To find the expression representing [tex]\( PS \)[/tex], we need to understand that [tex]\( PS \)[/tex] would be the sum of the segments [tex]\( PR \)[/tex] and [tex]\( RS \)[/tex]. Here are the step-by-step calculations:
1. We are given:
[tex]\[
PR = 4x - 2
\][/tex]
[tex]\[
RS = 3x - 5
\][/tex]
2. Since [tex]\( PS \)[/tex] is the total length consisting of both [tex]\( PR \)[/tex] and [tex]\( RS \)[/tex]:
[tex]\[
PS = PR + RS
\][/tex]
3. Substitute the given expressions for [tex]\( PR \)[/tex] and [tex]\( RS \)[/tex] into the equation for [tex]\( PS \)[/tex]:
[tex]\[
PS = (4x - 2) + (3x - 5)
\][/tex]
4. Combine the like terms:
[tex]\[
PS = 4x + 3x - 2 - 5
\][/tex]
5. Simplify the expression by adding the coefficients of [tex]\( x \)[/tex] and the constant terms:
[tex]\[
PS = 7x - 7
\][/tex]
After following these steps, the expression that represents [tex]\( PS \)[/tex] is:
[tex]\[
7x - 7
\][/tex]
So, the correct choice is:
[tex]\[
\boxed{7x - 7}
\][/tex]
