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Solve [tex]$4|x+6|=16$[/tex].

A. [tex]$x=-2$[/tex] and [tex][tex]$x=-10$[/tex][/tex]
B. [tex]$x=2$[/tex] and [tex]$x=-2$[/tex]
C. [tex][tex]$x=2$[/tex][/tex] and [tex]$x=-10$[/tex]
D. [tex]$x=-2$[/tex] and [tex][tex]$x=10$[/tex][/tex]

Answer :

To solve the equation [tex]\(4|x+6|=16\)[/tex], we need to follow these steps:

1. Simplify the equation:
Divide both sides by 4 to isolate the absolute value expression:
[tex]\[
|x+6| = \frac{16}{4}
\][/tex]
[tex]\[
|x+6| = 4
\][/tex]

2. Set up two separate cases:
The absolute value equation |A| = B can be broken into two cases:
- Case 1: [tex]\(x + 6 = 4\)[/tex]
- Case 2: [tex]\(x + 6 = -4\)[/tex]

3. Solve each case:

- For Case 1:
[tex]\[
x + 6 = 4
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = 4 - 6
\][/tex]
[tex]\[
x = -2
\][/tex]

- For Case 2:
[tex]\[
x + 6 = -4
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = -4 - 6
\][/tex]
[tex]\[
x = -10
\][/tex]

4. Check and state the solutions:
The solutions to the equation [tex]\(4|x+6|=16\)[/tex] are [tex]\(x = -2\)[/tex] and [tex]\(x = -10\)[/tex].

Thus, the correct choice from the provided options is A. [tex]\(x=-2\)[/tex] and [tex]\(x=-10\)[/tex].

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