Given that [tex]\( B \)[/tex] is between [tex]\( A \)[/tex] and [tex]\( C \)[/tex], [tex]\( AB = 2x + 6 \)[/tex], [tex]\( BC = 6x - 10 \)[/tex], and [tex]\( AC = 36 \)[/tex], find [tex]\( x \)[/tex].

A. [tex]\( x = 9.5 \)[/tex]

B. [tex]\( x = 4 \)[/tex]

C. [tex]\( x = 5 \)[/tex]

D. [tex]\( x = 7 \)[/tex]



Answer :

To find the value of [tex]\( x \)[/tex] given that [tex]\( AB = 2x + 6 \)[/tex], [tex]\( BC = 6x - 10 \)[/tex], and [tex]\( AC = 36 \)[/tex] with point B between points A and C, follow these steps:

1. Understand the Given Information:
- [tex]\( AB = 2x + 6 \)[/tex]
- [tex]\( BC = 6x - 10 \)[/tex]
- [tex]\( AC = 36 \)[/tex]

2. Establish the Relationship:
Since point B is between points A and C, we can express [tex]\( AC \)[/tex] as the sum of [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex]:
[tex]\[ AC = AB + BC \][/tex]

3. Set Up the Equation:
Substitute the given expressions for [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex]:
[tex]\[ 36 = (2x + 6) + (6x - 10) \][/tex]

4. Simplify the Equation:
Combine like terms on the right-hand side:
[tex]\[ 36 = 2x + 6 + 6x - 10 \][/tex]
[tex]\[ 36 = 8x - 4 \][/tex]

5. Solve for [tex]\( x \)[/tex]:
Add 4 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 36 + 4 = 8x \][/tex]
[tex]\[ 40 = 8x \][/tex]

Divide both sides by 8:
[tex]\[ x = \frac{40}{8} \][/tex]
[tex]\[ x = 5 \][/tex]

Therefore, the correct value of [tex]\( x \)[/tex] is [tex]\( 5 \)[/tex], which is indeed one of the given answer choices.

[tex]\[ \boxed{5} \][/tex]