High School

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

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

Answer :

Let's solve the equation [tex]\(4|x+7| + 8 = 32\)[/tex] step by step.

1. Isolate the Absolute Value:

Start by subtracting 8 from both sides of the equation:

[tex]\[
4|x+7| = 32 - 8
\][/tex]

Simplify:

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

2. Solve for the Absolute Value:

Divide both sides by 4 to isolate the absolute value:

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

Simplify:

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

3. Remove the Absolute Value:

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

a) [tex]\(x + 7 = 6\)[/tex]

b) [tex]\(x + 7 = -6\)[/tex]

4. Solve Each Equation:

- For the first equation [tex]\(x + 7 = 6\)[/tex]:

Subtract 7 from both sides:

[tex]\[
x = 6 - 7
\][/tex]

Simplify:

[tex]\[
x = -1
\][/tex]

- For the second equation [tex]\(x + 7 = -6\)[/tex]:

Subtract 7 from both sides:

[tex]\[
x = -6 - 7
\][/tex]

Simplify:

[tex]\[
x = -13
\][/tex]

Therefore, the solutions to the equation are [tex]\(x = -1\)[/tex] and [tex]\(x = -13\)[/tex].

So, the correct answer choice is:

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