To find the value of [tex]\( x \)[/tex] given the lengths of segments on line segment [tex]\(\overline{AC}\)[/tex], we proceed with the following steps:
1. Set up the relationship between the segments:
- [tex]\( AB = x + 6 \)[/tex]
- [tex]\( BC = x + 8 \)[/tex]
- The total length of [tex]\( AC \)[/tex] is given as 10.
2. Write the equation relating the segments:
Since [tex]\( B \)[/tex] is a point on line segment [tex]\(\overline{AC}\)[/tex], we know that:
[tex]\[
AB + BC = AC
\][/tex]
Substituting the given values, we have:
[tex]\[
(x + 6) + (x + 8) = 10
\][/tex]
3. Simplify the equation:
Combine like terms:
[tex]\[
x + 6 + x + 8 = 10
\][/tex]
[tex]\[
2x + 14 = 10
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Isolate [tex]\( x \)[/tex] by first subtracting 14 from both sides:
[tex]\[
2x + 14 - 14 = 10 - 14
\][/tex]
[tex]\[
2x = -4
\][/tex]
Then, divide both sides by 2:
[tex]\[
x = \frac{-4}{2}
\][/tex]
[tex]\[
x = -2
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( -2 \)[/tex].
