Answer :
To solve the equation [tex]\(4|x+5| + 8 = 24\)[/tex], we need to follow these steps:
1. Start by isolating the absolute value expression:
Subtract 8 from both sides of the equation:
[tex]\[
4|x+5| = 24 - 8
\][/tex]
Simplify the right side:
[tex]\[
4|x+5| = 16
\][/tex]
2. Divide both sides by 4 to solve for the absolute value:
[tex]\[
|x+5| = \frac{16}{4}
\][/tex]
Simplify:
[tex]\[
|x+5| = 4
\][/tex]
3. This leads to two separate equations because the absolute value represents the distance from zero:
- [tex]\(x + 5 = 4\)[/tex]
- [tex]\(x + 5 = -4\)[/tex]
4. Solve each equation:
- For [tex]\(x + 5 = 4\)[/tex]:
[tex]\[
x = 4 - 5
\][/tex]
[tex]\[
x = -1
\][/tex]
- For [tex]\(x + 5 = -4\)[/tex]:
[tex]\[
x = -4 - 5
\][/tex]
[tex]\[
x = -9
\][/tex]
5. Conclusion:
The solutions to the equation are [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex]. Therefore, the correct answer is option D: [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].
1. Start by isolating the absolute value expression:
Subtract 8 from both sides of the equation:
[tex]\[
4|x+5| = 24 - 8
\][/tex]
Simplify the right side:
[tex]\[
4|x+5| = 16
\][/tex]
2. Divide both sides by 4 to solve for the absolute value:
[tex]\[
|x+5| = \frac{16}{4}
\][/tex]
Simplify:
[tex]\[
|x+5| = 4
\][/tex]
3. This leads to two separate equations because the absolute value represents the distance from zero:
- [tex]\(x + 5 = 4\)[/tex]
- [tex]\(x + 5 = -4\)[/tex]
4. Solve each equation:
- For [tex]\(x + 5 = 4\)[/tex]:
[tex]\[
x = 4 - 5
\][/tex]
[tex]\[
x = -1
\][/tex]
- For [tex]\(x + 5 = -4\)[/tex]:
[tex]\[
x = -4 - 5
\][/tex]
[tex]\[
x = -9
\][/tex]
5. Conclusion:
The solutions to the equation are [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex]. Therefore, the correct answer is option D: [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].