Determine If (2,-1) And (-4,2) Satisfy The Inequality 2x-3y > 4

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Introduction

In this article, we will explore how to determine whether given points satisfy a linear inequality. Specifically, we will investigate if the points (2,-1) and (-4,2) satisfy the inequality 2x - 3y > 4. This involves substituting the coordinates of each point into the inequality and checking if the resulting statement is true. Understanding how to solve such problems is fundamental in algebra and coordinate geometry, as it helps in identifying regions on a coordinate plane that satisfy a given set of conditions. We aim to provide a clear and detailed explanation, making it easy to follow along and understand the underlying concepts. This skill is not only crucial for academic purposes but also has practical applications in various fields, including economics, engineering, and computer science, where optimization and constraint satisfaction are essential.

Understanding Linear Inequalities

Before we dive into the specifics, it's important to grasp what linear inequalities represent. A linear inequality is similar to a linear equation, but instead of an equals sign (=), it uses inequality symbols such as > (greater than), < (less than), ≥ (greater than or equal to), or ≤ (less than or equal to). A linear inequality in two variables, like the one we're dealing with (2x - 3y > 4), represents a region in the coordinate plane. This region consists of all the points (x, y) that make the inequality true. The boundary line, which is the line obtained by replacing the inequality symbol with an equals sign (2x - 3y = 4), divides the plane into two regions. One of these regions contains the solutions to the inequality. To determine which region, we can test points. If a point in a region satisfies the inequality, then the entire region does. This concept is crucial for visualizing solutions and understanding the broader implications of linear inequalities. For instance, in real-world scenarios, linear inequalities can represent constraints on resources, such as budget limitations or material availability. Solving these inequalities helps in making decisions that comply with these constraints. The ability to interpret and solve these inequalities is therefore an indispensable skill in many domains.

Step-by-Step Solution for Point (2, -1)

Let's start by determining whether the point (2, -1) satisfies the inequality 2x - 3y > 4. To do this, we substitute the x-coordinate (2) and the y-coordinate (-1) into the inequality. This substitution gives us 2(2) - 3(-1) > 4. Simplifying this expression, we get 4 + 3 > 4, which further simplifies to 7 > 4. The statement 7 > 4 is true, which means that the point (2, -1) does indeed satisfy the inequality 2x - 3y > 4. This step-by-step approach is crucial in solving such problems because it breaks down the process into manageable parts, reducing the chances of errors. Understanding this substitution method is key to verifying whether any given point is part of the solution set for an inequality. The process involves careful arithmetic and a clear understanding of inequality rules. Moreover, this exercise reinforces the concept that a point satisfies an inequality if its coordinates, when substituted, yield a true statement. This is a fundamental concept in linear algebra and is widely used in solving systems of inequalities and linear programming problems.

Step-by-Step Solution for Point (-4, 2)

Now, let's examine the point (-4, 2) and determine if it satisfies the inequality 2x - 3y > 4. We will use the same method as before, substituting the x-coordinate (-4) and the y-coordinate (2) into the inequality. This substitution results in the expression 2(-4) - 3(2) > 4. Simplifying this, we get -8 - 6 > 4, which further simplifies to -14 > 4. The statement -14 > 4 is false. This indicates that the point (-4, 2) does not satisfy the inequality 2x - 3y > 4. The result highlights the importance of accurate arithmetic in determining the truth of an inequality statement. When solving such problems, it's essential to pay close attention to signs and the order of operations. The process of substituting coordinates and simplifying expressions is a fundamental skill in algebra and is used extensively in graphing inequalities and solving systems of equations. This example also demonstrates that not all points will satisfy a given inequality, emphasizing the concept of a solution region where the inequality holds true.

Conclusion

In conclusion, after evaluating both points, we found that (2, -1) satisfies the inequality 2x - 3y > 4, while (-4, 2) does not. This was determined by substituting the coordinates of each point into the inequality and checking the validity of the resulting statement. The point (2, -1) yielded the true statement 7 > 4, confirming that it is a solution to the inequality. Conversely, the point (-4, 2) resulted in the false statement -14 > 4, indicating that it is not a solution. This exercise underscores the importance of understanding linear inequalities and how to test points to determine if they satisfy a given condition. Such skills are crucial not only in mathematics but also in various applications where constraints and optimization are involved. The ability to accurately substitute and simplify expressions is fundamental to solving a wide range of problems in algebra and beyond. This specific example also illustrates the practical application of these concepts, demonstrating how mathematical principles can be used to analyze and solve real-world problems.

Therefore, the final answer is:

A. (2,-1) satisfies the inequality, but (-4,2) does not.