Determining Solution Sets Which System Of Linear Inequalities Contains (3,-2)

Jul 11, 2025 by ADMIN 78 views

Which System of Linear Inequalities Contains the Point (3,-2) in Its Solution Set? A Comprehensive Guide

In the realm of mathematics, linear inequalities play a crucial role in defining regions within a coordinate plane. The solution set of a system of linear inequalities represents the intersection of these regions, and identifying whether a specific point lies within this solution set is a fundamental task. This article delves into the process of determining if a given point, specifically $(3, -2)$ , belongs to the solution set of the system of linear inequalities:

 y < -3
 y ≤ (2/3)x - 4

We will explore the underlying concepts, provide a step-by-step solution, and discuss the broader implications of this type of problem. Understanding how to solve these problems is essential for students and anyone working with mathematical models and real-world constraints.

Understanding Linear Inequalities and Solution Sets

Before we dive into the specific problem, let's establish a solid foundation by understanding the key concepts of linear inequalities and their solution sets. 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). Unlike linear equations, which represent a specific line on a graph, linear inequalities represent a region of the coordinate plane.

The solution set of a linear inequality is the collection of all points $(x, y)$ that satisfy the inequality. Graphically, this solution set is represented by a shaded region on the coordinate plane. The boundary line of this region is determined by the corresponding linear equation (obtained by replacing the inequality symbol with an equals sign). If the inequality includes ≤ or ≥, the boundary line is solid, indicating that points on the line are included in the solution set. If the inequality includes < or >, the boundary line is dashed, indicating that points on the line are not included in the solution set.

A system of linear inequalities consists of two or more linear inequalities considered together. The solution set of a system of linear inequalities is the intersection of the solution sets of the individual inequalities. Graphically, this is the region where the shaded areas of all the inequalities overlap. To determine if a point lies in the solution set of a system of linear inequalities, you must check if the point satisfies all inequalities in the system.

Step-by-Step Solution: Does (3,-2) Belong to the Solution Set?

Now, let's tackle the problem at hand. We need to determine if the point $(3, -2)$ lies in the solution set of the following system of linear inequalities:

 y < -3
 y ≤ (2/3)x - 4

To do this, we will substitute the coordinates of the point $(3, -2)$ into each inequality and check if the inequalities hold true.

Step 1: Substitute the point (3, -2) into the first inequality, y < -3.

Replace y with -2:

 -2 < -3

This statement is false because -2 is not less than -3. Therefore, the point $(3, -2)$ does not satisfy the first inequality.

Step 2: Substitute the point (3, -2) into the second inequality, y ≤ (2/3)x - 4.

Replace x with 3 and y with -2:

 -2 ≤ (2/3)(3) - 4
 -2 ≤ 2 - 4
 -2 ≤ -2

This statement is true because -2 is less than or equal to -2. The point $(3, -2)$ satisfies the second inequality.

Step 3: Determine if the point satisfies the entire system.

For a point to be in the solution set of a system of inequalities, it must satisfy all inequalities in the system. Since the point $(3, -2)$ does not satisfy the first inequality (y < -3), it does not belong to the solution set of the system.

Graphical Interpretation

To further illustrate this, let's consider the graphical representation of the inequalities.

y < -3: This inequality represents the region below the horizontal line y = -3. The line is dashed because the inequality is strictly less than. The point $(3, -2)$ lies on the line y = -2, which is above the line y = -3, so it is not in the solution set of this inequality.
y ≤ (2/3)x - 4: This inequality represents the region below and including the line y = (2/3)x - 4. To plot this line, we can find two points. For example, when x = 0, y = -4, and when x = 3, y = -2. The point $(3, -2)$ lies on this line, so it satisfies this inequality.

The solution set of the system is the region where the shaded areas of both inequalities overlap. Since the point $(3, -2)$ is not in the solution set of y < -3, it cannot be in the solution set of the system, even though it satisfies the second inequality. This graphical analysis confirms our algebraic solution.

