College

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 :

We start with the inequality:

[tex]$$
|x-2| + 3 > 17.
$$[/tex]

Step 1: Isolate the Absolute Value

Subtract 3 from both sides to get:

[tex]$$
|x-2| > 14.
$$[/tex]

Step 2: Solve the Absolute Value Inequality

An inequality of the form [tex]$|A| > B$[/tex], where [tex]$B > 0$[/tex], translates into two separate inequalities:

[tex]$$
A > B \quad \text{or} \quad A < -B.
$$[/tex]

In this case, [tex]$A = x-2$[/tex] and [tex]$B = 14$[/tex], so we have:

1. [tex]$$ x-2 > 14 $$[/tex]
2. [tex]$$ x-2 < -14 $$[/tex]

Step 3: Solve Each Inequality

1. For the first inequality:

[tex]\[
x-2 > 14 \quad \Longrightarrow \quad x > 14+2 \quad \Longrightarrow \quad x > 16.
\][/tex]

2. For the second inequality:

[tex]\[
x-2 < -14 \quad \Longrightarrow \quad x < -14+2 \quad \Longrightarrow \quad x < -12.
\][/tex]

Step 4: Combine the Solutions

The combined solution is:

[tex]$$
x < -12 \quad \text{or} \quad x > 16.
$$[/tex]

This corresponds to option 1: "[tex]$x < -12$[/tex] or [tex]$x > 16$[/tex]."