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

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

Answer :

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

1. Isolate the Absolute Value:
Begin by getting the absolute value expression on its own. You can do this by subtracting 7 from both sides of the equation:
[tex]\[
|x-5| + 7 - 7 = 17 - 7
\][/tex]
[tex]\[
|x-5| = 10
\][/tex]

2. Solve the Absolute Value Equation:
The equation [tex]\( |x-5| = 10 \)[/tex] means that the expression inside the absolute value, [tex]\( x-5 \)[/tex], can be either [tex]\( 10 \)[/tex] or [tex]\( -10 \)[/tex]:

- Case 1: [tex]\( x-5 = 10 \)[/tex]
[tex]\[
x = 10 + 5
\][/tex]
[tex]\[
x = 15
\][/tex]

- Case 2: [tex]\( x-5 = -10 \)[/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].

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

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