Solve |x-1|-|2x-5|=2x

Explanation

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 absolute value expressions. The critical points are where the expressions inside the absolute values equal zero, which are \(x = 1\) and \(x = \frac{5}{2}\).

### Step 1: Identify the critical points
The critical points are:
- \(x = 1\) (where \(|x - 1|\) changes)
- \(x = \frac{5}{2}\) (where \(|2x - 5|\) changes)

### Step 2: Consider the intervals
We will consider three intervals based on the critical points:
1. \(x < 1\)
2. \(1 \leq x < \frac{5}{2}\)
3. \(x \geq \frac{5}{2}\)

### Step 3: Case 1: \(x < 1\)
In this case:
- \(|x - 1| = -(x - 1) = -x + 1\)
- \(|2x - 5| = -(2x - 5) = -2x + 5\)

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

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

Substituting into the equation:
\[ x - 1 - (-2x + 5) = 2x \]
This simplifies to:
\[ x - 1 + 2x - 5 = 2x \]
\[ 3x - 6 = 2x \]
\[ x = 6 \]
Since \(6\) is not in the interval \([1, \frac{5}{2})\), this is not a valid solution.

### Step 5: Case 3: \(x \geq \frac{5}{2}\)
In this case:
- \(|x - 1| = x - 1\)
- \(|2x - 5| = 2x - 5\)

Substituting into the equation:
\[ x - 1 - (2x - 5) = 2x \]
This simplifies to:
\[ x - 1 - 2x + 5 = 2x \]
\[ - x + 4 = 2x \]
\[ 4 = 3x \implies x = \frac{4}{3} \]
Since \(\frac{4}{3} < \frac{5}{2}\), this is not a valid solution.

### Conclusion
The valid solutions from our cases are:
1. \(x = -4\)

Thus, the final solutions are:
\[ x = -4 \]