Since ray $BE$ bisects $\angle ABC$, we know that:

$m\angle ABE = m\angle CBE$

Given that:

$m\angle ABE = (3x + 8)^\circ \quad \text{and} \quad m\angle CBE = (7x - 8)^\circ$

We set these two expressions equal to each other:

$3x + 8 = 7x - 8$

Now, solve for $x$:

$8 + 8 = 7x - 3x$

$16 = 4x$

$x = 4$

Now substitute $x = 4$ back into the expression for $m\angle CBE$:

$m\angle CBE = 7x - 8 = 7(4) - 8 = 28 - 8 = 20^\circ$

So, $m\angle CBE = 20^\circ$.

Would you like more details on any part of the solution? Here are some related questions:

1. How do you prove that a ray bisects an angle?
2. What are the properties of an angle bisector?
3. How do you solve for $x$ when two expressions are equal?
4. What is the relationship between adjacent angles when a line bisects an angle?
5. How do we verify the solution once we find the value of $x$?

Tip: When solving equations involving bisected angles, setting the two angles equal to each other is often the key step.