High School

Solve [tex]$4|x+5|+8=24$[/tex]

A. [tex]$x=-1$[/tex] and [tex]$x=-9$[/tex]
B. [tex]$x=1$[/tex] and [tex]$x=-1$[/tex]
C. [tex]$x=-1$[/tex] and [tex]$x=9$[/tex]
D. [tex]$x=1$[/tex] and [tex]$x=-9$[/tex]

Answer :

To solve the equation [tex]\(4|x+5| + 8 = 24\)[/tex], follow these steps:

1. Isolate the Absolute Value Term:
- Start by subtracting 8 from both sides of the equation:
[tex]\[
4|x+5| + 8 - 8 = 24 - 8
\][/tex]
[tex]\[
4|x+5| = 16
\][/tex]

2. Divide by 4:
- To isolate [tex]\( |x+5| \)[/tex], divide both sides of the equation by 4:
[tex]\[
|x+5| = \frac{16}{4}
\][/tex]
[tex]\[
|x+5| = 4
\][/tex]

3. Set Up Two Cases for the Absolute Value:
- Since the absolute value of a number [tex]\( |a| \)[/tex] is [tex]\( a \)[/tex] or [tex]\(-a\)[/tex], we can set up two separate equations:
- Case 1: [tex]\( x+5 = 4 \)[/tex]
- Case 2: [tex]\( x+5 = -4 \)[/tex]

4. Solve Each Case:
- Case 1: Solve [tex]\( x + 5 = 4 \)[/tex]
[tex]\[
x = 4 - 5
\][/tex]
[tex]\[
x = -1
\][/tex]

- Case 2: Solve [tex]\( x + 5 = -4 \)[/tex]
[tex]\[
x = -4 - 5
\][/tex]
[tex]\[
x = -9
\][/tex]

5. Solution:
- The solutions to the equation are [tex]\( x = -1 \)[/tex] and [tex]\( x = -9 \)[/tex].

Therefore, the correct answer is option A: [tex]\( x = -1 \)[/tex] and [tex]\( x = -9 \)[/tex].