To solve the inequality [tex]\(1 < |x - 2| < 5\)[/tex], we will break it down into two separate inequalities because it involves an absolute value.
First, recall that [tex]\( |a| < b \)[/tex] means [tex]\(-b < a < b\)[/tex], and [tex]\( |a| > c \)[/tex] means [tex]\( a < - c \)[/tex] or [tex]\( a > c \)[/tex].
### Step 1: Solve [tex]\( |x - 2| < 5 \)[/tex]
[tex]\[
|x - 2| < 5 \implies -5 < x - 2 < 5
\][/tex]
Next, solve the compound inequality by isolating [tex]\(x\)[/tex].
[tex]\[
-5 < x - 2 \implies -5 + 2 < x \implies -3 < x
\][/tex]
[tex]\[
x - 2 < 5 \implies x < 5 + 2 \implies x < 7
\][/tex]
So, from [tex]\( |x - 2| < 5 \)[/tex] we get:
[tex]\[
-3 < x < 7
\][/tex]
### Step 2: Solve [tex]\( 1 < |x - 2| \)[/tex]
This absolute value inequality can be rewritten as two separate inequalities:
[tex]\[
|x - 2| > 1 \implies (x - 2 < -1 \quad \text{or} \quad x - 2 > 1)
\][/tex]
For [tex]\(x - 2 < -1\)[/tex]:
[tex]\[
x - 2 < -1 \implies x < -1 + 2 \implies x < 1
\][/tex]
For [tex]\(x - 2 > 1\)[/tex]:
[tex]\[
x - 2 > 1 \implies x > 1 + 2 \implies x > 3
\][/tex]
### Step 3: Combine the results
We've determined two conditions:
1. [tex]\( -3 < x < 7 \)[/tex]
2. [tex]\( x < 1 \)[/tex] or [tex]\( x > 3 \)[/tex]
We'll now find the intersection of these conditions.
[tex]\[
(-3 < x < 7) \cap (x < 1 \cup x > 3)
\][/tex]
This provides two intervals where both conditions are satisfied:
[tex]\[
(-3 < x < 1) \cup (3 < x < 7)
\][/tex]
So the solution set of the inequality [tex]\(1 < |x - 2| < 5\)[/tex] is:
[tex]\[
(-3, 1) \cup (3, 7)
\][/tex]
Thus, the correct values to fill in the blanks are:
[tex]\[
(-3, 1) \cup (3, 7)
\][/tex]
So, we have:
### Box 1: 1
### Box 2: 3
### Box 3: 7
