Answer :
To solve the equation [tex]\(|x - 5| + 7 = 17\)[/tex], let's go through it step-by-step:
1. Isolate the Absolute Value:
First, subtract 7 from both sides of the equation to isolate the absolute value expression:
[tex]\[
|x - 5| + 7 - 7 = 17 - 7
\][/tex]
[tex]\[
|x - 5| = 10
\][/tex]
2. Solve for Two Cases:
The equation [tex]\(|x - 5| = 10\)[/tex] means that the expression inside the absolute value, [tex]\(x - 5\)[/tex], can be either 10 or -10. This gives us two cases to consider:
- Case 1: [tex]\(x - 5 = 10\)[/tex]
- Add 5 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 10 + 5 = 15
\][/tex]
- Case 2: [tex]\(x - 5 = -10\)[/tex]
- Add 5 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -10 + 5 = -5
\][/tex]
3. Conclusion:
The solutions to the equation [tex]\(|x - 5| + 7 = 17\)[/tex] are [tex]\(x = 15\)[/tex] and [tex]\(x = -5\)[/tex].
Therefore, the correct answer is option D: [tex]\(x = 15\)[/tex] and [tex]\(x = -5\)[/tex].
1. Isolate the Absolute Value:
First, subtract 7 from both sides of the equation to isolate the absolute value expression:
[tex]\[
|x - 5| + 7 - 7 = 17 - 7
\][/tex]
[tex]\[
|x - 5| = 10
\][/tex]
2. Solve for Two Cases:
The equation [tex]\(|x - 5| = 10\)[/tex] means that the expression inside the absolute value, [tex]\(x - 5\)[/tex], can be either 10 or -10. This gives us two cases to consider:
- Case 1: [tex]\(x - 5 = 10\)[/tex]
- Add 5 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 10 + 5 = 15
\][/tex]
- Case 2: [tex]\(x - 5 = -10\)[/tex]
- Add 5 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -10 + 5 = -5
\][/tex]
3. Conclusion:
The solutions to the equation [tex]\(|x - 5| + 7 = 17\)[/tex] are [tex]\(x = 15\)[/tex] and [tex]\(x = -5\)[/tex].
Therefore, the correct answer is option D: [tex]\(x = 15\)[/tex] and [tex]\(x = -5\)[/tex].
