Solving Systems Of Inequalities Finding Ordered Pair Solutions
In mathematics, a system of inequalities is a set of two or more inequalities involving the same variables. The solution to a system of inequalities is the region in the coordinate plane that satisfies all the inequalities simultaneously. Identifying the correct ordered pair within the solution set is a fundamental skill in algebra. This article delves into how to determine whether a given ordered pair is a solution to a system of inequalities. We'll walk through a specific example and discuss the underlying concepts, providing a clear understanding of the solution process. This article aims to clarify the process of identifying ordered pair solutions within systems of inequalities. By understanding how to test ordered pairs against multiple inequalities, readers can build a solid foundation for solving more complex mathematical problems. The ability to solve systems of inequalities is a crucial skill in various fields, including economics, engineering, and computer science, where optimization problems often involve multiple constraints. This article's detailed explanation and step-by-step approach will equip readers with the necessary tools to tackle these problems confidently. We will explore how to verify solutions graphically and algebraically, ensuring a comprehensive understanding of the concepts. Throughout this discussion, we will emphasize the importance of accuracy and attention to detail, which are essential for success in mathematics and related disciplines.
The System of Inequalities
We are given the following system of inequalities:
$ y ext{≤} rac{2}{3}x + 1 $
$ y > - rac{1}{4}x + 2 $
Our task is to determine which of the given ordered pairs—(6, -2), (6, 0.5), (6, 5), and (6, 8)—is a solution to this system. An ordered pair is a solution if it satisfies both inequalities simultaneously. To verify this, we will substitute the x and y values of each ordered pair into both inequalities and check if the resulting statements are true. This process involves careful calculation and attention to the inequality signs. Each ordered pair must satisfy both inequalities to be considered a solution to the system. If an ordered pair fails to satisfy even one of the inequalities, it is not a solution. This method ensures that we accurately identify the pairs that lie within the region defined by the system of inequalities. In the following sections, we will perform these substitutions and evaluate each ordered pair to determine the correct solution. This step-by-step approach will illustrate the practical application of solving systems of inequalities.
Testing Ordered Pairs
To determine which ordered pair is a solution to the system of inequalities, we need to test each pair in both inequalities. This involves substituting the x and y values from the ordered pair into the inequalities and checking if the resulting statements are true. This process requires careful attention to detail and accurate arithmetic. For an ordered pair to be a solution, it must satisfy both inequalities simultaneously. If an ordered pair fails to satisfy even one of the inequalities, it is not a solution to the system. Each ordered pair will be evaluated systematically to ensure no errors are made. This method provides a clear and logical way to identify the correct solution from the given options. Understanding how to test ordered pairs is crucial for solving systems of inequalities and has practical applications in various mathematical and real-world scenarios. We will now proceed to test each ordered pair, providing a detailed explanation for each step.
Testing (6, -2)
First, let's test the ordered pair (6, -2). We will substitute x = 6 and y = -2 into both inequalities.
Inequality 1: y ≤ rac{2}{3}x + 1
Substitute x = 6 and y = -2:
$ -2 ≤ rac{2}{3}(6) + 1 $
Simplify the expression:
$ -2 ≤ 4 + 1 $
$ -2 ≤ 5 $
This statement is true, so (6, -2) satisfies the first inequality. However, we must also check the second inequality to confirm if it is a solution.
Inequality 2: y > - rac{1}{4}x + 2
Substitute x = 6 and y = -2:
$ -2 > - rac{1}{4}(6) + 2 $
Simplify the expression:
$ -2 > - rac{3}{2} + 2 $
$ -2 > -1.5 + 2 $
$ -2 > 0.5 $
This statement is false. Since (6, -2) does not satisfy both inequalities, it is not a solution to the system. This illustrates the importance of checking both inequalities to ensure the ordered pair is a valid solution. If an ordered pair fails to meet the conditions of even one inequality, it cannot be considered a solution to the entire system. We will now move on to testing the next ordered pair.
Testing (6, 0.5)
Next, we test the ordered pair (6, 0.5) by substituting x = 6 and y = 0.5 into both inequalities.
Inequality 1: y ≤ rac{2}{3}x + 1
Substitute x = 6 and y = 0.5:
$ 0.5 ≤ rac{2}{3}(6) + 1 $
Simplify the expression:
$ 0. 5 ≤ 4 + 1 $
$ 0.5 ≤ 5 $
This statement is true, so (6, 0.5) satisfies the first inequality. Now we need to check the second inequality.
Inequality 2: y > - rac{1}{4}x + 2
Substitute x = 6 and y = 0.5:
$ 0.5 > - rac{1}{4}(6) + 2 $
Simplify the expression:
$ 0.5 > - rac{3}{2} + 2 $
$ 0.5 > -1.5 + 2 $
$ 0.5 > 0.5 $
This statement is false because 0.5 is not strictly greater than 0.5. Therefore, (6, 0.5) is not a solution to the system, as it does not satisfy both inequalities. It is crucial to recognize the difference between “greater than” (>) and “greater than or equal to” (≥) when evaluating inequalities. This example highlights the importance of precision in mathematical analysis. We will now proceed to test the next ordered pair.
Testing (6, 5)
Now, let's test the ordered pair (6, 5) by substituting x = 6 and y = 5 into both inequalities.
Inequality 1: y ≤ rac{2}{3}x + 1
Substitute x = 6 and y = 5:
$ 5 ≤ rac{2}{3}(6) + 1 $
Simplify the expression:
$ 5 ≤ 4 + 1 $
$ 5 ≤ 5 $
This statement is true because 5 is less than or equal to 5. So, (6, 5) satisfies the first inequality. We must now check the second inequality to confirm.
Inequality 2: y > - rac{1}{4}x + 2
Substitute x = 6 and y = 5:
$ 5 > - rac{1}{4}(6) + 2 $
Simplify the expression:
$ 5 > - rac{3}{2} + 2 $
$ 5 > -1.5 + 2 $
$ 5 > 0.5 $
This statement is also true. Since (6, 5) satisfies both inequalities, it is a solution to the system. We have found a valid solution, but for completeness, we will also test the last ordered pair.
Testing (6, 8)
Finally, we test the ordered pair (6, 8) by substituting x = 6 and y = 8 into both inequalities.
Inequality 1: y ≤ rac{2}{3}x + 1
Substitute x = 6 and y = 8:
$ 8 ≤ rac{2}{3}(6) + 1 $
Simplify the expression:
$ 8 ≤ 4 + 1 $
$ 8 ≤ 5 $
This statement is false. Since 8 is not less than or equal to 5, the ordered pair (6, 8) does not satisfy the first inequality. Therefore, it is not a solution to the system. We do not need to check the second inequality because if an ordered pair fails one inequality, it cannot be a solution to the system.
Conclusion
After testing each ordered pair, we found that only the ordered pair (6, 5) satisfies both inequalities in the system. Therefore, (6, 5) is the solution. This process demonstrates the importance of systematically testing each potential solution against all inequalities in the system. Understanding how to solve systems of inequalities is crucial for various mathematical applications. The ability to accurately substitute and simplify expressions is key to determining whether an ordered pair is a solution. By following these steps, one can confidently solve similar problems involving systems of inequalities. This comprehensive approach ensures that the correct solution is identified through a logical and methodical process. The skills developed in this exercise are valuable for further studies in mathematics and related fields.
The ordered pair included in the solution to the system is (6, 5).
