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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.
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.
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