Answer :
To solve the equation [tex]\(4|x+5|=24\)[/tex], we need to first isolate the absolute value expression and then consider the two cases that arise from it. Let's go through the steps:
1. Isolate the Absolute Value:
Divide both sides of the equation by 4 to isolate [tex]\(|x+5|\)[/tex]:
[tex]\[
|x+5| = \frac{24}{4} = 6
\][/tex]
2. Set Up the Two Cases:
The absolute value equation [tex]\(|x+5| = 6\)[/tex] means that the expression inside the absolute value, [tex]\(x+5\)[/tex], can either be 6 or -6. So we have two cases to consider:
- Case 1: [tex]\(x + 5 = 6\)[/tex]
- Case 2: [tex]\(x + 5 = -6\)[/tex]
3. Solve Each Case:
- Case 1: [tex]\(x + 5 = 6\)[/tex]
Subtract 5 from both sides:
[tex]\[
x = 6 - 5 = 1
\][/tex]
- Case 2: [tex]\(x + 5 = -6\)[/tex]
Subtract 5 from both sides:
[tex]\[
x = -6 - 5 = -11
\][/tex]
4. Write the Solutions:
The solutions to the equation [tex]\(4|x+5|=24\)[/tex] are [tex]\(x = 1\)[/tex] and [tex]\(x = -11\)[/tex].
Comparing the solutions with the given options, the correct answer is:
C. [tex]\(x=-11\)[/tex] and [tex]\(x=1\)[/tex]
1. Isolate the Absolute Value:
Divide both sides of the equation by 4 to isolate [tex]\(|x+5|\)[/tex]:
[tex]\[
|x+5| = \frac{24}{4} = 6
\][/tex]
2. Set Up the Two Cases:
The absolute value equation [tex]\(|x+5| = 6\)[/tex] means that the expression inside the absolute value, [tex]\(x+5\)[/tex], can either be 6 or -6. So we have two cases to consider:
- Case 1: [tex]\(x + 5 = 6\)[/tex]
- Case 2: [tex]\(x + 5 = -6\)[/tex]
3. Solve Each Case:
- Case 1: [tex]\(x + 5 = 6\)[/tex]
Subtract 5 from both sides:
[tex]\[
x = 6 - 5 = 1
\][/tex]
- Case 2: [tex]\(x + 5 = -6\)[/tex]
Subtract 5 from both sides:
[tex]\[
x = -6 - 5 = -11
\][/tex]
4. Write the Solutions:
The solutions to the equation [tex]\(4|x+5|=24\)[/tex] are [tex]\(x = 1\)[/tex] and [tex]\(x = -11\)[/tex].
Comparing the solutions with the given options, the correct answer is:
C. [tex]\(x=-11\)[/tex] and [tex]\(x=1\)[/tex]