Answer :
To solve the equation [tex]\(4|x+7|+8=32\)[/tex], follow these steps:
1. Isolate the Absolute Value Expression:
Start by subtracting 8 from both sides of the equation:
[tex]\[
4|x + 7| + 8 - 8 = 32 - 8 \\
4|x + 7| = 24
\][/tex]
2. Divide to Remove the Coefficient:
Divide both sides of the equation by 4 to isolate the absolute value:
[tex]\[
|x + 7| = \frac{24}{4} \\
|x + 7| = 6
\][/tex]
3. Solve the Absolute Value Equation:
The equation [tex]\( |x + 7| = 6 \)[/tex] can be split into two separate equations:
- Case 1: [tex]\( x + 7 = 6 \)[/tex]
[tex]\[
x = 6 - 7 \\
x = -1
\][/tex]
- Case 2: [tex]\( x + 7 = -6 \)[/tex]
[tex]\[
x = -6 - 7 \\
x = -13
\][/tex]
4. Write the Solution:
Therefore, the solutions to the equation are [tex]\(x = -1\)[/tex] and [tex]\(x = -13\)[/tex].
From the options given, the correct choice is:
C. [tex]\(x = -1\)[/tex] and [tex]\(x = -13\)[/tex]
1. Isolate the Absolute Value Expression:
Start by subtracting 8 from both sides of the equation:
[tex]\[
4|x + 7| + 8 - 8 = 32 - 8 \\
4|x + 7| = 24
\][/tex]
2. Divide to Remove the Coefficient:
Divide both sides of the equation by 4 to isolate the absolute value:
[tex]\[
|x + 7| = \frac{24}{4} \\
|x + 7| = 6
\][/tex]
3. Solve the Absolute Value Equation:
The equation [tex]\( |x + 7| = 6 \)[/tex] can be split into two separate equations:
- Case 1: [tex]\( x + 7 = 6 \)[/tex]
[tex]\[
x = 6 - 7 \\
x = -1
\][/tex]
- Case 2: [tex]\( x + 7 = -6 \)[/tex]
[tex]\[
x = -6 - 7 \\
x = -13
\][/tex]
4. Write the Solution:
Therefore, the solutions to the equation are [tex]\(x = -1\)[/tex] and [tex]\(x = -13\)[/tex].
From the options given, the correct choice is:
C. [tex]\(x = -1\)[/tex] and [tex]\(x = -13\)[/tex]