High School

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

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

Answer :

Sure, I can help with that! Let's solve the equation [tex]\(4|x+6| = 16\)[/tex].

1. Start by simplifying the equation:

Divide both sides of the equation by 4 to make it easier to handle:

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

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

2. Set up the cases for the absolute value:

When you have an absolute value equation like [tex]\(|x+6| = 4\)[/tex], it means that [tex]\(x+6\)[/tex] can be either 4 or -4, because the absolute value of a number is its distance from zero, so it could be the positive or negative of the value.

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

3. Solve each case separately:

- For Case 1:

[tex]\[
x + 6 = 4
\][/tex]

Subtract 6 from both sides to solve for [tex]\(x\)[/tex]:

[tex]\[
x = 4 - 6
\][/tex]

[tex]\[
x = -2
\][/tex]

- For Case 2:

[tex]\[
x + 6 = -4
\][/tex]

Again, subtract 6 from both sides to solve for [tex]\(x\)[/tex]:

[tex]\[
x = -4 - 6
\][/tex]

[tex]\[
x = -10
\][/tex]

4. Conclusion:

We have two solutions for the equation [tex]\(4|x+6| = 16\)[/tex]:

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

Therefore, the correct choice is [tex]\(\boxed{A. \, x=-2 \text{ and } x=-10}\)[/tex].