What is the value of x if `|x-1|+|x|=2:`

Options

A. `-(1)/(2)`
B. `(3)/(2)`
C. both (a) and (b)
D. none of these

Answer

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

### Step 1: Identify the critical points
The critical points for the absolute values are where the expressions inside the absolute values are zero.
- For \( |x-1| \), the critical point is \( x = 1 \).
- For \( |x| \), the critical point is \( x = 0 \).

Thus, we will consider three intervals based on these critical points:
1. \( x < 0 \)
2. \( 0 \leq x < 1 \)
3. \( x \geq 1 \)

### Step 2: Case 1: \( x < 0 \)
In this case, both \( x-1 \) and \( x \) are negative. Therefore:
\[ |x-1| = -(x-1) = -x + 1 \]
\[ |x| = -x \]
Substituting these into the equation gives:
\[ (-x + 1) + (-x) = 2 \]
Simplifying this, we get:
\[ -2x + 1 = 2 \]
\[ -2x = 1 \]
\[ x = -\frac{1}{2} \]

### Step 3: Case 2: \( 0 \leq x < 1 \)
In this case, \( |x-1| \) is negative and \( |x| \) is positive. Therefore:
\[ |x-1| = -(x-1) = -x + 1 \]
\[ |x| = x \]
Substituting these into the equation gives:
\[ (-x + 1) + x = 2 \]
Simplifying this, we get:
\[ 1 = 2 \]
This is a contradiction, so there are no solutions in this interval.

### Step 4: Case 3: \( x \geq 1 \)
In this case, both \( x-1 \) and \( x \) are positive. Therefore:
\[ |x-1| = x-1 \]
\[ |x| = x \]
Substituting these into the equation gives:
\[ (x - 1) + x = 2 \]
Simplifying this, we get:
\[ 2x - 1 = 2 \]
\[ 2x = 3 \]
\[ x = \frac{3}{2} \]

### Step 5: Conclusion
The solutions we found are:
1. From Case 1: \( x = -\frac{1}{2} \)
2. From Case 3: \( x = \frac{3}{2} \)

Thus, the values of \( x \) that satisfy the equation \( |x-1| + |x| = 2 \) are:
\[ x = -\frac{1}{2} \quad \text{and} \quad x = \frac{3}{2} \]