To find the value of \( x \) given the points \( A, B, \) and \( C \) are collinear with \( B \) between \( A \) and \( C \) and the distances \( AB = 2x \), \( BC = x - 2 \), and \( AC = 28 \), follow these steps:
1. Set Up the Equation:
Since points \( A \), \( B \), and \( C \) are collinear and \( B \) is between \( A \) and \( C \), the total distance from \( A \) to \( C \) is the sum of the distances from \( A \) to \( B \) and from \( B \) to \( C \).
[tex]\[
AB + BC = AC
\][/tex]
Substitute the given distances:
[tex]\[
2x + (x - 2) = 28
\][/tex]
2. Combine Like Terms:
Combine the \( x \) terms on the left-hand side of the equation:
[tex]\[
2x + x - 2 = 28
\][/tex]
[tex]\[
3x - 2 = 28
\][/tex]
3. Solve for \( x \):
Isolate \( x \) by adding 2 to both sides of the equation:
[tex]\[
3x - 2 + 2 = 28 + 2
\][/tex]
[tex]\[
3x = 30
\][/tex]
Now, divide by 3:
[tex]\[
x = \frac{30}{3}
\][/tex]
[tex]\[
x = 10
\][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\(\boxed{10}\)[/tex].
