Finding Solutions To Systems Of Inequalities A Step By Step Guide

Solving systems of inequalities is a fundamental concept in mathematics, particularly in algebra and precalculus. When we are presented with a system of inequalities, we are essentially looking for the region in the coordinate plane where all the inequalities are simultaneously satisfied. This region represents the set of all possible solutions to the system. To determine whether a specific point is a solution to a system of inequalities, we need to check if the coordinates of the point satisfy all the inequalities in the system. In this article, we will explore the process of identifying solutions to systems of inequalities, using a specific example to illustrate the method. We will delve into the steps involved in verifying whether a given point is a solution by substituting its coordinates into the inequalities and checking for their validity. This comprehensive guide aims to provide a clear understanding of how to tackle such problems, ensuring you can confidently solve similar questions in the future.

Understanding Systems of Inequalities

In mathematics, a system of inequalities is a set of two or more inequalities involving the same variables. Unlike equations, which have specific solutions, inequalities define a range of values that satisfy the expression. When dealing with a system of inequalities, we are interested in finding the set of all points that satisfy all the inequalities simultaneously. This set of points forms a region in the coordinate plane, often referred to as the solution region. The solution region is the intersection of the regions defined by each individual inequality. Understanding the concept of a solution region is crucial for solving systems of inequalities. Each inequality represents a half-plane, which is the region on one side of a line. The line itself is determined by the equality form of the inequality. For example, the inequality $4x + 2y < 8$ corresponds to the region below the line $4x + 2y = 8$. Similarly, the inequality $x - 6y < -2$ corresponds to the region above the line $x - 6y = -2$. The solution region for the system of inequalities is the area where these half-planes overlap. This region contains all the points whose coordinates satisfy both inequalities. To determine if a point is a solution, we simply need to check if it lies within this solution region. This involves substituting the coordinates of the point into each inequality and verifying that the inequalities hold true. In the following sections, we will apply this concept to a specific system of inequalities and identify which of the given points is a solution.

Problem Statement

Let's consider the following system of inequalities:

1. $4x + 2y < 8$
2. $x - 6y < -2$

We are given four points and we need to determine which of these points is a solution to the system:

A. $(3, 6)$ B. $(0, 0)$ C. $(1, 1)$ D. $(-6, -1)$

To solve this problem, we will substitute the coordinates of each point into both inequalities and check if the inequalities hold true. If a point satisfies both inequalities, then it is a solution to the system. This method involves a straightforward application of the definitions of inequalities and systems of inequalities. Each point is tested individually against the inequalities, and the results are analyzed to determine if the point lies within the solution region. This process is a fundamental technique in solving systems of inequalities and provides a clear and concise way to identify solutions. In the subsequent sections, we will perform these substitutions for each point and determine the correct answer. This step-by-step approach will illustrate the practical application of the concepts discussed earlier and provide a solid understanding of how to solve similar problems.

Testing Point A: (3, 6)

To determine if the point $(3, 6)$ is a solution to the system of inequalities, we need to substitute $x = 3$ and $y = 6$ into both inequalities and check if they hold true. This process is a direct application of the definition of a solution to a system of inequalities. If both inequalities are satisfied by the coordinates of the point, then the point is indeed a solution. If either inequality is not satisfied, then the point is not a solution. This method is a fundamental technique in solving systems of inequalities and provides a clear and concise way to verify potential solutions. By systematically testing each point, we can identify the correct solution to the system. Let's begin by substituting the coordinates of point A into the first inequality.

Substituting into the First Inequality

The first inequality is $4x + 2y < 8$. Substituting $x = 3$ and $y = 6$, we get:

$4(3) + 2(6) < 8$ $12 + 12 < 8$ $24 < 8$

This statement is false. Since $24$ is not less than $8$, the first inequality is not satisfied by the point $(3, 6)$. Because the first inequality is not satisfied, we do not need to check the second inequality. If a point fails to satisfy even one inequality in the system, it cannot be a solution to the entire system. Therefore, we can conclude that point A is not a solution to the system of inequalities. This result highlights the importance of satisfying all inequalities in the system for a point to be considered a solution. In the next section, we will test point B to see if it satisfies both inequalities.

Testing Point B: (0, 0)

Now, let's test point B $(0, 0)$ to see if it is a solution to the system of inequalities. As we did with point A, we will substitute the coordinates $x = 0$ and $y = 0$ into both inequalities and check if they hold true. This process is a crucial step in identifying solutions to systems of inequalities. By substituting the coordinates of the point into each inequality, we can directly verify whether the point lies within the solution region. If both inequalities are satisfied, then the point is a solution. If either inequality is not satisfied, then the point is not a solution. This method provides a systematic way to determine the solution set for the system. Let's start by substituting the coordinates of point B into the first inequality.

Substituting into the First Inequality

The first inequality is $4x + 2y < 8$. Substituting $x = 0$ and $y = 0$, we get:

$4(0) + 2(0) < 8$ $0 + 0 < 8$ $0 < 8$

This statement is true. The first inequality is satisfied by the point $(0, 0)$. Now we need to check the second inequality to confirm if the point is a solution to the entire system. If the point also satisfies the second inequality, then we can conclude that point B is a solution. If not, then point B is not a solution. This step-by-step approach ensures that we accurately determine the solution set for the system of inequalities.

