Short Answer 
The solution involves establishing a ratio between segments AB and AC using the angle bisector theorem, leading to the equation AB/AC = 2/3. By cross-multiplying and solving the resulting equation, the value of x is found to be 10.
Step 1: Establish Ratios
Begin by recognizing that the line from point A serves as an angle bisector, which creates a specific ratio between the segments AB and AC. The ratios are set as follows:
- AB = x + 4
- AC = 2x + 1
Thus, we equate the ratios: AB/AC = 8/12 simplifies to 2/3.
Step 2: Cross-Multiply and Distribute
Next, apply the cross-multiplication technique to solve for x. This involves:
- Cross-multiplying: 8(2x + 1) = 12(x + 4)
- Distributing the terms on each side:
- Left side: 16x + 8
- Right side: 12x + 48
This gives you the equation: 16x + 8 = 12x + 48.
Step 3: Solve for x
To isolate x, perform algebraic operations on the equation derived from the previous step:
- Subtract 12x from both sides to yield 4x + 8 = 48.
- Next, subtract 8 from both sides, resulting in 4x = 40.
- Finally, divide both sides by 4 to find the value: x = 10.