To determine the expression for [tex]\( PS \)[/tex] given that [tex]\( PR = 4x - 2 \)[/tex] and [tex]\( RS = 3x - 5 \)[/tex], we need to add the two expressions [tex]\( PR \)[/tex] and [tex]\( RS \)[/tex].
Let's proceed step-by-step:
1. Write down the given expressions:
[tex]\[
PR = 4x - 2
\][/tex]
[tex]\[
RS = 3x - 5
\][/tex]
2. To find [tex]\( PS \)[/tex], add the expressions for [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]:
[tex]\[
PS = (4x - 2) + (3x - 5)
\][/tex]
4. Combine the like terms (terms involving [tex]\( x \)[/tex] with terms involving [tex]\( x \)[/tex], and constant terms with constant terms):
[tex]\[
PS = 4x + 3x - 2 - 5
\][/tex]
5. Simplify the expression:
[tex]\[
PS = (4x + 3x) + (-2 - 5)
\][/tex]
[tex]\[
PS = 7x - 7
\][/tex]
So, the expression that represents [tex]\( PS \)[/tex] is:
[tex]\[
7x - 7
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{7x - 7}
\][/tex]
