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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], let's go through the steps systematically:

1. Isolate the absolute value expression:
[tex]\[
4|x + 5| + 8 = 24
\][/tex]
Subtract 8 from both sides of the equation to isolate the absolute value term:
[tex]\[
4|x + 5| + 8 - 8 = 24 - 8
\][/tex]
[tex]\[
4|x + 5| = 16
\][/tex]

2. Divide by 4 to solve for the absolute value:
[tex]\[
\frac{4|x + 5|}{4} = \frac{16}{4}
\][/tex]
[tex]\[
|x + 5| = 4
\][/tex]

3. Consider the definition of absolute value:
The equation [tex]\( |x + 5| = 4 \)[/tex] means [tex]\( x + 5 \)[/tex] can be either 4 or -4.

Thus, we have two cases to solve:
[tex]\[
x + 5 = 4 \quad \text{or} \quad x + 5 = -4
\][/tex]

4. Solve each equation:
- 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]

5. Combine the solutions:
The solutions to the equation [tex]\( 4|x + 5| + 8 = 24 \)[/tex] are [tex]\( x = -1 \)[/tex] and [tex]\( x = -9 \)[/tex].

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

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