Let's break down the given information to solve the problem.

We are told that:

• $\overline{BD}$ bisects $\angle ABC$,
• $m\angle DBC = 79^\circ$,
• $m\angle ABC = (9x - 4)^\circ$.

Since $BD$ bisects $\angle ABC$, it divides $\angle ABC$ into two equal parts. This means: $m\angle ABC = 2 \times m\angle DBC.$ Therefore, we have the equation: $9x - 4 = 2 \times 79.$

Now, let's solve for $x$.

$9x - 4 = 158$ $9x = 158 + 4$ $9x = 162$ $x = \frac{162}{9} = 18$

Thus, the value of $x$ is $\boxed{18}$.

Would you like further details or have any questions?

Here are 5 related questions to expand on this concept:

1. What is the measure of $\angle ABC$ after solving for $x$?
2. How would the problem change if $BD$ did not bisect $\angle ABC$?
3. Can you calculate the sum of the interior angles at point $B$?
4. What would the value of $x$ be if $m\angle ABC = (10x - 5)^\circ$ and $m\angle DBC = 85^\circ$?
5. How does the concept of angle bisectors relate to the incenter of a triangle?

Tip: When solving angle bisector problems, remember that the bisected angle is always split into two equal halves, which can help simplify your calculations.