The solution set of the inequation `||x|-1| < | 1 – x|, x in RR,` is (i)`(-1,1)` (ii)`(0,infty)` (iii)`(-1,infty)` (iv) none of these

Options

A. `(-1, 1)`
B. `(0, oo)`
C. `(-1, oo)`
D. none of these

Answer

To solve the inequality \( ||x|-1| < |1-x| \), we will analyze it step by step.

### Step 1: Analyze the absolute values
We start with the inequality:
\[ ||x|-1| < |1-x| \]

### Step 2: Consider cases for \( x \)
We will consider two main cases based on the value of \( x \):

1. **Case 1:** \( x \geq 0 \)
2. **Case 2:** \( x < 0 \)

### Case 1: \( x \geq 0 \)
In this case, we have:
\[ |x| = x \]
Thus, the inequality becomes:
\[ |x - 1| < |1 - x| \]

Now, since \( |1 - x| = |x - 1| \), we rewrite the inequality as:
\[ |x - 1| < |x - 1| \]
This is not possible since a number cannot be less than itself. Therefore, there are no solutions for \( x \geq 0 \).

### Step 3: Move to Case 2
Now we consider:
**Case 2:** \( x < 0 \)

In this case, we have:
\[ |x| = -x \]
Thus, the inequality becomes:
\[ |-x - 1| < |1 - x| \]

### Step 4: Simplify the inequality
Now we simplify both sides:
- The left side becomes:
\[ |-x - 1| = -x - 1 \quad (\text{since } -x - 1 < 0 \text{ for } x < 0) \]
- The right side becomes:
\[ |1 - x| = 1 - x \quad (\text{since } 1 - x > 0 \text{ for } x < 0) \]

So we have:
\[ -x - 1 < 1 - x \]

### Step 5: Solve the simplified inequality
Now we solve:
\[ -x - 1 < 1 - x \]
Adding \( x \) to both sides gives:
\[ -1 < 1 \]
This is always true, indicating that all values of \( x < 0 \) are solutions.

### Step 6: Combine the results
From both cases, we found:
- No solutions for \( x \geq 0 \)
- All \( x < 0 \) are solutions

Thus, the solution set is:
\[ (-\infty, 0) \]

### Step 7: Check against the options
The options provided were:
(i) \((-1, 1)\)
(ii) \((0, \infty)\)
(iii) \((-1, \infty)\)
(iv) none of these

Since our solution set is \((- \infty, 0)\), which is not listed among the options, the correct answer is:
**(iv) none of these.**