Identify The Coordinates That Are Solutions To The System Of Inequalities:$\[ \left\{ \begin{array}{c} 3x + 2y \ \textgreater \ 24 \\ 4x - 5y \leq -20 \end{array} \right. \\]Note: The Additional Expressions Provided Do Not Contribute To

Feb 26, 2025 by ADMIN 239 views

**Solving a System of Inequalities: A Step-by-Step Guide**

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Introduction

In mathematics, solving a system of inequalities involves finding the values of variables that satisfy multiple inequalities simultaneously. In this article, we will focus on solving a system of two linear inequalities and identifying the coordinates that are solutions to the system. We will use a step-by-step approach to solve the system and provide a detailed explanation of each step.

Understanding the System of Inequalities

The given system of inequalities is:

{ \left\{ \begin{array}{c} 3x + 2y \ \textgreater \ 24 \\ 4x - 5y \leq -20 \end{array} \right. \}

This system consists of two linear inequalities:

$3x + 2y > 24$
$4x - 5y \leq -20$

Graphing the Inequalities

To solve the system of inequalities, we need to graph each inequality on a coordinate plane. We will use a solid line to represent the boundary of the inequality and a shaded region to represent the solution set.

Graphing the First Inequality

The first inequality is $3x + 2y > 24$ . To graph this inequality, we need to find the boundary line by setting the inequality to an equation:

$3x + 2y = 24$

We can solve for $y$ by isolating it on one side of the equation:

$y = -\frac{3}{2}x + 12$

This is the equation of a line with a slope of $-\frac{3}{2}$ and a $y$ -intercept of $12$ . We can graph this line on a coordinate plane.

To determine the direction of the inequality, we need to choose a test point that is not on the boundary line. Let's choose the point $(0, 0)$ . Plugging this point into the inequality, we get:

$3(0) + 2(0) > 24$

This is false, so the point $(0, 0)$ is not in the solution set. Since the inequality is greater than, we know that the solution set is above the boundary line.

Graphing the Second Inequality

The second inequality is $4x - 5y \leq -20$ . To graph this inequality, we need to find the boundary line by setting the inequality to an equation:

$4x - 5y = -20$

We can solve for $y$ by isolating it on one side of the equation:

$y = \frac{4}{5}x - 4$

This is the equation of a line with a slope of $\frac{4}{5}$ and a $y$ -intercept of $-4$ . We can graph this line on a coordinate plane.

To determine the direction of the inequality, we need to choose a test point that is not on the boundary line. Let's choose the point $(0, 0)$ . Plugging this point into the inequality, we get:

$4(0) - 5(0) \leq -20$

This is false, so the point $(0, 0)$ is not in the solution set. Since the inequality is less than or equal to, we know that the solution set is below or on the boundary line.

Finding the Intersection of the Solution Sets

To find the intersection of the solution sets, we need to find the region where both inequalities are satisfied. We can do this by finding the intersection of the two shaded regions.

The first inequality has a solution set above the boundary line $y = -\frac{3}{2}x + 12$ . The second inequality has a solution set below or on the boundary line $y = \frac{4}{5}x - 4$ .

To find the intersection of the two solution sets, we need to find the region where both inequalities are satisfied. We can do this by finding the intersection of the two shaded regions.

Solving the System of Inequalities

To solve the system of inequalities, we need to find the values of $x$ and $y$ that satisfy both inequalities simultaneously.

We can do this by finding the intersection of the two solution sets. The intersection of the two solution sets is the region where both inequalities are satisfied.

To find the intersection of the two solution sets, we need to find the values of $x$ and $y$ that satisfy both inequalities simultaneously. We can do this by solving the system of equations:

$3x + 2y = 24$ $4x - 5y = -20$

We can solve this system of equations using substitution or elimination. Let's use substitution.

