High School

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

Answer :

Sure! Let's solve the equation [tex]\( |x-5| + 7 = 17 \)[/tex] step by step.

1. Isolate the Absolute Value:
Start by subtracting 7 from both sides of the equation to get rid of the constant term outside the absolute value:
[tex]\[
|x-5| + 7 - 7 = 17 - 7
\][/tex]
Simplifying both sides, we have:
[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 10 or -10 because absolute value denotes distance from zero, which can be in either direction.

- 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. Write the Solution:
The solutions to the equation [tex]\( |x-5| + 7 = 17 \)[/tex] are [tex]\( x = 15 \)[/tex] and [tex]\( x = -5 \)[/tex].

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

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