Answer :
Let's solve the equation [tex]\( 4|x+6| = 16 \)[/tex] step by step.
1. Isolate the absolute value term:
First, divide both sides of the equation by 4.
[tex]\[
4|x+6| = 16 \implies |x+6| = \frac{16}{4} = 4
\][/tex]
2. Set up the two cases for the absolute value:
The expression [tex]\( |x+6| = 4 \)[/tex] means that the quantity inside the absolute value can be either 4 or -4.
So, we have two equations to solve:
[tex]\[
x + 6 = 4 \quad \text{(Case 1)}
\][/tex]
[tex]\[
x + 6 = -4 \quad \text{(Case 2)}
\][/tex]
3. Solve Case 1:
[tex]\[
x + 6 = 4
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = 4 - 6
\][/tex]
[tex]\[
x = -2
\][/tex]
4. Solve Case 2:
[tex]\[
x + 6 = -4
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = -4 - 6
\][/tex]
[tex]\[
x = -10
\][/tex]
Therefore, the solutions to the equation [tex]\( 4|x+6| = 16 \)[/tex] are [tex]\( x = -2 \)[/tex] and [tex]\( x = -10 \)[/tex].
So, the correct answer is:
B. [tex]\( x = -2 \)[/tex] and [tex]\( x = -10 \)[/tex]
1. Isolate the absolute value term:
First, divide both sides of the equation by 4.
[tex]\[
4|x+6| = 16 \implies |x+6| = \frac{16}{4} = 4
\][/tex]
2. Set up the two cases for the absolute value:
The expression [tex]\( |x+6| = 4 \)[/tex] means that the quantity inside the absolute value can be either 4 or -4.
So, we have two equations to solve:
[tex]\[
x + 6 = 4 \quad \text{(Case 1)}
\][/tex]
[tex]\[
x + 6 = -4 \quad \text{(Case 2)}
\][/tex]
3. Solve Case 1:
[tex]\[
x + 6 = 4
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = 4 - 6
\][/tex]
[tex]\[
x = -2
\][/tex]
4. Solve Case 2:
[tex]\[
x + 6 = -4
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = -4 - 6
\][/tex]
[tex]\[
x = -10
\][/tex]
Therefore, the solutions to the equation [tex]\( 4|x+6| = 16 \)[/tex] are [tex]\( x = -2 \)[/tex] and [tex]\( x = -10 \)[/tex].
So, the correct answer is:
B. [tex]\( x = -2 \)[/tex] and [tex]\( x = -10 \)[/tex]