The given equation is:

$|x - 1| = 2x + 1$

To solve this, we need to consider two cases for the absolute value expression.

Case 1: $x - 1 \geq 0$ (which means $x \geq 1$)

In this case, we can remove the absolute value directly:

$x - 1 = 2x + 1$

Solve for $x$:

$x - 1 = 2x + 1$ $x - 2x = 1 + 1$ $-x = 2$ $x = -2$

Since $x = -2$ is not greater than or equal to 1, this solution is not valid for Case 1.

Case 2: $x - 1 < 0$ (which means $x < 1$)

In this case, the absolute value becomes $-(x - 1)$, so we have:

$-(x - 1) = 2x + 1$

Simplify and solve for $x$:

$-x + 1 = 2x + 1$ $1 = 3x + 1$ $0 = 3x$ $x = 0$

Since $x = 0$ is less than 1, this solution is valid for Case 2.

Final Answer:

The solution to the equation is:

$x = 0$

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. What happens if the right-hand side of the equation is a negative value?
2. How would you solve $|x + 3| = 4x - 5$?
3. Can absolute value equations have no solution? Why or why not?
4. How do you graph the solution to $|x - 1| = 2x + 1$?
5. What are the properties of absolute value functions?

Tip: Always check the validity of solutions in both cases when solving absolute value equations to avoid extraneous solutions!