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]."
[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]."
