Ordered Pairs And Inequalities Determine Solutions

In mathematics, understanding inequalities is crucial for various applications, from solving real-world problems to grasping advanced concepts. This article delves into the process of identifying ordered pairs that satisfy multiple inequalities simultaneously. Specifically, we will explore the given system of inequalities:

$ \begin{array}{l} y < 5x + 2 \ y \geq \frac{1}{2}x + 1 \end{array} $

and determine which of the provided ordered pairs, (-1, 3) and (0, 2), make both inequalities true. This involves substituting the x and y values of each ordered pair into the inequalities and verifying if the resulting statements hold. This detailed exploration will not only provide a solution to the problem but also enhance your understanding of how to work with inequalities and ordered pairs.

Understanding Inequalities and Ordered Pairs

Before diving into the specific problem, it's essential to grasp the fundamental concepts of inequalities and ordered pairs. Inequalities are mathematical expressions that compare two values using symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which assert the equality of two expressions, inequalities define a range of values that satisfy a given condition. Understanding inequalities is pivotal in various mathematical and real-world applications, as they allow us to model situations where exact equality is not required or possible.

An ordered pair, denoted as (x, y), represents a point in the Cartesian coordinate system. The first value, x, represents the horizontal position, and the second value, y, represents the vertical position. Ordered pairs are fundamental in graphing and analyzing relationships between two variables. When dealing with inequalities, ordered pairs can represent solutions that satisfy the inequality. For instance, if an inequality describes a region in the coordinate plane, any ordered pair within that region is a solution to the inequality. The ability to identify and interpret ordered pairs is a crucial skill in algebra and beyond, enabling us to visualize and understand mathematical relationships in a geometric context. In the context of our problem, we will be testing whether specific ordered pairs fall within the regions defined by the given inequalities.

Step-by-Step Solution: Testing Ordered Pairs

To determine which ordered pairs satisfy both inequalities, we need to test each pair individually against each inequality. This involves substituting the x and y values of the ordered pair into the inequality and checking if the resulting statement is true. This process ensures that we accurately identify the ordered pairs that lie within the solution set of the system of inequalities. Let's start by examining the first ordered pair, (-1, 3), and then proceed to the second pair, (0, 2).

Testing the Ordered Pair (-1, 3)

First, substitute x = -1 and y = 3 into the first inequality, y < 5x + 2:3 < 5(-1) + 2 simplifies to 3 < -5 + 2, which further simplifies to 3 < -3. This statement is false, as 3 is not less than -3. Therefore, the ordered pair (-1, 3) does not satisfy the first inequality. Since an ordered pair must satisfy both inequalities to be a solution, we can conclude that (-1, 3) is not a solution to the system of inequalities. This step highlights the importance of verifying each inequality to ensure the ordered pair is a valid solution.

Next, let's substitute x = -1 and y = 3 into the second inequality, y ≥ (1/2)x + 1:3 ≥ (1/2)(-1) + 1 simplifies to 3 ≥ -0.5 + 1, which further simplifies to 3 ≥ 0.5. This statement is true, as 3 is greater than or equal to 0.5. However, since the ordered pair (-1, 3) did not satisfy the first inequality, it cannot be a solution to the system of inequalities. This reiterates the requirement that an ordered pair must satisfy all inequalities in the system to be considered a solution. Therefore, we can confidently conclude that (-1, 3) is not a valid solution.

Testing the Ordered Pair (0, 2)

Now, let's test the ordered pair (0, 2) against both inequalities. First, substitute x = 0 and y = 2 into the first inequality, y < 5x + 2:2 < 5(0) + 2 simplifies to 2 < 0 + 2, which further simplifies to 2 < 2. This statement is false, as 2 is not less than 2. Therefore, the ordered pair (0, 2) does not satisfy the first inequality. This outcome demonstrates that even if an ordered pair satisfies one inequality, it must also satisfy the other(s) to be a solution to the system.

Next, substitute x = 0 and y = 2 into the second inequality, y ≥ (1/2)x + 1:2 ≥ (1/2)(0) + 1 simplifies to 2 ≥ 0 + 1, which further simplifies to 2 ≥ 1. This statement is true, as 2 is greater than or equal to 1. However, similar to the previous case, the ordered pair (0, 2) does not satisfy the first inequality, so it cannot be a solution to the system of inequalities. This reinforces the critical point that a solution to a system of inequalities must satisfy all inequalities in the system. Thus, we can conclude that (0, 2) is not a solution to the given system.

Conclusion: Identifying Solutions to Systems of Inequalities

In conclusion, after testing both ordered pairs (-1, 3) and (0, 2) against the given system of inequalities:

$ \begin{array}{l} y < 5x + 2 \ y \geq \frac{1}{2}x + 1 \end{array} $

we found that neither ordered pair satisfies both inequalities simultaneously. For the ordered pair (-1, 3), while it satisfied the second inequality (y ≥ (1/2)x + 1), it did not satisfy the first inequality (y < 5x + 2). Similarly, for the ordered pair (0, 2), it satisfied the second inequality but not the first. This exercise underscores the importance of verifying that a solution satisfies all inequalities in a system, not just one or some. Understanding how to test ordered pairs against inequalities is a fundamental skill in algebra, with applications in graphing, linear programming, and various real-world problem-solving scenarios. By mastering this skill, you can confidently tackle more complex problems involving systems of inequalities and their solutions. The process involves careful substitution and verification, ensuring that all conditions are met for a solution to be valid. This thorough approach is essential for accuracy and a deeper understanding of mathematical concepts.