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.