High School

Solve [tex]4|x+7|+8=32[/tex].

A. [tex]x=1[/tex] and [tex]x=-13[/tex]
B. [tex]x=-1[/tex] and [tex]x=13[/tex]
C. [tex]x=-1[/tex] and [tex]x=-13[/tex]
D. [tex]x=1[/tex] and [tex]x=-1[/tex]

Answer :

To solve the equation [tex]\(4|x+7| + 8 = 32\)[/tex], let's follow these steps:

1. Isolate the absolute value expression:
[tex]\[
4|x+7| + 8 = 32
\][/tex]
Subtract 8 from both sides:
[tex]\[
4|x+7| = 24
\][/tex]
Divide both sides by 4:
[tex]\[
|x+7| = 6
\][/tex]

2. Solve the absolute value equation:
Recall that if [tex]\(|A| = B\)[/tex], then [tex]\(A = B\)[/tex] or [tex]\(A = -B\)[/tex]. Therefore, we have:
[tex]\[
x+7 = 6 \quad \text{or} \quad x+7 = -6
\][/tex]
Solve each equation separately:

- For [tex]\(x + 7 = 6\)[/tex]:
[tex]\[
x = 6 - 7
\][/tex]
[tex]\[
x = -1
\][/tex]

- For [tex]\(x + 7 = -6\)[/tex]:
[tex]\[
x = -6 - 7
\][/tex]
[tex]\[
x = -13
\][/tex]

The solutions to the equation [tex]\(4|x+7| + 8 = 32\)[/tex] are [tex]\(x = -1\)[/tex] and [tex]\(x = -13\)[/tex].

So, the correct answer is:
[tex]\[
\boxed{\text{C. } x = -1 \text{ and } x = -13}
\][/tex]

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