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The problem states that BD bisects ∠ABC. We're given the measures of ∠ABD and ∠CBD as expressions involving \(x\), and we're tasked with solving for \(x\) and finding the measure of ∠ABC.

1. Since BD bisects ∠ABC, it means that:
   \[
   m\angle ABD = m\angle CBD
   \]
   Using the expressions provided:
   \[
   5x - 3 = 3x + 3
   \]
   
2. To solve for \(x\):
   - Subtract \(3x\) from both sides:
     \[
     5x - 3x - 3 = 3 \quad \Rightarrow \quad 2x - 3 = 3
     \]
   - Add 3 to both sides:
     \[
     2x = 6
     \]
   - Divide both sides by 2:
     \[
     x = 3
     \]

3. Now that we have \(x = 3\), substitute it back into the expressions for the angles:
   - \(m\angle ABD = 5(3) - 3 = 15 - 3 = 12^\circ\)
   - \(m\angle CBD = 3(3) + 3 = 9 + 3 = 12^\circ\)

4. Since BD bisects ∠ABC, we can find:
   \[
   m\angle ABC = m\angle ABD + m\angle CBD = 12^\circ + 12^\circ = 24^\circ
   \]

Thus, \(x = 3\) and \(m\angle ABC = 24^\circ\).

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Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What happens if BD doesn't bisect ∠ABC? How would you solve it?
2. If ∠ABD were twice ∠CBD, how would the equation change?
3. How would you solve if one of the angle expressions involved a quadratic?
4. Can bisectors be used to solve problems in three-dimensional geometry?
5. How does the angle bisector theorem help in solving problems involving triangles?

**Tip:** In angle bisector problems, remember that the two bisected angles are always equal unless other conditions are specified.

BD bisects ∠ABC. Solve for x and find m∠ABC.
m∠ABD = 5x - 3, m∠CBD = 3x + 3

Solve for x and find the measure of ∠ABC when BD bisects ∠ABC. Using algebraic equations, this problem explores the Angle Bisector Theorem with step-by-step guidance suitable for students in Grades 8-10.

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