Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ 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]

We start with the inequality:
[tex]$$
|x-2| + 3 > 17.
$$[/tex]

Step 1. Isolate the absolute value

Subtract 3 from both sides:
[tex]$$
|x-2| > 17 - 3 \quad \Rightarrow \quad |x-2| > 14.
$$[/tex]

Step 2. Solve the absolute value inequality

The inequality [tex]$|x-2| > 14$[/tex] means that the expression inside the absolute value is either greater than 14 or less than -14.

Case 1:
[tex]$$
x-2 > 14 \quad \Rightarrow \quad x > 14+2 \quad \Rightarrow \quad x > 16.
$$[/tex]

Case 2:
[tex]$$
x-2 < -14 \quad \Rightarrow \quad x < -14+2 \quad \Rightarrow \quad x < -12.
$$[/tex]

Step 3. Write the final solution

The solution to the inequality is:
[tex]$$
x < -12 \quad \text{or} \quad x > 16.
$$[/tex]

This corresponds to the answer choice:
[tex]$$
x < -12 \text{ or } x > 16.
$$[/tex]