Does (3,-2) Solve The System Of Linear Inequalities Y < -3 And Y ≤ (2/3)x - 4?

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In mathematics, systems of linear inequalities play a crucial role in modeling real-world constraints and finding feasible solutions. When working with these systems, a key question often arises: Which system of linear inequalities has the point (3,-2) in its solution set? To answer this question, we must understand what it means for a point to be a solution to a system of inequalities and how to verify if a given point satisfies all the inequalities in the system. This article will delve into the process of identifying the correct system by substituting the point's coordinates into the inequalities and checking for validity.

Understanding Systems of Linear Inequalities

Before we dive into the specifics of the problem, let's first establish a solid understanding of systems of linear inequalities. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). A system of linear inequalities is a set of two or more linear inequalities involving the same variables. The solution set of a system of inequalities is the region in the coordinate plane that satisfies all the inequalities simultaneously. Graphically, this region is often represented as the intersection of the shaded regions corresponding to each individual inequality. To determine if a point lies within the solution set, we substitute its coordinates into each inequality and check if the resulting statements are true. If the point satisfies all inequalities, it is part of the solution set; otherwise, it is not.

The Significance of Solution Sets in Linear Inequalities

Understanding solution sets in linear inequalities is crucial for various applications across different fields. In economics, for instance, businesses use systems of inequalities to model production constraints, resource allocation, and cost optimization. The solution set represents the feasible region where the company can operate while meeting all constraints, such as budget limitations, supply constraints, and demand requirements. Similarly, in engineering, solution sets help in designing structures, circuits, and systems that meet specific performance criteria and safety standards. For example, engineers might use inequalities to ensure that a bridge can withstand certain loads or that an electrical circuit operates within safe voltage and current limits. In computer science, linear inequalities are used in optimization algorithms, machine learning models, and network flow problems. The solution set defines the space of possible solutions, and algorithms search within this space to find the optimal solution that maximizes or minimizes a particular objective function. Furthermore, in everyday life, we encounter situations where linear inequalities are applicable, such as budgeting, time management, and decision-making under constraints. Understanding how to solve and interpret systems of linear inequalities empowers us to make informed decisions and manage resources effectively.

Analyzing the Given Inequalities: y < -3 and y ≤ (2/3)x - 4

The problem presents us with a specific point, (3,-2), and asks us to determine if it belongs to the solution set of the given system of linear inequalities. The system includes two inequalities:

  1. y < -3
  2. y ≤ (2/3)x - 4

To check if the point (3,-2) is a solution, we need to substitute the x and y coordinates of the point into each inequality and see if the resulting statements are true. For the first inequality, y < -3, we substitute y = -2, which gives us -2 < -3. This statement is false because -2 is greater than -3. Since the point (3,-2) does not satisfy the first inequality, it cannot be part of the solution set for the entire system. It is essential to understand that for a point to be a solution to a system of inequalities, it must satisfy all inequalities in the system. If even one inequality is not satisfied, the point is not a solution.

For the sake of completeness, let's also check the second inequality, y ≤ (2/3)x - 4. Substituting x = 3 and y = -2, we get -2 ≤ (2/3)(3) - 4, which simplifies to -2 ≤ 2 - 4, and further to -2 ≤ -2. This statement is true because -2 is equal to -2. However, since the point (3,-2) did not satisfy the first inequality, the fact that it satisfies the second inequality is irrelevant. The point is not a solution to the system.

Step-by-Step Verification Process

To systematically determine if a point is a solution to a system of linear inequalities, follow these steps:

  1. Identify the inequalities: List all the inequalities in the system.
  2. Identify the point: Note the coordinates (x, y) of the point you want to test.
  3. Substitute the coordinates: Replace the variables x and y in each inequality with the corresponding coordinates of the point.
  4. Simplify each inequality: Perform the necessary arithmetic operations to simplify both sides of each inequality.
  5. Check for truth: Determine if the resulting statement for each inequality is true or false.
  6. Evaluate the solution:
    • If the point satisfies all inequalities (i.e., all resulting statements are true), then the point is a solution to the system.
    • If the point fails to satisfy even one inequality (i.e., at least one resulting statement is false), then the point is not a solution to the system.

By following this step-by-step process, you can efficiently and accurately verify whether a given point is part of the solution set for any system of linear inequalities.

Common Mistakes to Avoid

When working with systems of linear inequalities, several common mistakes can lead to incorrect conclusions. One frequent error is failing to check all inequalities in the system. Remember that a point must satisfy every inequality to be considered a solution. Overlooking even one inequality can result in misidentifying the solution set. Another common mistake is making arithmetic errors when substituting and simplifying the inequalities. Carefully double-check your calculations to ensure accuracy. Pay close attention to signs (positive and negative) and the order of operations. A minor arithmetic error can change the outcome of the comparison and lead to a wrong conclusion. Additionally, students sometimes confuse the inequality symbols, such as mistaking < for ≤ or > for ≥. Understanding the difference between strict inequalities (< and >) and inclusive inequalities (≤ and ≥) is crucial for correctly interpreting the results. Strict inequalities do not include the boundary line, while inclusive inequalities do. Lastly, it's important to remember that the solution set of a system of linear inequalities is a region, not just a single point. While verifying whether a specific point is a solution is a useful exercise, it is also essential to understand how to graph the inequalities and identify the entire region that represents the solution set.

Conclusion: Mastering Linear Inequalities

In conclusion, determining whether a point belongs to the solution set of a system of linear inequalities involves a straightforward process of substitution and verification. By substituting the coordinates of the point into each inequality and checking for truth, we can efficiently identify if the point satisfies all conditions. In the case of the inequalities y < -3 and y ≤ (2/3)x - 4, we found that the point (3,-2) does not belong to the solution set because it fails to satisfy the first inequality. Understanding how to work with linear inequalities is not only essential for academic success in mathematics but also provides valuable tools for problem-solving in various real-world scenarios. Mastering these concepts empowers us to make informed decisions, optimize resources, and tackle complex challenges across diverse fields.