Which of the following expressions is equal to 0 for some value of x ?

Options

A. ` | x - 1 | - 1 `
B. ` | x + 1 | + 1 `
C. ` | 1 - x | + 1 `
D. `| x - 1 | + 1 `

Explanation

To determine which of the given expressions is equal to 0 for some value of \( x \), we will analyze each option step by step.

### Step 1: Analyze Option 1
**Expression:** \( |x - 1| - 1 \)

Set the expression equal to 0:
\[ |x - 1| - 1 = 0 \]
Rearranging gives:
\[ |x - 1| = 1 \]

This absolute value equation can be split into two cases:
1. \( x - 1 = 1 \)
2. \( x - 1 = -1 \)

**Solving Case 1:**
\[ x - 1 = 1 \implies x = 2 \]

**Solving Case 2:**
\[ x - 1 = -1 \implies x = 0 \]

Thus, for Option 1, the expression is equal to 0 for \( x = 2 \) and \( x = 0 \).

### Step 2: Analyze Option 2
**Expression:** \( |x + 1| + 1 \)

Set the expression equal to 0:
\[ |x + 1| + 1 = 0 \]
Rearranging gives:
\[ |x + 1| = -1 \]

Since the absolute value cannot be negative, there are no values of \( x \) that satisfy this equation. Thus, Option 2 does not equal 0 for any \( x \).

### Step 3: Analyze Option 3
**Expression:** \( |1 - x| + 1 \)

Set the expression equal to 0:
\[ |1 - x| + 1 = 0 \]
Rearranging gives:
\[ |1 - x| = -1 \]

Again, since the absolute value cannot be negative, there are no values of \( x \) that satisfy this equation. Thus, Option 3 does not equal 0 for any \( x \).

### Step 4: Analyze Option 4
**Expression:** \( |x - 1| + 1 \)

Set the expression equal to 0:
\[ |x - 1| + 1 = 0 \]
Rearranging gives:
\[ |x - 1| = -1 \]

As with the previous options, the absolute value cannot be negative, so there are no values of \( x \) that satisfy this equation. Thus, Option 4 does not equal 0 for any \( x \).

### Conclusion
After analyzing all the options:
- **Option 1** is the only expression that equals 0 for some values of \( x \) (specifically \( x = 2 \) and \( x = 0 \)).
- **Options 2, 3, and 4** do not equal 0 for any value of \( x \).

Thus, the answer is **Option 1**.

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