The number of solutions of the equation `|x-1|-|2x-5|=2x`

Answer

To solve the equation \( |x - 1| - |2x - 5| = 2x \), we need to consider the different cases based on the values of \( x \) that affect the modulus functions. The critical points where the expressions inside the modulus change signs are \( x = 1 \) and \( x = \frac{5}{2} \).

### Step 1: Identify Critical Points
The critical points are:
- \( x = 1 \)
- \( x = \frac{5}{2} \)

These points will help us break the number line into intervals to analyze the equation.

### Step 2: Break into Cases
We will analyze three cases based on the intervals determined by the critical points:

1. **Case 1:** \( x < 1 \)
2. **Case 2:** \( 1 \leq x < \frac{5}{2} \)
3. **Case 3:** \( x \geq \frac{5}{2} \)

### Step 3: Solve Case 1: \( x < 1 \)
In this case, both expressions inside the modulus are negative:
- \( |x - 1| = -(x - 1) = -x + 1 \)
- \( |2x - 5| = -(2x - 5) = -2x + 5 \)

Substituting these into the equation:
\[ -x + 1 - (-2x + 5) = 2x \]
This simplifies to:
\[ -x + 1 + 2x - 5 = 2x \]
\[ x - 4 = 2x \]
Rearranging gives:
\[ -x = 4 \quad \Rightarrow \quad x = -4 \]
Since \( -4 < 1 \), this solution is valid.

### Step 4: Solve Case 2: \( 1 \leq x < \frac{5}{2} \)
In this case:
- \( |x - 1| = x - 1 \)
- \( |2x - 5| = -(2x - 5) = -2x + 5 \)

Substituting these into the equation:
\[ x - 1 - (-2x + 5) = 2x \]
This simplifies to:
\[ x - 1 + 2x - 5 = 2x \]
\[ 3x - 6 = 2x \]
Rearranging gives:
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]
However, \( 6 \) is not in the interval \( [1, \frac{5}{2}) \), so this case yields no valid solutions.

### Step 5: Solve Case 3: \( x \geq \frac{5}{2} \)
In this case, both expressions inside the modulus are positive:
- \( |x - 1| = x - 1 \)
- \( |2x - 5| = 2x - 5 \)

Substituting these into the equation:
\[ x - 1 - (2x - 5) = 2x \]
This simplifies to:
\[ x - 1 - 2x + 5 = 2x \]
\[ -x + 4 = 2x \]
Rearranging gives:
\[ 4 = 3x \quad \Rightarrow \quad x = \frac{4}{3} \]
However, \( \frac{4}{3} \) is not in the interval \( [\frac{5}{2}, \infty) \), so this case also yields no valid solutions.

### Conclusion
The only valid solution we found is from Case 1:
\[ x = -4 \]
Thus, the number of solutions to the equation \( |x - 1| - |2x - 5| = 2x \) is **1**.