College

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

Answer :

We start with the equation

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

Step 1: Isolate the absolute value.

Divide both sides of the equation by 4:

[tex]$$
|x+6| = \frac{16}{4} = 4.
$$[/tex]

Step 2: Solve the absolute value equation.

The equation

[tex]$$
|x+6| = 4
$$[/tex]

splits into two separate equations:

1. [tex]$$ x+6 = 4 $$[/tex]
2. [tex]$$ x+6 = -4 $$[/tex]

Step 3: Solve each equation.

For the first equation:

[tex]$$
x+6 = 4 \quad \Longrightarrow \quad x = 4-6 = -2.
$$[/tex]

For the second equation:

[tex]$$
x+6 = -4 \quad \Longrightarrow \quad x = -4-6 = -10.
$$[/tex]

Conclusion:

The solutions to the equation are

[tex]$$
x = -2 \quad \text{and} \quad x = -10.
$$[/tex]

Thus, the correct answer is option B.