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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]

To solve the equation [tex]\( |x-5| + 7 = 17 \)[/tex], we can follow these steps:

1. Isolate the absolute value:
Start by getting the absolute value by itself on one side of the equation. Subtract 7 from both sides:
[tex]\[ |x-5| = 17 - 7 \][/tex]
[tex]\[ |x-5| = 10 \][/tex]

2. Set up the two possible equations:
An absolute value equation of the form [tex]\( |A| = B \)[/tex] results in two possible equations: [tex]\( A = B \)[/tex] and [tex]\( A = -B \)[/tex]. Here, [tex]\( A = x - 5 \)[/tex] and [tex]\( B = 10 \)[/tex].

- First equation:
[tex]\[ x - 5 = 10 \][/tex]

- Second equation:
[tex]\[ x - 5 = -10 \][/tex]

3. Solve each equation for [tex]\( x \)[/tex]:

- For the first equation [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]

- For the second equation [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]

4. Identify the solutions:
The solutions to the equation 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]