To determine the value of [tex]\( x \)[/tex] given the points on the line segment [tex]\( AC \)[/tex] with [tex]\( AB = x + 6 \)[/tex], [tex]\( BC = x + 8 \)[/tex], and [tex]\( AC = 10 \)[/tex]:
1. Understand the Problem:
- We have a line segment [tex]\( AC \)[/tex] with point [tex]\( B \)[/tex] lying on it.
- The total length of [tex]\( AC \)[/tex] is given as 10 units.
- The lengths [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex] are given as expressions involving [tex]\( x \)[/tex].
2. Formulate the Equation:
Since [tex]\( B \)[/tex] is a point on the line segment [tex]\( AC \)[/tex], the lengths [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex] must sum up to the length [tex]\( AC \)[/tex]. Therefore, we set up the equation:
[tex]\[
AB + BC = AC
\][/tex]
3. Substitute the Given Expressions:
Substitute [tex]\( AB = x + 6 \)[/tex], [tex]\( BC = x + 8 \)[/tex], and [tex]\( AC = 10 \)[/tex]:
[tex]\[
(x + 6) + (x + 8) = 10
\][/tex]
4. Simplify the Equation:
Combine like terms on the left side of the equation:
[tex]\[
x + 6 + x + 8 = 10
\][/tex]
[tex]\[
2x + 14 = 10
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
- Isolate the term with [tex]\( x \)[/tex]:
[tex]\[
2x + 14 = 10
\][/tex]
Subtract 14 from both sides:
[tex]\[
2x = 10 - 14
\][/tex]
[tex]\[
2x = -4
\][/tex]
- Divide both sides by 2:
[tex]\[
x = \frac{-4}{2}
\][/tex]
[tex]\[
x = -2
\][/tex]
6. Conclusion:
The value of [tex]\( x \)[/tex] is:
[tex]\[
\boxed{-2}
\][/tex]
