Answer :
To solve the equation [tex]\(|x-5| + 7 = 17\)[/tex], we need to isolate the absolute value expression first. Here's how you can do it step by step:
1. Subtract 7 from both sides of the equation:
[tex]\[
|x-5| + 7 - 7 = 17 - 7
\][/tex]
This simplifies to:
[tex]\[
|x-5| = 10
\][/tex]
2. Solve the absolute value equation:
The absolute value equation [tex]\(|x-5| = 10\)[/tex] means that [tex]\(x-5\)[/tex] can be either 10 or -10. This gives us two separate equations to solve:
- Case 1: [tex]\(x - 5 = 10\)[/tex]
[tex]\[
x - 5 = 10
\][/tex]
Add 5 to both sides:
[tex]\[
x = 10 + 5
\][/tex]
[tex]\[
x = 15
\][/tex]
- Case 2: [tex]\(x - 5 = -10\)[/tex]
[tex]\[
x - 5 = -10
\][/tex]
Add 5 to both sides:
[tex]\[
x = -10 + 5
\][/tex]
[tex]\[
x = -5
\][/tex]
3. Conclusion:
The solutions to the equation [tex]\(|x-5| + 7 = 17\)[/tex] are [tex]\(x = 15\)[/tex] and [tex]\(x = -5\)[/tex].
Thus, the correct choice is:
C. [tex]\(x=15\)[/tex] and [tex]\(x=-5\)[/tex]
1. Subtract 7 from both sides of the equation:
[tex]\[
|x-5| + 7 - 7 = 17 - 7
\][/tex]
This simplifies to:
[tex]\[
|x-5| = 10
\][/tex]
2. Solve the absolute value equation:
The absolute value equation [tex]\(|x-5| = 10\)[/tex] means that [tex]\(x-5\)[/tex] can be either 10 or -10. This gives us two separate equations to solve:
- Case 1: [tex]\(x - 5 = 10\)[/tex]
[tex]\[
x - 5 = 10
\][/tex]
Add 5 to both sides:
[tex]\[
x = 10 + 5
\][/tex]
[tex]\[
x = 15
\][/tex]
- Case 2: [tex]\(x - 5 = -10\)[/tex]
[tex]\[
x - 5 = -10
\][/tex]
Add 5 to both sides:
[tex]\[
x = -10 + 5
\][/tex]
[tex]\[
x = -5
\][/tex]
3. Conclusion:
The solutions to the equation [tex]\(|x-5| + 7 = 17\)[/tex] are [tex]\(x = 15\)[/tex] and [tex]\(x = -5\)[/tex].
Thus, the correct choice is:
C. [tex]\(x=15\)[/tex] and [tex]\(x=-5\)[/tex]
