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Solve [tex]$4|x+6| + 8 = 28$[/tex].

A. [tex]$x=-1$[/tex] and [tex]$x=11$[/tex]
B. [tex]$x=1$[/tex] and [tex]$x=-11$[/tex]
C. [tex]$x=-1$[/tex] and [tex]$x=-11$[/tex]
D. [tex]$x=1$[/tex] and [tex]$x=-1$[/tex]

Answer :

Sure! Let's solve the equation [tex]\( 4 |x+6| + 8 = 28 \)[/tex] step-by-step.

1. Start with the given equation:
[tex]\[
4 |x+6| + 8 = 28
\][/tex]

2. Isolate the absolute value expression:
[tex]\[
4 |x+6| = 28 - 8
\][/tex]
[tex]\[
4 |x+6| = 20
\][/tex]

3. Divide both sides by 4:
[tex]\[
|x+6| = \frac{20}{4}
\][/tex]
[tex]\[
|x+6| = 5
\][/tex]

4. Now, consider the definition of the absolute value function, which states that if [tex]\( |A| = B \)[/tex], then [tex]\( A = B \)[/tex] or [tex]\( A = -B \)[/tex]:
[tex]\[
x + 6 = 5 \quad \text{or} \quad x + 6 = -5
\][/tex]

5. Solve each of these equations separately:

- For [tex]\( x + 6 = 5 \)[/tex]:
[tex]\[
x = 5 - 6
\][/tex]
[tex]\[
x = -1
\][/tex]

- For [tex]\( x + 6 = -5 \)[/tex]:
[tex]\[
x = -5 - 6
\][/tex]
[tex]\[
x = -11
\][/tex]

6. So, the solutions for the equation [tex]\( 4 |x+6| + 8 = 28 \)[/tex] are [tex]\( x = -1 \)[/tex] and [tex]\( x = -11 \)[/tex].

Therefore, the correct answer is:
[tex]\[ \boxed{\text{C. } x = -1 \text{ and } x = -11} \][/tex]

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