Answer :
To solve the equation [tex]\(4|x+7|+8=32\)[/tex], let's go through the steps to find the value of [tex]\(x\)[/tex].
1. Isolate the Absolute Value:
Start by getting the absolute value by itself on one side of the equation. Subtract 8 from both sides:
[tex]\[
4|x+7| = 24
\][/tex]
2. Divide to Remove the Coefficient:
Divide both sides by 4 to further isolate the absolute value:
[tex]\[
|x+7| = 6
\][/tex]
3. Consider the Two Cases for Absolute Value:
Since [tex]\(|x+7| = 6\)[/tex], this means [tex]\(x+7\)[/tex] can be either 6 or -6 because absolute value measures the distance from zero, which could have been reached in either direction.
- Case 1: [tex]\(x + 7 = 6\)[/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 6 - 7
\][/tex]
[tex]\[
x = -1
\][/tex]
- Case 2: [tex]\(x + 7 = -6\)[/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = -6 - 7
\][/tex]
[tex]\[
x = -13
\][/tex]
4. Conclusion:
The solutions to the equation are [tex]\(x = -1\)[/tex] and [tex]\(x = -13\)[/tex].
Thus, the correct choice is:
[tex]\[ D. \, x = -1 \text{ and } x = -13 \][/tex]
1. Isolate the Absolute Value:
Start by getting the absolute value by itself on one side of the equation. Subtract 8 from both sides:
[tex]\[
4|x+7| = 24
\][/tex]
2. Divide to Remove the Coefficient:
Divide both sides by 4 to further isolate the absolute value:
[tex]\[
|x+7| = 6
\][/tex]
3. Consider the Two Cases for Absolute Value:
Since [tex]\(|x+7| = 6\)[/tex], this means [tex]\(x+7\)[/tex] can be either 6 or -6 because absolute value measures the distance from zero, which could have been reached in either direction.
- Case 1: [tex]\(x + 7 = 6\)[/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 6 - 7
\][/tex]
[tex]\[
x = -1
\][/tex]
- Case 2: [tex]\(x + 7 = -6\)[/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = -6 - 7
\][/tex]
[tex]\[
x = -13
\][/tex]
4. Conclusion:
The solutions to the equation are [tex]\(x = -1\)[/tex] and [tex]\(x = -13\)[/tex].
Thus, the correct choice is:
[tex]\[ D. \, x = -1 \text{ and } x = -13 \][/tex]