Answer :
We start with the equation:
[tex]$$
4|x+6| + 8 = 28.
$$[/tex]
Step 1: Subtract 8 from both sides to isolate the term with the absolute value.
[tex]$$
4|x+6| + 8 - 8 = 28 - 8 \quad \Rightarrow \quad 4|x+6| = 20.
$$[/tex]
Step 2: Divide both sides by 4 to solve for the absolute expression.
[tex]$$
|x+6| = \frac{20}{4} = 5.
$$[/tex]
Step 3: Solve the absolute value equation, which gives two cases.
- Case 1: Assume the expression inside the absolute value is positive.
[tex]$$
x+6 = 5 \quad \Rightarrow \quad x = 5 - 6 = -1.
$$[/tex]
- Case 2: Assume the expression inside the absolute value is negative.
[tex]$$
x+6 = -5 \quad \Rightarrow \quad x = -5 - 6 = -11.
$$[/tex]
Thus, the solutions to the equation are:
[tex]$$
x = -1 \quad \text{and} \quad x = -11.
$$[/tex]
Comparing these results with the given answer choices, the correct answer is:
C. [tex]\( x = -1 \)[/tex] and [tex]\( x = -11 \)[/tex].
[tex]$$
4|x+6| + 8 = 28.
$$[/tex]
Step 1: Subtract 8 from both sides to isolate the term with the absolute value.
[tex]$$
4|x+6| + 8 - 8 = 28 - 8 \quad \Rightarrow \quad 4|x+6| = 20.
$$[/tex]
Step 2: Divide both sides by 4 to solve for the absolute expression.
[tex]$$
|x+6| = \frac{20}{4} = 5.
$$[/tex]
Step 3: Solve the absolute value equation, which gives two cases.
- Case 1: Assume the expression inside the absolute value is positive.
[tex]$$
x+6 = 5 \quad \Rightarrow \quad x = 5 - 6 = -1.
$$[/tex]
- Case 2: Assume the expression inside the absolute value is negative.
[tex]$$
x+6 = -5 \quad \Rightarrow \quad x = -5 - 6 = -11.
$$[/tex]
Thus, the solutions to the equation are:
[tex]$$
x = -1 \quad \text{and} \quad x = -11.
$$[/tex]
Comparing these results with the given answer choices, the correct answer is:
C. [tex]\( x = -1 \)[/tex] and [tex]\( x = -11 \)[/tex].