To solve the equation \(|2x - 3| = |-x|\), we need to consider the absolute value properties. Absolute values represent the distance from zero, which means:

\[
|a| = |b| \implies a = b \quad \text{or} \quad a = -b
\]

Let's apply this property to solve the equation.

### Step 1: Set up two cases based on absolute value

The equation \(|2x - 3| = |-x|\) translates to:

1. \(2x - 3 = -x\)
2. \(2x - 3 = x\)

### Step 2: Solve each case

**Case 1: \(2x - 3 = -x\)**

\[
2x + x = 3 \implies 3x = 3 \implies x = 1
\]

**Case 2: \(2x - 3 = x\)**

\[
2x - x = 3 \implies x = 3
\]

### Step 3: Verify the solutions

We substitute the values back into the original equation to check if they satisfy it:

- For \(x = 1\):

\[
|2(1) - 3| = |-1| \implies |2 - 3| = 1 \implies | -1 | = 1 \quad (\text{True})
\]

- For \(x = 3\):

\[
|2(3) - 3| = |-3| \implies |6 - 3| = 3 \implies | -3 | = 3 \quad (\text{True})
\]

### Conclusion

Both solutions satisfy the original equation, so the solutions are:

\[
\boxed{x = 1 \text{ or } x = 3}
\]

Would you like more details, or do you have any questions?

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Here are 5 related questions to expand your understanding:

1. How do you solve absolute value inequalities, such as \(|x - 2| \leq 5\)?
2. What are the general properties of absolute values that can help simplify equations?
3. Can absolute value equations have no solutions? If so, under what conditions?
4. How would the solution change if the equation were \(|2x - 3| = |x + 1|\)?
5. How can you graph the equation \(|2x - 3| = |-x|\) to visualize the solutions?

**Tip:** When solving absolute value equations, always consider both the positive and negative cases to find all potential solutions.

Learn how to solve the absolute value equation |2x - 3| = |-x| with a detailed step-by-step solution. This problem covers key algebraic concepts involving absolute values and linear equations. Suitable for students in Grades 9-12.

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