College

What is the solution set of the system shown below?

[tex]\left\{\begin{array}{c} x-y\ \textless \ 7 \\ 3x+2y \geq 15 \end{array}\right.[/tex]

Choose the correct answer below.

A. The solution set is the set of ordered pairs that satisfy [tex]x-y\ \textless \ 7[/tex] or [tex]3x+2y \geq 15[/tex].

B. The solution set is the set of ordered pairs that satisfy [tex]x-y\ \textgreater \ 7[/tex] or [tex]3x+2y \geq 15[/tex].

C. The solution set is the set of ordered pairs that satisfy [tex]x-y\ \textless \ 7[/tex] and [tex]3x+2y \geq 15[/tex].

D. The solution set is the set of ordered pairs that satisfy [tex]x-y\ \textgreater \ 7[/tex] and [tex]3x+2y \geq 15[/tex].

Answer :

To solve the system of inequalities and determine the solution set, let's analyze each part of the system:

1. First Inequality: [tex]\(x - y < 7\)[/tex]
- This inequality represents a region below the line [tex]\(x - y = 7\)[/tex]. The solution set includes all points below this line.

2. Second Inequality: [tex]\(3x + 2y \geq 15\)[/tex]
- This inequality represents a region above the line [tex]\(3x + 2y = 15\)[/tex]. The solution set includes all points on or above this line.

To find the overall solution set, we need the set of ordered pairs [tex]\((x, y)\)[/tex] that satisfy both inequalities simultaneously:

- For the solution set, any ordered pair [tex]\((x, y)\)[/tex] must satisfy both:
- [tex]\(x - y < 7\)[/tex] (meaning the point is below the line [tex]\(x - y = 7\)[/tex])
- [tex]\(3x + 2y \geq 15\)[/tex] (meaning the point is on or above the line [tex]\(3x + 2y = 15\)[/tex])

Therefore, the correct choice is:
- C. The solution set is the set of ordered pairs that satisfy [tex]\(x-y<7\)[/tex] and [tex]\(3x+2y \geq 15\)[/tex].

This choice correctly identifies the requirement that both conditions must be met simultaneously for a point to be within the solution set of the system.