To find the value of [tex]\( x \)[/tex] given that points [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] are collinear and point [tex]\( B \)[/tex] is between points [tex]\( A \)[/tex] and [tex]\( C \)[/tex], we need to establish a relationship between the distances [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( AC \)[/tex].
We are given the following information:
- [tex]\( AB = 6x \)[/tex]
- [tex]\( BC = x - 5 \)[/tex]
- [tex]\( AC = 23 \)[/tex]
Since [tex]\( B \)[/tex] is between [tex]\( A \)[/tex] and [tex]\( C \)[/tex], the total distance from [tex]\( A \)[/tex] to [tex]\( C \)[/tex] is the sum of the distances from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] and from [tex]\( B \)[/tex] to [tex]\( C \)[/tex]. Hence, we can write the equation:
[tex]\[
AB + BC = AC
\][/tex]
Substituting the given expressions for [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( AC \)[/tex] into the equation, we get:
[tex]\[
6x + (x - 5) = 23
\][/tex]
Now, combine like terms:
[tex]\[
6x + x - 5 = 23
\][/tex]
[tex]\[
7x - 5 = 23
\][/tex]
Next, solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex]. Add 5 to both sides of the equation to get:
[tex]\[
7x = 28
\][/tex]
Finally, divide both sides by 7:
[tex]\[
x = 4
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[
\boxed{4}
\][/tex]
