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.