Solve |x|= 2x-1

Explanation

To solve the equation |x| = 2x - 1, we will consider two cases based on the definition of absolute value.

### Step 1: Consider Case 1 (x < 0)
In this case, the absolute value of x is negative, so we can rewrite the equation as:
\[ -x = 2x - 1 \]
Now, we will solve for x.

### Step 2: Rearranging the equation
Adding x to both sides gives:
\[ 0 = 3x - 1 \]
Now, we can isolate x by adding 1 to both sides:
\[ 3x = 1 \]
Dividing both sides by 3 gives:
\[ x = \frac{1}{3} \]

### Step 3: Check the condition for Case 1
However, we initially assumed that x < 0. Since \(\frac{1}{3}\) is not less than 0, there is no solution in this case.

### Step 4: Consider Case 2 (x ≥ 0)
In this case, the absolute value of x is simply x, so we can rewrite the equation as:
\[ x = 2x - 1 \]
Now, we will solve for x.

### Step 5: Rearranging the equation
Subtracting 2x from both sides gives:
\[ -x = -1 \]
Now, multiplying both sides by -1 gives:
\[ x = 1 \]

### Step 6: Check the condition for Case 2
Since x = 1 is greater than or equal to 0, this solution is valid.

### Conclusion
The solution to the equation |x| = 2x - 1 is:
\[ x = 1 \]

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