The problem states that $B$ is the midpoint of $AC$, and asks to find the length of $BC$. The given distances are:

• $AB = 3(3x - 1)$
• $BC = 5(2x + 2)$

Since $B$ is the midpoint of $AC$, the length of $AB$ must equal the length of $BC$. Therefore, we can set up the following equation:

$3(3x - 1) = 5(2x + 2)$

Solving the equation:

1. Expand both sides: $9x - 3 = 10x + 10$

2. Rearrange the terms: $9x - 10x = 10 + 3$

$-x = 13$

3. Solve for $x$: $x = -13$

Finding the length of $BC$:

Now substitute $x = -13$ into the expression for $BC$:

$BC = 5(2(-13) + 2) = 5(-26 + 2) = 5(-24) = -120$

Since length can't be negative, the absolute value is taken:

$BC = 120$

Thus, the length of $BC$ is 120 units.

Would you like further details or have any questions?

Here are 5 related questions you might find useful:

1. How do you find the length of a segment when given its midpoint?
2. What happens if $B$ is not the midpoint of $AC$?
3. How do you interpret a negative result in distance problems?
4. Can we generalize the midpoint formula for any pair of points?
5. What other applications can you find for this type of problem?

Tip: Always check the conditions of the problem, like midpoint or symmetry, as they often simplify the calculation!