To solve for [tex]\( x \)[/tex] given the collinear points [tex]\( P \)[/tex], [tex]\( Q \)[/tex], and [tex]\( R \)[/tex] with [tex]\( Q \)[/tex] lying between [tex]\( P \)[/tex] and [tex]\( R \)[/tex], we start with the given distances:
[tex]\[ PR = 5x \][/tex]
[tex]\[ PQ = 10 \][/tex]
[tex]\[ QR = 3x - 2 \][/tex]
Since [tex]\( Q \)[/tex] lies between [tex]\( P \)[/tex] and [tex]\( R \)[/tex], the distance [tex]\( PR \)[/tex] can be expressed as the sum of the distances [tex]\( PQ \)[/tex] and [tex]\( QR \)[/tex]:
[tex]\[ PR = PQ + QR \][/tex]
Substituting the given distances into the equation:
[tex]\[ 5x = 10 + (3x - 2) \][/tex]
First, simplify the right-hand side of the equation:
[tex]\[ 5x = 10 + 3x - 2 \][/tex]
[tex]\[ 5x = 3x + 8 \][/tex]
Next, isolate [tex]\( x \)[/tex] by subtracting [tex]\( 3x \)[/tex] from both sides:
[tex]\[ 5x - 3x = 8 \][/tex]
[tex]\[ 2x = 8 \][/tex]
To find the value of [tex]\( x \)[/tex], divide both sides by 2:
[tex]\[ x = \frac{8}{2} \][/tex]
[tex]\[ x = 4 \][/tex]
Thus, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{4} \)[/tex].
