High School

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

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

Answer :

Sure, let's solve the given equation step by step:

The given equation is:
[tex]\[ 4|x+5| + 8 = 24 \][/tex]

1. Isolate the absolute value expression:
First, subtract 8 from both sides of the equation:
[tex]\[ 4|x+5| + 8 - 8 = 24 - 8 \][/tex]
Simplifying, we get:
[tex]\[ 4|x+5| = 16 \][/tex]

2. Divide both sides by 4:
[tex]\[ \frac{4|x+5|}{4} = \frac{16}{4} \][/tex]
Simplifying, we get:
[tex]\[ |x+5| = 4 \][/tex]

3. Break it into two separate equations:
The absolute value equation [tex]\(|x+5| = 4\)[/tex] means there are two possible cases:

- Case 1: [tex]\( x + 5 = 4 \)[/tex]
- Case 2: [tex]\( x + 5 = -4 \)[/tex]

4. Solve both equations:

- For [tex]\( x + 5 = 4 \)[/tex]:
[tex]\[ x + 5 = 4 \][/tex]
Subtract 5 from both sides:
[tex]\[ x = 4 - 5 \][/tex]
[tex]\[ x = -1 \][/tex]

- For [tex]\( x + 5 = -4 \)[/tex]:
[tex]\[ x + 5 = -4 \][/tex]
Subtract 5 from both sides:
[tex]\[ x = -4 - 5 \][/tex]
[tex]\[ x = -9 \][/tex]

So, the solutions to the equation are [tex]\( x = -1 \)[/tex] and [tex]\( x = -9 \)[/tex].

Thus, the correct answer is:
[tex]\[ \boxed{B. \ x=-1 \ \text{and} \ x=-9} \][/tex]