High School

Solve [tex]4|x+5|+8=24[/tex].

A. [tex]x=-1[/tex] and [tex]x=9[/tex]
B. [tex]x=-1[/tex] and [tex]x=-9[/tex]
C. [tex]x=1[/tex] and [tex]x=-1[/tex]
D. [tex]x=1[/tex] and [tex]x=-9[/tex]

Answer :

Sure, let's solve the equation [tex]\(4|x+5| + 8 = 24\)[/tex].

### Step-by-Step Solution:

1. Isolate the absolute value expression:
[tex]\[
4|x+5| + 8 = 24
\][/tex]
Subtract 8 from both sides:
[tex]\[
4|x+5| = 16
\][/tex]

2. Divide by 4:
[tex]\[
|x+5| = 4
\][/tex]

3. Consider the definition of absolute value:
The absolute value equation [tex]\(|x+5| = 4\)[/tex] has two cases:
- [tex]\( x + 5 = 4 \)[/tex]
- [tex]\( x + 5 = -4 \)[/tex]

4. Solve each case separately:

Case 1: [tex]\(x + 5 = 4\)[/tex]:
[tex]\[
x + 5 = 4
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = -1
\][/tex]

Case 2: [tex]\(x + 5 = -4\)[/tex]:
[tex]\[
x + 5 = -4
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = -9
\][/tex]

5. Conclusion:
The solutions to the equation [tex]\(4|x+5| + 8 = 24\)[/tex] are [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].

So, the correct answer is:

B. [tex]\( x = -1 \)[/tex] and [tex]\( x = -9 \)[/tex]