College

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

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

Answer :

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

1. Isolate the absolute value:
Start by dividing both sides of the equation by 4:
[tex]\[
4|x+6| = 16 \quad \Rightarrow \quad |x+6| = \frac{16}{4} \quad \Rightarrow \quad |x+6| = 4
\][/tex]

2. Set up the two possible cases for [tex]\( |x+6| = 4 \)[/tex]:
The definition of absolute value tells us that [tex]\( |x+6| = 4 \)[/tex] means [tex]\( x+6 \)[/tex] can be either 4 or -4.

- Case 1: [tex]\( x+6 = 4 \)[/tex]
- Case 2: [tex]\( x+6 = -4 \)[/tex]

3. Solve each case separately:

- Case 1: [tex]\( x+6 = 4 \)[/tex]
[tex]\[
x + 6 = 4 \quad \Rightarrow \quad x = 4 - 6 \quad \Rightarrow \quad x = -2
\][/tex]

- Case 2: [tex]\( x+6 = -4 \)[/tex]
[tex]\[
x + 6 = -4 \quad \Rightarrow \quad x = -4 - 6 \quad \Rightarrow \quad x = -10
\][/tex]

4. Combine the results:
The solutions from both cases are [tex]\( x = -2 \)[/tex] and [tex]\( x = -10 \)[/tex].

So, the correct solutions are [tex]\( x = -2 \)[/tex] and [tex]\( x = -10 \)[/tex].

Comparing this with the given options:

A. [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex]
B. [tex]\( x = -2 \)[/tex] and [tex]\( x = 10 \)[/tex]
C. [tex]\( x = -2 \)[/tex] and [tex]\( x = -10 \)[/tex]
D. [tex]\( x = 2 \)[/tex] and [tex]\( x = -10 \)[/tex]

The correct answer is:
C. [tex]\( x = -2 \)[/tex] and [tex]\( x = -10 \)[/tex]