College

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 :

Sure, let's solve the equation step-by-step:

The equation is [tex]\(4|x+7| + 8 = 32\)[/tex].

1. Isolate the absolute value expression:

Subtract 8 from both sides:

[tex]\[
4|x+7| = 24
\][/tex]

2. Divide by 4 to further isolate the absolute value:

[tex]\[
|x+7| = 6
\][/tex]

3. Solve the equation by considering both cases of the absolute value:

The absolute value equation [tex]\( |x+7| = 6 \)[/tex] can be split into two separate equations:

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

Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 6 - 7 = -1
\][/tex]

- Case 2: [tex]\(x + 7 = -6\)[/tex]

Solve for [tex]\(x\)[/tex]:
[tex]\[
x = -6 - 7 = -13
\][/tex]

4. Solutions:

The solutions are [tex]\(x = -1\)[/tex] and [tex]\(x = -13\)[/tex].

So, the correct answer is:

C. [tex]\(x = -1\)[/tex] and [tex]\(x = -13\)[/tex]