College

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 :

Sure, let's solve the equation [tex]\(4|x+5|+8=24\)[/tex] step by step.

1. Isolate the absolute value term:
[tex]\[
4|x+5| + 8 = 24
\][/tex]

2. Subtract 8 from both sides of the equation:
[tex]\[
4|x+5| + 8 - 8 = 24 - 8
\][/tex]
This simplifies to:
[tex]\[
4|x+5| = 16
\][/tex]

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

4. Solve for the variable [tex]\(x\)[/tex] considering the definition of absolute value:
The equation [tex]\( |x+5| = 4 \)[/tex] implies two scenarios:

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

- Scenario 2: [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. Summarize the solutions:
The solutions are [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].

To verify which option matches our solutions:

A. [tex]\(x = 1\)[/tex] and [tex]\(x = -9\)[/tex] — Incorrect
B. [tex]\(x = 1\)[/tex] and [tex]\(x = -1\)[/tex] — Incorrect
C. [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex] — Correct
D. [tex]\(x = -1\)[/tex] and [tex]\(x = 9\)[/tex] — Incorrect

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