To solve for [tex]\( x \)[/tex] given the collinear points [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] where [tex]\( B \)[/tex] is between [tex]\( A \)[/tex] and [tex]\( C \)[/tex], we can use a property of collinear points: the sum of segments [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex] should equal the segment [tex]\( AC \)[/tex].
Given:
- [tex]\( AB = 5x - 1 \)[/tex]
- [tex]\( BC = 14 \)[/tex]
- [tex]\( AC = 25 - x \)[/tex]
We can write the equation:
[tex]\[ AB + BC = AC \][/tex]
Substituting the given lengths:
[tex]\[ (5x - 1) + 14 = 25 - x \][/tex]
Next, combine the constant terms on the left side:
[tex]\[ 5x - 1 + 14 = 25 - x \][/tex]
[tex]\[ 5x + 13 = 25 - x \][/tex]
Now, move the [tex]\( x \)[/tex] terms to one side of the equation and the constants to the other side to isolate [tex]\( x \)[/tex]:
[tex]\[ 5x + x = 25 - 13 \][/tex]
[tex]\[ 6x = 12 \][/tex]
Divide both sides by 6:
[tex]\[ x = 2 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[ \boxed{2} \][/tex]
