Which of the following systems of inequalities would produce a solution set that includes the point (2, 3)? This question is a classic example of a problem that requires an understanding of how to solve systems of inequalities. In this article, we will explore different systems of inequalities and analyze which one would result in the specified solution set. By the end, you will have a clearer understanding of how to approach such problems and identify the correct system of inequalities that leads to the desired outcome.
In order to determine which system of inequalities would produce the solution set containing the point (2, 3), we must first understand the concept of systems of inequalities. A system of inequalities consists of two or more inequalities that are solved simultaneously to find the solution set, which is the region where all inequalities are satisfied.
Let’s consider the following systems of inequalities:
1. x + y > 5
2. x – y < 3
System 1: x + y > 5
System 2: x – y < 3
To determine if this system would produce the solution set containing the point (2, 3), we must substitute the values of x and y into each inequality and check if they are satisfied.
For System 1, substituting x = 2 and y = 3 gives us:
2 + 3 > 5
5 > 5
Since 5 is not greater than 5, the point (2, 3) does not satisfy this inequality.
For System 2, substituting x = 2 and y = 3 gives us:
2 – 3 < 3
-1 < 3
In this case, the point (2, 3) satisfies the inequality.
Now, let's consider another system of inequalities:
1. x + y ≤ 5
2. x - y ≥ 3
System 3: x + y ≤ 5
System 4: x - y ≥ 3
Substituting x = 2 and y = 3 into each inequality, we get:
For System 3: 2 + 3 ≤ 5
5 ≤ 5
The point (2, 3) satisfies this inequality.
For System 4: 2 - 3 ≥ 3
-1 ≥ 3
The point (2, 3) does not satisfy this inequality.
From our analysis, we can conclude that the system of inequalities that would produce the solution set containing the point (2, 3) is:
1. x + y ≤ 5
2. x - y ≥ 3
This example demonstrates the importance of carefully analyzing each inequality in a system to determine if a given point is part of the solution set. By following this approach, you can effectively solve problems involving systems of inequalities and identify the correct system that yields the desired result.
