College

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 :

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

1. Isolate the absolute value expression.

Start by subtracting 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 by 4 to further isolate the absolute value.

Divide both sides by 4:

[tex]\[
\frac{4|x+5|}{4} = \frac{16}{4}
\][/tex]

Simplifying, we have:

[tex]\[
|x+5| = 4
\][/tex]

3. Solve the absolute value equation.

The equation [tex]\( |x+5| = 4 \)[/tex] means that the expression inside the absolute value can be either 4 or -4. So we have two cases to consider:

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

Solve for [tex]\( x \)[/tex]:

[tex]\[
x + 5 = 4
\][/tex]

Subtract 5 from both sides:

[tex]\[
x = 4 - 5
\][/tex]

[tex]\[
x = -1
\][/tex]

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

Solve for [tex]\( x \)[/tex]:

[tex]\[
x + 5 = -4
\][/tex]

Subtract 5 from both sides:

[tex]\[
x = -4 - 5
\][/tex]

[tex]\[
x = -9
\][/tex]

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

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