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What is the solution to [tex]|x-2|+3 > 17[/tex]?

A. [tex]x < -12[/tex] or [tex]x > 16[/tex]
B. [tex]x < -14[/tex] or [tex]x > 7[/tex]
C. [tex]-12 < x < 16[/tex]
D. [tex]-14 < x < 7[/tex]

Answer :

To solve the inequality [tex]\( |x-2| + 3 > 17 \)[/tex], we can follow these steps:

1. Isolate the Absolute Value: Start by moving the constant term to the other side of the inequality.
[tex]\[
|x-2| + 3 > 17
\][/tex]
Subtract 3 from both sides:
[tex]\[
|x-2| > 14
\][/tex]

2. Consider the Two Cases for the Absolute Value: The expression [tex]\( |x-2| > 14 \)[/tex] implies two possible scenarios:

- Case 1: [tex]\( x - 2 > 14 \)[/tex]
- Solve for [tex]\( x \)[/tex] by adding 2 to both sides:
[tex]\[
x > 16
\][/tex]

- Case 2: [tex]\( x - 2 < -14 \)[/tex]
- Solve for [tex]\( x \)[/tex] by adding 2 to both sides:
[tex]\[
x < -12
\][/tex]

3. Combine the Solutions: Since the original inequality involves an absolute value, the solution is a combination of the two cases above, representing all values of [tex]\( x \)[/tex] that satisfy either condition.
[tex]\[
x < -12 \quad \text{or} \quad x > 16
\][/tex]

Thus, the solution to the inequality [tex]\( |x-2| + 3 > 17 \)[/tex] is [tex]\( x < -12 \)[/tex] or [tex]\( x > 16 \)[/tex].

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