Let's solve the equation [tex]\( 4|x+6| = 32 \)[/tex] step-by-step:
1. Isolate the absolute value expression:
[tex]\[
4|x+6| = 32
\][/tex]
Divide both sides by 4.
[tex]\[
|x+6| = \frac{32}{4}
\][/tex]
[tex]\[
|x+6| = 8
\][/tex]
2. Set up the two cases for the absolute value equation:
The equation [tex]\( |x+6| = 8 \)[/tex] translates to two linear equations:
- Case 1: [tex]\( x + 6 = 8 \)[/tex]
- Case 2: [tex]\( x + 6 = -8 \)[/tex]
3. Solve for [tex]\( x \)[/tex] in each case:
- For Case 1:
[tex]\[
x + 6 = 8
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = 8 - 6
\][/tex]
[tex]\[
x = 2
\][/tex]
- For Case 2:
[tex]\[
x + 6 = -8
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = -8 - 6
\][/tex]
[tex]\[
x = -14
\][/tex]
4. Conclude the solutions:
The solutions to the equation [tex]\( 4|x+6| = 32 \)[/tex] are:
[tex]\[
x = -14 \quad \text{and} \quad x = 2
\][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{C. \; x = -14 \text{ and } x = 2} \][/tex]
