To solve this problem, let's break it down step-by-step:

We are given:

• AC = $10x + 6$
• AB = $5x$
• BC = $4x + 12$

Since point B is between points A and C, the sum of AB and BC must be equal to AC: $AB + BC = AC$

Substitute the given values: $5x + (4x + 12) = 10x + 6$

Now, simplify and solve for $x$: $5x + 4x + 12 = 10x + 6$ $9x + 12 = 10x + 6$

Move all terms involving $x$ to one side and constants to the other side: $9x - 10x = 6 - 12$ $-x = -6$

Divide by -1: $x = 6$

Thus, the value of $x$ is 6.

Answer: $x = 6$.

Would you like further details or have any questions about this solution?

Here are 5 related questions for practice:

1. If AC = $15x + 2$, AB = $3x$, and BC = $12x + 2$, what is the value of $x$?
2. What happens to the equation if $AB = 5x + 3$ instead of $5x$?
3. How do you solve for $x$ if AC, AB, and BC were in terms of quadratic expressions?
4. Can you solve a similar problem if B is not exactly between A and C but some distance away?
5. How would this problem change if the segments were part of a geometric figure like a triangle?

Tip:

Always simplify both sides of the equation as much as possible before moving terms around to avoid mistakes.