The solution of `| |x|-1|lt|1-x|,x in R` is

Options

A. `(-1,1)`
B. `(o,oo)`
C. `(-1,oo)`
D. None of these

Answer

To solve the inequality \( ||x| - 1| < |1 - x| \), let's break it down step by step.

### Step 1: Understand the expressions involved
We have two expressions involving absolute values:
1. \( ||x| - 1| \)
2. \( |1 - x| \)

### Step 2: Analyze the first expression \( ||x| - 1| \)
The expression \( |x| \) represents the distance of \( x \) from 0 on the number line. The expression \( ||x| - 1| \) can be interpreted as the distance of \( |x| \) from 1.

### Step 3: Analyze the second expression \( |1 - x| \)
The expression \( |1 - x| \) represents the distance of \( x \) from 1.

### Step 4: Set up cases based on the definition of absolute values
To solve the inequality, we need to consider different cases for \( x \) based on the critical points where the expressions change, which are at \( x = -1, 0, 1 \).

#### Case 1: \( x < -1 \)
- Here, \( |x| = -x \) (since \( x \) is negative).
- Thus, \( ||x| - 1| = |-x - 1| = -x - 1 \).
- For \( |1 - x| = |1 + (-x)| = 1 + (-x) = 1 - x \).

The inequality becomes:
\[ -x - 1 < 1 - x \]
This simplifies to:
\[ -1 < 1 \quad \text{(which is always true)} \]
Thus, for \( x < -1 \), the inequality holds.

#### Case 2: \( -1 \leq x < 0 \)
- Here, \( |x| = -x \).
- Thus, \( ||x| - 1| = |-x - 1| = -x - 1 \).
- For \( |1 - x| = 1 - x \).

The inequality becomes:
\[ -x - 1 < 1 - x \]
This simplifies to:
\[ -1 < 1 \quad \text{(which is always true)} \]
Thus, for \( -1 \leq x < 0 \), the inequality holds.

#### Case 3: \( 0 \leq x < 1 \)
- Here, \( |x| = x \).
- Thus, \( ||x| - 1| = |x - 1| = 1 - x \).
- For \( |1 - x| = 1 - x \).

The inequality becomes:
\[ 1 - x < 1 - x \]
This is never true (as it simplifies to \( 0 < 0 \)), so there are no solutions in this interval.

#### Case 4: \( x \geq 1 \)
- Here, \( |x| = x \).
- Thus, \( ||x| - 1| = |x - 1| = x - 1 \).
- For \( |1 - x| = x - 1 \).

The inequality becomes:
\[ x - 1 < x - 1 \]
This is never true (as it simplifies to \( 0 < 0 \)), so there are no solutions in this interval.

### Step 5: Combine the results
From our analysis:
- The inequality holds for \( x < 0 \) (i.e., \( x \in (-\infty, 0) \)).
- The inequality does not hold for \( x \geq 0 \).

### Final Solution
Thus, the solution to the inequality \( ||x| - 1| < |1 - x| \) is:
\[ x \in (-\infty, 0) \]