To solve the inequality \( ||x| - 1| < 1 - x \) for \( x \in \mathbb{R} \), we will break it down step by step.
### Step 1: Identify critical points
We need to identify the points where the expressions inside the absolute values change. The critical points occur when:
1. \( |x| - 1 = 0 \) → \( |x| = 1 \) → \( x = -1 \) or \( x = 1 \)
2. \( 1 - x = 0 \) → \( x = 1 \)
Thus, the critical points are \( -1, 0, 1 \).
### Step 2: Set up intervals
The critical points divide the real line into the following intervals:
1. \( (-\infty, -1) \)
2. \( [-1, 0) \)
3. \( [0, 1) \)
4. \( [1, \infty) \)
### Step 3: Test each interval
**Interval 1: \( (-\infty, -1) \)**
Choose \( x = -2 \):
\[
||-2| - 1| < 1 - (-2)
\]
Calculating:
\[
|2 - 1| < 3 \implies 1 < 3 \quad \text{(True)}
\]
Thus, this interval satisfies the inequality.
**Interval 2: \( [-1, 0) \)**
Choose \( x = -0.5 \):
\[
||-0.5| - 1| < 1 - (-0.5)
\]
Calculating:
\[
|0.5 - 1| < 1.5 \implies 0.5 < 1.5 \quad \text{(True)}
\]
Thus, this interval satisfies the inequality.
**Interval 3: \( [0, 1) \)**
Choose \( x = 0.5 \):
\[
||0.5| - 1| < 1 - 0.5
\]
Calculating:
\[
|0.5 - 1| < 0.5 \implies 0.5 < 0.5 \quad \text{(False)}
\]
Thus, this interval does not satisfy the inequality.
**Interval 4: \( [1, \infty) \)**
Choose \( x = 2 \):
\[
||2| - 1| < 1 - 2
\]
Calculating:
\[
|2 - 1| < -1 \implies 1 < -1 \quad \text{(False)}
\]
Thus, this interval does not satisfy the inequality.
### Step 4: Combine results
The solution set of the inequality \( ||x| - 1| < 1 - x \) is:
\[
(-\infty, -1) \cup [-1, 0)
\]
### Final Answer
The solution set is:
\[
(-\infty, 0)
\]