Substituting into the Second Inequality

The second inequality is $x - 6y < -2$. Substituting $x = 0$ and $y = 0$, we get:

$0 - 6(0) < -2$ $0 - 0 < -2$ $0 < -2$

This statement is false. Since $0$ is not less than $-2$, the second inequality is not satisfied by the point $(0, 0)$. Because point B fails to satisfy the second inequality, it is not a solution to the system of inequalities. This result emphasizes the importance of satisfying all inequalities in the system for a point to be considered a solution. In the next section, we will test point C to see if it satisfies both inequalities.

Testing Point C: (1, 1)

Next, we will test point C $(1, 1)$ to see if it satisfies the system of inequalities. As with the previous points, we will substitute the coordinates $x = 1$ and $y = 1$ into both inequalities and check if they hold true. This process is a fundamental technique for verifying solutions to systems of inequalities. By substituting the coordinates of the point into each inequality, we can directly determine whether the point lies within the solution region. If both inequalities are satisfied, then the point is a solution. If either inequality is not satisfied, then the point is not a solution. This method provides a clear and systematic way to identify the solution set for the system. Let's begin by substituting the coordinates of point C into the first inequality.

Substituting into the First Inequality

The first inequality is $4x + 2y < 8$. Substituting $x = 1$ and $y = 1$, we get:

$4(1) + 2(1) < 8$ $4 + 2 < 8$ $6 < 8$

This statement is true. The first inequality is satisfied by the point $(1, 1)$. Now, we need to check the second inequality to confirm if the point is a solution to the entire system. If the point also satisfies the second inequality, then we can conclude that point C is a solution. If not, then point C is not a solution. This step-by-step approach ensures that we accurately determine the solution set for the system of inequalities.

Substituting into the Second Inequality

The second inequality is $x - 6y < -2$. Substituting $x = 1$ and $y = 1$, we get:

$1 - 6(1) < -2$ $1 - 6 < -2$ $-5 < -2$

This statement is also true. Since $-5$ is less than $-2$, the second inequality is satisfied by the point $(1, 1)$. Because point C satisfies both inequalities, it is a solution to the system of inequalities. This result highlights the importance of satisfying all inequalities in the system for a point to be considered a solution. In the next section, we will test point D to ensure we have identified all possible solutions and to further illustrate the method.

Testing Point D: (-6, -1)

Finally, let's test point D $(-6, -1)$ to determine if it is a solution to the system of inequalities. As with the previous points, we will substitute the coordinates $x = -6$ and $y = -1$ into both inequalities and check if they hold true. This process is a crucial step in verifying potential solutions to systems of inequalities. By substituting the coordinates of the point into each inequality, we can directly assess whether the point lies within the solution region. If both inequalities are satisfied, then the point is a solution. If either inequality is not satisfied, then the point is not a solution. This method provides a systematic approach to identifying the solution set for the system. We will begin by substituting the coordinates of point D into the first inequality.

Substituting into the First Inequality

The first inequality is $4x + 2y < 8$. Substituting $x = -6$ and $y = -1$, we get:

$4(-6) + 2(-1) < 8$ $-24 - 2 < 8$ $-26 < 8$

This statement is true. The first inequality is satisfied by the point $(-6, -1)$. Now we need to check the second inequality to confirm if the point is a solution to the entire system. If the point also satisfies the second inequality, then we can conclude that point D is a solution. If not, then point D is not a solution. This step-by-step approach ensures that we accurately determine the solution set for the system of inequalities.

Substituting into the Second Inequality

The second inequality is $x - 6y < -2$. Substituting $x = -6$ and $y = -1$, we get:

$-6 - 6(-1) < -2$ $-6 + 6 < -2$ $0 < -2$

This statement is false. Since $0$ is not less than $-2$, the second inequality is not satisfied by the point $(-6, -1)$. Because point D fails to satisfy the second inequality, it is not a solution to the system of inequalities. This result further emphasizes the importance of satisfying all inequalities in the system for a point to be considered a solution. Having tested all four points, we can now confidently state the solution to the problem.

Conclusion

After testing all four points, we found that only point C $(1, 1)$ satisfies both inequalities in the system:

1. $4x + 2y < 8$
2. $x - 6y < -2$

Therefore, the solution to the system of inequalities among the given options is $(1, 1)$. This comprehensive analysis demonstrates the step-by-step process of verifying solutions to systems of inequalities. By substituting the coordinates of each point into the inequalities and checking for their validity, we can accurately determine which points are solutions. This method is a fundamental technique in algebra and precalculus and provides a solid foundation for solving more complex problems involving inequalities. Understanding how to solve systems of inequalities is crucial for various applications in mathematics and other fields. This article has provided a detailed guide on how to approach such problems, ensuring that you can confidently identify solutions to systems of inequalities.