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Solve [tex]|x-5|+7=17[/tex]

A. [tex]x=15[/tex] and [tex]x=-15[/tex]
B. [tex]x=-15[/tex] and [tex]x=5[/tex]
C. [tex]x=-15[/tex] and [tex]x=-5[/tex]
D. [tex]x=15[/tex] and [tex]x=-5[/tex]

Answer :

To solve the equation [tex]\( |x-5| + 7 = 17 \)[/tex], let's follow the steps below:

1. Isolate the absolute value:
- Start with the original equation: [tex]\( |x-5| + 7 = 17 \)[/tex].
- Subtract 7 from both sides to isolate the expression with the absolute value:
[tex]\[
|x-5| = 17 - 7
\][/tex]
[tex]\[
|x-5| = 10
\][/tex]

2. Consider the two cases for the absolute value:
- The equation [tex]\( |x-5| = 10 \)[/tex] means the expression inside the absolute value can be either 10 or -10. So, we have two cases to consider:

Case 1:
[tex]\[
x - 5 = 10
\][/tex]
- Add 5 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 10 + 5
\][/tex]
[tex]\[
x = 15
\][/tex]

Case 2:
[tex]\[
x - 5 = -10
\][/tex]
- Add 5 to both sides to solve for [tex]\( x \)[/tex]:
[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].

Therefore, the correct answer is:
D. [tex]\(x = 15\)[/tex] and [tex]\(x = -5\)[/tex]

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