Which Ordered Pair Is A Solution To The System Of Inequalities?${ \begin{array}{l} y \geq 2x - 1 \ y \ \textless \ 4x \end{array} }$A. { (2, 6)$}$B. { (2, 8)$}$C. { (3, 4)$}$D. { (3, 2)$}$

Introduction

In mathematics, systems of inequalities are a set of two or more inequalities that are related to each other. Solving a system of inequalities involves finding the values of the variables that satisfy all the inequalities in the system. In this article, we will focus on solving a system of two linear inequalities and determine which ordered pair is a solution to the system.

Understanding the System of Inequalities

The given system of inequalities is:

{ \begin{array}{l} y \geq 2x - 1 \\ y \ \textless \ 4x \end{array} \}

This system consists of two linear inequalities:

1. $y \geq 2x - 1$
2. $y \ \textless \ 4x$

Graphing the Inequalities

To solve the system of inequalities, we need to graph the two inequalities on a coordinate plane. The first inequality, $y \geq 2x - 1$, can be graphed by drawing a line with a slope of 2 and a y-intercept of -1. Since the inequality is greater than or equal to, we need to shade the region above the line.

The second inequality, $y \ \textless \ 4x$, can be graphed by drawing a line with a slope of 4 and a y-intercept of 0. Since the inequality is less than, we need to shade the region below the line.

Finding the Solution Region

To find the solution region, we need to find the intersection of the two shaded regions. The solution region is the area where both inequalities are satisfied.

Analyzing the Options

We are given four options:

A. $(2, 6)$ B. $(2, 8)$ C. $(3, 4)$ D. $(3, 2)$

We need to determine which of these options is a solution to the system of inequalities.

Checking Option A

Let's check if option A, $(2, 6)$, is a solution to the system of inequalities.

For the first inequality, $y \geq 2x - 1$, we substitute $x = 2$ and $y = 6$:

$6 \geq 2(2) - 1$ $6 \geq 4 - 1$ $6 \geq 3$

Since $6 \geq 3$, option A satisfies the first inequality.

For the second inequality, $y \ \textless \ 4x$, we substitute $x = 2$ and $y = 6$:

$6 \ \textless \ 4(2)$ $6 \ \textless \ 8$

Since $6 \ \textless \ 8$, option A satisfies the second inequality.

Checking Option B

Let's check if option B, $(2, 8)$, is a solution to the system of inequalities.

For the first inequality, $y \geq 2x - 1$, we substitute $x = 2$ and $y = 8$:

$8 \geq 2(2) - 1$ $8 \geq 4 - 1$ $8 \geq 3$

Since $8 \geq 3$, option B satisfies the first inequality.

For the second inequality, $y \ \textless \ 4x$, we substitute $x = 2$ and $y = 8$:

$8 \ \textless \ 4(2)$ $8 \ \textless \ 8$

Since $8 \ \textless \ 8$ is not true, option B does not satisfy the second inequality.

Checking Option C

Let's check if option C, $(3, 4)$, is a solution to the system of inequalities.

For the first inequality, $y \geq 2x - 1$, we substitute $x = 3$ and $y = 4$:

$4 \geq 2(3) - 1$ $4 \geq 6 - 1$ $4 \geq 5$

Since $4 \geq 5$ is not true, option C does not satisfy the first inequality.

Checking Option D

Let's check if option D, $(3, 2)$, is a solution to the system of inequalities.

For the first inequality, $y \geq 2x - 1$, we substitute $x = 3$ and $y = 2$:

$2 \geq 2(3) - 1$ $2 \geq 6 - 1$ $2 \geq 5$

Since $2 \geq 5$ is not true, option D does not satisfy the first inequality.

Conclusion

Based on our analysis, we can conclude that option A, $(2, 6)$, is a solution to the system of inequalities.

Final Answer

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Introduction

In our previous article, we discussed how to solve a system of two linear inequalities and determine which ordered pair is a solution to the system. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving systems of inequalities.

Q: What is a system of inequalities?

A: A system of inequalities is a set of two or more inequalities that are related to each other. Solving a system of inequalities involves finding the values of the variables that satisfy all the inequalities in the system.

Q: How do I graph a system of inequalities?

A: To graph a system of inequalities, you need to graph each inequality on a coordinate plane. The first inequality is graphed by drawing a line with a slope of 2 and a y-intercept of -1. Since the inequality is greater than or equal to, you need to shade the region above the line. The second inequality is graphed by drawing a line with a slope of 4 and a y-intercept of 0. Since the inequality is less than, you need to shade the region below the line.

Q: How do I find the solution region?

A: To find the solution region, you need to find the intersection of the two shaded regions. The solution region is the area where both inequalities are satisfied.

Q: What is the difference between a system of linear inequalities and a system of nonlinear inequalities?

A: A system of linear inequalities consists of two or more linear inequalities, while a system of nonlinear inequalities consists of two or more nonlinear inequalities. Solving a system of linear inequalities is generally easier than solving a system of nonlinear inequalities.

Q: Can I use the same techniques to solve a system of inequalities with more than two variables?

A: Yes, you can use the same techniques to solve a system of inequalities with more than two variables. However, the number of variables can make the problem more complex and require more advanced techniques.

Q: How do I check if an ordered pair is a solution to a system of inequalities?

A: To check if an ordered pair is a solution to a system of inequalities, you need to substitute the values of the variables into each inequality and check if the inequality is satisfied.

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

A: Some common mistakes to avoid when solving systems of inequalities include:

• Not graphing the inequalities correctly
• Not finding the intersection of the two shaded regions
• Not checking if an ordered pair is a solution to the system
• Not using the correct techniques to solve the system

Q: How can I practice solving systems of inequalities?

A: You can practice solving systems of inequalities by working through examples and exercises in a textbook or online resource. You can also try creating your own systems of inequalities and solving them.

Conclusion

Solving systems of inequalities can be a challenging but rewarding topic in mathematics. By understanding the concepts and techniques involved, you can become proficient in solving systems of inequalities and apply them to real-world problems.

Final Tips

• Make sure to graph the inequalities correctly
• Find the intersection of the two shaded regions
• Check if an ordered pair is a solution to the system
• Use the correct techniques to solve the system
• Practice, practice, practice!

Additional Resources

• Khan Academy: Systems of Inequalities
• Mathway: Systems of Inequalities
• Wolfram Alpha: Systems of Inequalities

We hope this Q&A guide has been helpful in understanding the concepts and techniques involved in solving systems of inequalities. If you have any further questions or need additional help, please don't hesitate to ask.