Solve for x in the following equation  | 2𝑥 + 1 3𝑥 − 1 | = 1

The given equation is:

\[
| 2x + 1 | = 3x - 1
\]

To solve this equation, we need to consider two cases, since the absolute value function splits the equation into two possible scenarios.

### Case 1: \( 2x + 1 = 3x - 1 \)

Solve for \( x \):

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

Subtract \( 2x \) from both sides:

\[
1 = x - 1
\]

Add 1 to both sides:

\[
x = 2
\]

### Case 2: \( 2x + 1 = -(3x - 1) \)

Simplify the right side:

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

Add \( 3x \) to both sides:

\[
5x + 1 = 1
\]

Subtract 1 from both sides:

\[
5x = 0
\]

Solve for \( x \):

\[
x = 0
\]

### Solution:
The two possible solutions are \( x = 2 \) and \( x = 0 \).

However, we need to check if these solutions satisfy the original equation by plugging them back in:

- For \( x = 2 \): 
\[
| 2(2) + 1 | = | 4 + 1 | = 5
\]
\[
3(2) - 1 = 6 - 1 = 5
\]
Since both sides are equal, \( x = 2 \) is a valid solution.

- For \( x = 0 \):
\[
| 2(0) + 1 | = | 0 + 1 | = 1
\]
\[
3(0) - 1 = 0 - 1 = -1
\]
Since the left side is not equal to the right side, \( x = 0 \) is **not** a valid solution.

Thus, the only solution is:

\[
x = 2
\]

Would you like more details or have any questions?

### Related questions:
1. What happens if we change the equation to \( | 2x + 1 | = 4x - 2 \)?
2. Can an absolute value equation have no solution? If so, under what conditions?
3. How does the solution change if \( | 2x + 1 | = -1 \)?
4. How do we solve absolute value inequalities?
5. What are the graphical interpretations of absolute value equations?

### Tip: 
Always check if solutions from absolute value equations are valid by plugging them back into the original equation.

Solve for x in the following equation | 2x + 1 | = 3x - 1

Learn how to solve the absolute value equation | 2x + 1 | = 3x - 1 with step-by-step instructions. This problem covers key algebraic concepts such as absolute value and linear equations, including checking the validity of solutions. Suitable for students in Grades 8-10.

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