What are the possible values of x in the absolute value equation |2x + 1| = |3x - 4|?

To solve the equation |2x + 1| = |3x - 4|, we must consider what absolute value means. The expression |A| = |B| indicates that A can equal B or A can equal -B.

Thus, we will set up two scenarios:

1. 2x + 1 = 3x - 4

2. 2x + 1 = -(3x - 4), which can also be rearranged.

For the first equation:

2x + 1 = 3x - 4

By rearranging this equation we can isolate x:

2x - 3x = -4 - 1

-x = -5

x = 5

For the second scenario:

2x + 1 = -3x + 4

Again we rearrange:

2x + 3x = 4 - 1

5x = 3

x = 3/5

The solutions derived from these calculations are x = 5 and x = 3/5, which aligns perfectly with the values presented in the first choice. By validating these solutions against the original absolute value equation, we can confirm that both satisfy the condition set by