Answer :
Sure! Let's solve the equation [tex]\(4|x+5|+8=24\)[/tex] step by step.
1. Start by isolating the absolute value term:
[tex]\[
4|x+5| + 8 = 24
\][/tex]
Subtract 8 from both sides:
[tex]\[
4|x+5| = 16
\][/tex]
2. Divide both sides by 4 to solve for [tex]\(|x+5|\)[/tex]:
[tex]\[
|x+5| = 4
\][/tex]
3. Now we have an absolute value equation, which means there are two cases to consider:
- [tex]\(x + 5 = 4\)[/tex]
- [tex]\(x + 5 = -4\)[/tex]
4. Solve each case separately:
- Case 1: [tex]\(x + 5 = 4\)[/tex]
[tex]\[
x = 4 - 5
\][/tex]
[tex]\[
x = -1
\][/tex]
- Case 2: [tex]\(x + 5 = -4\)[/tex]
[tex]\[
x = -4 - 5
\][/tex]
[tex]\[
x = -9
\][/tex]
5. Therefore, the solutions to the equation are [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].
So the correct answer is:
[tex]\[
\boxed{D.\ x=-1\ \text{and}\ x=-9}
\][/tex]
1. Start by isolating the absolute value term:
[tex]\[
4|x+5| + 8 = 24
\][/tex]
Subtract 8 from both sides:
[tex]\[
4|x+5| = 16
\][/tex]
2. Divide both sides by 4 to solve for [tex]\(|x+5|\)[/tex]:
[tex]\[
|x+5| = 4
\][/tex]
3. Now we have an absolute value equation, which means there are two cases to consider:
- [tex]\(x + 5 = 4\)[/tex]
- [tex]\(x + 5 = -4\)[/tex]
4. Solve each case separately:
- Case 1: [tex]\(x + 5 = 4\)[/tex]
[tex]\[
x = 4 - 5
\][/tex]
[tex]\[
x = -1
\][/tex]
- Case 2: [tex]\(x + 5 = -4\)[/tex]
[tex]\[
x = -4 - 5
\][/tex]
[tex]\[
x = -9
\][/tex]
5. Therefore, the solutions to the equation are [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].
So the correct answer is:
[tex]\[
\boxed{D.\ x=-1\ \text{and}\ x=-9}
\][/tex]