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]
