High School

Solve [tex]$4|x+5|=24$[/tex].

A. [tex]$x=11$[/tex] and [tex]$x=-11$[/tex]
B. [tex]$x=11$[/tex] and [tex]$x=-1$[/tex]
C. [tex]$x=-11$[/tex] and [tex]$x=-1$[/tex]
D. [tex]$x=-11$[/tex] and [tex]$x=1$[/tex]

Answer :

To solve the equation [tex]\(4|x+5|=24\)[/tex], follow these steps:

1. Isolate the absolute value expression:
[tex]\[
4|x+5| = 24
\][/tex]
Divide both sides by 4:
[tex]\[
|x+5| = \frac{24}{4} = 6
\][/tex]

2. Set up the equations for the absolute value expression:
Remember that [tex]\(|A| = B\)[/tex] implies [tex]\(A = B\)[/tex] or [tex]\(A = -B\)[/tex]. Therefore, for [tex]\(|x+5| = 6\)[/tex], we have two cases:
- [tex]\(x + 5 = 6\)[/tex]
- [tex]\(x + 5 = -6\)[/tex]

3. Solve each equation:

- For [tex]\(x + 5 = 6\)[/tex]:
[tex]\[
x + 5 = 6
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = 6 - 5
\][/tex]
[tex]\[
x = 1
\][/tex]

- For [tex]\(x + 5 = -6\)[/tex]:
[tex]\[
x + 5 = -6
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = -6 - 5
\][/tex]
[tex]\[
x = -11
\][/tex]

4. Combine the solutions:
The solutions to the equation [tex]\(4|x+5|=24\)[/tex] are [tex]\(x=1\)[/tex] and [tex]\(x=-11\)[/tex].

5. Match the solutions to the choices given:
Comparing this to the options:
- A. [tex]\(x=11\)[/tex] and [tex]\(x=-11\)[/tex]
- B. [tex]\(x=11\)[/tex] and [tex]\(x=-1\)[/tex]
- C. [tex]\(x=-11\)[/tex] and [tex]\(x=-1\)[/tex]
- D. [tex]\(x=-11\)[/tex] and [tex]\(x=1\)[/tex]

The correct answer is:
[tex]\[
\boxed{D}
\][/tex]