Why is This Important? Applications and Broader Context

Understanding how to determine if a point belongs to the solution set of a system of linear inequalities is not just an academic exercise. It has practical applications in various fields, including:

Linear Programming: This is a mathematical technique used to optimize a linear objective function subject to linear constraints (which are often expressed as inequalities). Identifying the feasible region (the solution set of the constraints) is a crucial step in linear programming.
Resource Allocation: Businesses often face constraints on resources such as time, money, and materials. Systems of inequalities can be used to model these constraints and determine feasible production plans.
Decision Making: In many real-world scenarios, decisions must be made within certain limitations. Linear inequalities can help model these limitations and identify the set of possible choices.
Computer Graphics: Inequalities are used to define shapes and regions in computer graphics. For example, determining if a point is inside a polygon can be done using inequalities.

Moreover, the process of solving systems of inequalities reinforces essential mathematical skills such as algebraic manipulation, graphing, and logical reasoning. It also helps develop problem-solving strategies that can be applied in other areas of mathematics and beyond.

Common Mistakes and How to Avoid Them

When working with systems of linear inequalities, students often make certain common mistakes. Being aware of these pitfalls can help you avoid them.

Forgetting to Check All Inequalities: A point must satisfy all inequalities in the system to be in the solution set. It is not sufficient for the point to satisfy only some of the inequalities. Make sure to substitute the point into every inequality and verify that it holds true.
Incorrectly Interpreting Inequality Symbols: Pay close attention to the inequality symbols (<, >, ≤, ≥). A strict inequality (< or >) means that the boundary line is not included in the solution set, while a non-strict inequality (≤ or ≥) means that the boundary line is included. This distinction is crucial when graphing the inequalities.
Making Arithmetic Errors: Even a small arithmetic error can lead to an incorrect conclusion. Double-check your calculations, especially when substituting values and simplifying expressions.
Misunderstanding the Concept of a Solution Set: Remember that the solution set of a system of inequalities is the intersection of the solution sets of the individual inequalities. It is the region where all the shaded areas overlap. Visualizing the inequalities graphically can help you understand this concept.
Not Testing Points: When graphing inequalities, it is a good practice to test a point (e.g., (0, 0)) in each region to determine which side of the boundary line should be shaded. This helps avoid errors in shading the correct region.

By being mindful of these common mistakes and practicing regularly, you can improve your accuracy and confidence in solving problems involving systems of linear inequalities.

Practice Problems

To solidify your understanding, let's consider a few additional practice problems.

Problem 1: Determine if the point (-1, 4) is in the solution set of the following system:
```
y > 2x + 1
y ≤ -x + 5
```
Problem 2: Which of the following points is a solution to the system?
```
x + y < 3
2x - y ≥ 1
```
a) (0, 0) b) (2, 1) c) (1, 2) d) (3, -1)
Problem 3: Determine if the point (5, -3) is a solution to the system:
```
y ≥ -2x + 7
y < (1/2)x - 1
```

Working through these problems will help you develop your skills and solidify your understanding of how to determine if a point belongs to the solution set of a system of linear inequalities.

Conclusion

In this article, we explored the process of determining whether the point $(3, -2)$ lies in the solution set of the system of linear inequalities:

 y < -3
 y ≤ (2/3)x - 4

We learned that by substituting the coordinates of the point into each inequality and checking if the inequalities hold true, we can determine if the point is in the solution set. In this specific case, we found that the point $(3, -2)$ does not belong to the solution set because it does not satisfy the first inequality, y < -3. We also discussed the graphical interpretation of the solution, the importance of this type of problem in various applications, common mistakes to avoid, and provided additional practice problems.

Mastering the concepts and techniques discussed in this article will empower you to confidently tackle problems involving systems of linear inequalities and their solution sets. Remember to practice regularly and seek clarification whenever needed. With consistent effort, you can develop a strong understanding of this important mathematical topic and its applications.

By grasping these fundamentals, you will not only excel in your mathematical studies but also develop critical thinking skills applicable to a wide array of real-world scenarios. Understanding systems of linear inequalities is a crucial stepping stone for more advanced mathematical concepts and practical applications in various fields. Keep practicing, and you'll master this essential skill!