses:

\[
|2x + 5| \geq 1
\]

### Step 3: Break down the absolute value inequality
The expression \( |2x + 5| \geq 1 \) can be broken into two separate inequalities:

\[
2x + 5 \geq 1 \quad \text{or} \quad 2x + 5 \leq -1
\]

### Step 4: Solve each inequality
**For the first inequality:**

\[
2x + 5 \geq 1
\]
Subtract 5 from both sides:

\[
2x \geq -4
\]
Divide by 2:

\[
x \geq -2
\]

**For the second inequality:**

\[
2x + 5 \leq -1
\]
Subtract 5 from both sides:

\[
2x \leq -6
\]
Divide by 2:

\[
x \leq -3
\]

### Step 5: Combine the solutions
The solution to the inequality is the union of the two intervals:

\[
x \leq -3 \quad \text{or} \quad x \geq -2
\]

### Step 6: Write the solution in interval notation
The interval notation for the solution is:

\[
(-\infty, -3] \cup [-2, \infty)
\]

This is the solution to the inequality.

Would you like more details or have any questions?

Here are 5 related questions to further explore:
1. How would the solution change if the inequality was \( |2x + 5| \leq 1 \) instead?
2. What would be the graphical representation of the solution on a number line?
3. How do you solve absolute value equations compared to inequalities?
4. What happens if the absolute value inequality involves a quadratic expression?
5. Can you explain how to solve inequalities involving two absolute values?

**Tip:** Always check your solution by substituting values from each interval back into the original inequality.

Learn how to solve the absolute value inequality 9 − |2x + 5| ≤ 8 with a detailed step-by-step solution. This problem involves absolute value inequalities and their applications in algebra. Suitable for students in Grades 9-12.

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