Rearranging the first equation to isolate $y$ , we get:

$y = -\frac{3}{2}x + 12$

Substituting this expression for $y$ into the second equation, we get:

$4x - 5(-\frac{3}{2}x + 12) = -20$

Simplifying this equation, we get:

$4x + \frac{15}{2}x - 60 = -20$

Combine like terms:

$\frac{23}{2}x - 60 = -20$

Add 60 to both sides:

$\frac{23}{2}x = 40$

Multiply both sides by $\frac{2}{23}$ :

$x = \frac{80}{23}$

Now that we have found the value of $x$ , we can substitute it into one of the original equations to find the value of $y$ . Let's substitute it into the first equation:

$3(\frac{80}{23}) + 2y = 24$

Simplifying this equation, we get:

$\frac{240}{23} + 2y = 24$

Subtract $\frac{240}{23}$ from both sides:

$2y = 24 - \frac{240}{23}$

Simplify the right-hand side:

$2y = \frac{552 - 240}{23}$

$2y = \frac{312}{23}$

Divide both sides by 2:

$y = \frac{156}{23}$

Therefore, the solution to the system of inequalities is:

$x = \frac{80}{23}$ $y = \frac{156}{23}$

Conclusion

In this article, we have solved a system of two linear inequalities and identified the coordinates that are solutions to the system. We have used a step-by-step approach to solve the system and provided a detailed explanation of each step.

The solution to the system of inequalities is:

$x = \frac{80}{23}$ $y = \frac{156}{23}$

This solution represents the point of intersection of the two solution sets.

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Introduction

In our previous article, we solved a system of two linear inequalities and identified the coordinates that are solutions to the system. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used to solve the system of inequalities.

Q: What is a system of inequalities?

A system of inequalities is a set of two or more inequalities that are related to each other. In this article, we solved a system of two linear inequalities.

A: How do I graph a system of inequalities?

To graph a system of inequalities, you need to graph each inequality on a coordinate plane. You can use a solid line to represent the boundary of the inequality and a shaded region to represent the solution set.

Q: What is the difference between a solid line and a dashed line in graphing inequalities?

A solid line represents the boundary of the inequality, while a dashed line represents the boundary of the inequality that is not included in the solution set.

Q: How do I determine the direction of the inequality?

To determine the direction of the inequality, you need to choose a test point that is not on the boundary line. If the test point satisfies the inequality, then the solution set is on the same side of the boundary line as the test point.

Q: What is the intersection of the solution sets?

The intersection of the solution sets is the region where both inequalities are satisfied.

Q: How do I find the intersection of the solution sets?

To find the intersection of the solution sets, you need to find the values of $x$ and $y$ that satisfy both inequalities simultaneously. You can do this by solving the system of equations.

Q: What is the solution to the system of inequalities?

The solution to the system of inequalities is the point of intersection of the two solution sets.

Q: How do I check if a point is in the solution set?

To check if a point is in the solution set, you need to plug the coordinates of the point into both inequalities. If the point satisfies both inequalities, then it is in the solution set.

Q: What are some common mistakes to avoid when solving a system of inequalities?

Some common mistakes to avoid when solving a system of inequalities include:

Graphing the wrong boundary line
Choosing the wrong test point
Not checking if the point is in the solution set
Not solving the system of equations correctly

Q: What are some real-world applications of solving a system of inequalities?

Solving a system of inequalities has many real-world applications, including:

Finding the optimal solution to a problem
Determining the feasibility of a solution
Analyzing the behavior of a system
Making decisions based on data

Conclusion

In this article, we have provided a Q&A guide to help you understand the concepts and techniques used to solve a system of inequalities. We hope that this guide has been helpful in answering your questions and providing a better understanding of the topic.

Additional Resources

If you are interested in learning more about solving a system of inequalities, we recommend the following resources:

Khan Academy: Solving Systems of Linear Inequalities
Mathway: Solving Systems of Linear Inequalities
Wolfram Alpha: Solving Systems of Linear Inequalities

We hope that this guide has been helpful in providing a better understanding of the topic. If you have any further questions or need additional help, please don't hesitate to ask.