Sure, let's solve the equation [tex]\(4|x+5| + 8 = 24\)[/tex] step by step.
1. Isolate the Absolute Value:
First, we need to isolate the absolute value term [tex]\(4|x+5|\)[/tex]. To do so, we subtract 8 from both sides of the equation:
[tex]\[
4|x+5| + 8 - 8 = 24 - 8
\][/tex]
Simplifying this, we get:
[tex]\[
4|x+5| = 16
\][/tex]
2. Solve for the Absolute Value:
Next, divide both sides by 4 to isolate the absolute value term:
[tex]\[
\frac{4|x+5|}{4} = \frac{16}{4}
\][/tex]
Simplifying this, we have:
[tex]\[
|x+5| = 4
\][/tex]
3. Solve the Absolute Value Equation:
The equation [tex]\( |x+5| = 4 \)[/tex] means that [tex]\( x+5 \)[/tex] can be either 4 or -4. Therefore, we have two separate equations to solve:
[tex]\[
x + 5 = 4 \quad \text{and} \quad x + 5 = -4
\][/tex]
4. Solve Each Equation:
- For the equation [tex]\(x + 5 = 4\)[/tex]:
[tex]\[
x + 5 - 5 = 4 - 5
\][/tex]
Simplifying this, we get:
[tex]\[
x = -1
\][/tex]
- For the equation [tex]\(x + 5 = -4\)[/tex]:
[tex]\[
x + 5 - 5 = -4 - 5
\][/tex]
Simplifying this, we get:
[tex]\[
x = -9
\][/tex]
5. Identify the Solutions:
Therefore, the solutions to the original equation are [tex]\( x = -1 \)[/tex] and [tex]\( x = -9 \)[/tex].
6. Match with Given Choices:
Comparing these solutions with the given choices:
[tex]\[
D. \quad x = -1 \quad \text{and} \quad x = -9
\][/tex]
So, the correct option is
[tex]\[
\boxed{D}
\][/tex]
