https://file.aiquickdraw.com/mathbot/file/1726799475412u35p1aq7.jpg

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!

Learn how to solve the absolute value equation |x - 1| = 2x + 1 with a detailed solution. This problem covers key algebraic concepts and absolute value properties, suitable for students in Grades 9-10.

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