To solve the inequality [tex]\(1 < |x - 2| < 5\)[/tex], we need to break it down into two parts:
1. [tex]\(1 < |x - 2|\)[/tex]
2. [tex]\(|x - 2| < 5\)[/tex]
Let's deal with each part separately.
### Part 1: [tex]\(1 < |x - 2|\)[/tex]
An absolute value inequality of the form [tex]\(1 < |A|\)[/tex] translates to two separate inequalities:
[tex]\[ A < -1 \quad \text{or} \quad A > 1 \][/tex]
For [tex]\(1 < |x - 2|\)[/tex], this means:
[tex]\[ x - 2 < -1 \quad \text{or} \quad x - 2 > 1 \][/tex]
Solving these separately:
- [tex]\(x - 2 < -1\)[/tex]
[tex]\[
x < -1 + 2 \implies x < 1
\][/tex]
- [tex]\(x - 2 > 1\)[/tex]
[tex]\[
x > 1 + 2 \implies x > 3
\][/tex]
So, the solution set for this part is:
[tex]\[ x < 1 \quad \text{or} \quad x > 3 \][/tex]
### Part 2: [tex]\(|x - 2| < 5\)[/tex]
An absolute value inequality of this form translates to:
[tex]\[ -5 < x - 2 < 5 \][/tex]
Solving this inequality:
Add 2 to all parts of the inequality:
[tex]\[
-5 + 2 < x < 5 + 2
\][/tex]
[tex]\[
-3 < x < 7
\][/tex]
### Combine the Solutions
We need the values of [tex]\(x\)[/tex] that satisfy both conditions simultaneously. The first condition gives [tex]\(x < 1\)[/tex] or [tex]\(x > 3\)[/tex], and the second condition states [tex]\(-3 < x < 7\)[/tex].
- For [tex]\(x < 1\)[/tex], it must also lie within [tex]\(-3 < x < 7\)[/tex], hence [tex]\( -3 < x < 1 \)[/tex].
- For [tex]\(x > 3\)[/tex], it must also lie within [tex]\(-3 < x < 7\)[/tex], hence [tex]\( 3 < x < 7 \)[/tex].
So, combining these intervals, the solution set of the inequality [tex]\(1 < |x - 2| < 5\)[/tex] is:
[tex]\[(-3, 1) \cup (3, 7)\][/tex]
Therefore, the correct answer in each box is:
[tex]\[
(-3, 1) \cup (3, 7)
\][/tex]
