The possible value of `|x| + |x - 1| = 2` is :

Options

A. `(3/2, -1/2}`
B. `{3/2, 1/2}`
C. `{3/2, - 1/2}`
D. None of these

Answer

To solve the equation \( |x| + |x - 1| = 2 \), we will consider different cases based on the value of \( x \).

### Step 1: Consider the case when \( x < 0 \)

In this case, both \( |x| \) and \( |x - 1| \) can be expressed as:
- \( |x| = -x \) (since \( x \) is negative)
- \( |x - 1| = -(x - 1) = -x + 1 \) (since \( x - 1 \) is also negative)

Substituting these into the equation gives:
\[ -x + (-x + 1) = 2 \]
This simplifies to:
\[ -2x + 1 = 2 \]
Subtracting 1 from both sides:
\[ -2x = 1 \]
Dividing by -2:
\[ x = -\frac{1}{2} \]

### Step 2: Consider the case when \( 0 \leq x < 1 \)

In this case:
- \( |x| = x \) (since \( x \) is non-negative)
- \( |x - 1| = -(x - 1) = -x + 1 \) (since \( x - 1 \) is negative)

Substituting these into the equation gives:
\[ x + (-x + 1) = 2 \]
This simplifies to:
\[ 1 = 2 \]
This is a contradiction, so there are no solutions in this case.

### Step 3: Consider the case when \( x \geq 1 \)

In this case:
- \( |x| = x \)
- \( |x - 1| = x - 1 \) (since \( x - 1 \) is non-negative)

Substituting these into the equation gives:
\[ x + (x - 1) = 2 \]
This simplifies to:
\[ 2x - 1 = 2 \]
Adding 1 to both sides:
\[ 2x = 3 \]
Dividing by 2:
\[ x = \frac{3}{2} \]

### Conclusion

The possible values of \( x \) that satisfy the equation \( |x| + |x - 1| = 2 \) are:
- \( x = -\frac{1}{2} \)
- \( x = \frac{3}{2} \)

### Final Answer
The possible values of \( x \) are \( -\frac{1}{2} \) and \( \frac{3}{2} \).

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