Question

solve the equation for all values of x.\
ewline $|2x - 3| = x$

Explanation:

Step1: Consider the definition of absolute value

The absolute value equation \(|A| = B\) (where \(B\geq0\)) implies \(A = B\) or \(A=-B\). So for \(|2x - 3|=x\), we have two cases:
Case 1: \(2x - 3=x\)
Case 2: \(2x - 3=-x\)

Step2: Solve Case 1: \(2x - 3=x\)

Subtract \(x\) from both sides: \(2x - x- 3=x - x\)
Simplify: \(x - 3 = 0\)
Add 3 to both sides: \(x=3\)

Step3: Solve Case 2: \(2x - 3=-x\)

Add \(x\) to both sides: \(2x+x - 3=-x + x\)
Simplify: \(3x - 3=0\)
Add 3 to both sides: \(3x=3\)
Divide both sides by 3: \(x = 1\)

Step4: Check the solutions

For \(x = 3\): Left - hand side of the original equation \(|2(3)-3|=|6 - 3|=|3| = 3\), Right - hand side is \(3\). So \(x = 3\) is valid.
For \(x = 1\): Left - hand side of the original equation \(|2(1)-3|=|2 - 3|=|-1| = 1\), Right - hand side is \(1\). So \(x = 1\) is valid.
We also need to ensure that \(x\geq0\) (since the right - hand side of \(|2x - 3|=x\) must be non - negative as it is equal to an absolute value). Both \(x = 1\) and \(x = 3\) are non - negative.

Answer:

The solutions are \(x = 1\) and \(x = 3\)