To solve for [tex]\( x \)[/tex], we need to use the fact that points [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] are collinear, and point [tex]\( B \)[/tex] is between [tex]\( A \)[/tex] and [tex]\( C \)[/tex].
Given:
[tex]\[
AB = 5
\][/tex]
[tex]\[
BC = 3x + 7
\][/tex]
[tex]\[
AC = 5x - 2
\][/tex]
Since point [tex]\( B \)[/tex] is between [tex]\( A \)[/tex] and [tex]\( C \)[/tex], the total length [tex]\( AC \)[/tex] can be expressed as the sum of the segments [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex]:
[tex]\[
AB + BC = AC
\][/tex]
Substituting the given lengths into this equation:
[tex]\[
5 + (3x + 7) = 5x - 2
\][/tex]
Next, combine the constants and the terms involving [tex]\( x \)[/tex]:
[tex]\[
5 + 3x + 7 = 5x - 2
\][/tex]
Simplify the left side of the equation:
[tex]\[
12 + 3x = 5x - 2
\][/tex]
To isolate the variable [tex]\( x \)[/tex], move all terms involving [tex]\( x \)[/tex] to one side of the equation and constants to the other:
[tex]\[
12 + 2 = 5x - 3x
\][/tex]
Simplify both sides:
[tex]\[
14 = 2x
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{14}{2} = 7
\][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( 7 \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{7}
\][/tex]
