For The System Of Inequalities Below, Which Ordered Pair Is A Solution?${ \begin{array}{l} 2x - Y \ \textless \ 4 \ x + Y \ \textgreater \ 3 \end{array} }$A. (3, 4) B. (1, -3) C. (0, 0) D. (-2, 1)

Understanding Systems of Inequalities

A system of inequalities is a set of two or more inequalities that are related to each other through a common variable or variables. In this case, we have two inequalities: $2x - y < 4$ and $x + y > 3$. Our goal is to find the ordered pair that satisfies both inequalities.

Analyzing the First Inequality

The first inequality is $2x - y < 4$. To analyze this inequality, we can start by isolating the variable $y$. We can do this by subtracting $2x$ from both sides of the inequality, which gives us $-y < 4 - 2x$. Next, we can multiply both sides of the inequality by $-1$, which flips the direction of the inequality sign. This gives us $y > 2x - 4$.

Analyzing the Second Inequality

The second inequality is $x + y > 3$. To analyze this inequality, we can start by isolating the variable $y$. We can do this by subtracting $x$ from both sides of the inequality, which gives us $y > 3 - x$.

Finding the Solution

Now that we have analyzed both inequalities, we can find the solution by finding the intersection of the two inequalities. To do this, we can set the two inequalities equal to each other and solve for $x$. We have $2x - 4 = 3 - x$, which simplifies to $3x = 7$. Dividing both sides of the equation by $3$, we get $x = \frac{7}{3}$.

Finding the Value of $y$

Now that we have found the value of $x$, we can substitute it into one of the inequalities to find the value of $y$. We can use the first inequality $y > 2x - 4$. Substituting $x = \frac{7}{3}$, we get $y > 2(\frac{7}{3}) - 4$. Simplifying this expression, we get $y > \frac{14}{3} - 4$, which further simplifies to $y > \frac{14}{3} - \frac{12}{3}$. This gives us $y > \frac{2}{3}$.

Checking the Solution

Now that we have found the values of $x$ and $y$, we can check to see if they satisfy both inequalities. We can plug in $x = \frac{7}{3}$ and $y = \frac{2}{3}$ into both inequalities to see if they are true. For the first inequality, we have $2(\frac{7}{3}) - \frac{2}{3} < 4$, which simplifies to $\frac{14}{3} - \frac{2}{3} < 4$. This gives us $\frac{12}{3} < 4$, which is true. For the second inequality, we have $\frac{7}{3} + \frac{2}{3} > 3$, which simplifies to $\frac{9}{3} > 3$. This is also true.

Conclusion

Based on our analysis, we have found that the ordered pair $(\frac{7}{3}, \frac{2}{3})$ satisfies both inequalities. However, this ordered pair is not among the answer choices. Let's go back and re-examine the answer choices to see if any of them satisfy both inequalities.

Re-examining the Answer Choices

Let's start by plugging in the answer choice $(3, 4)$ into both inequalities. For the first inequality, we have $2(3) - 4 < 4$, which simplifies to $6 - 4 < 4$. This gives us $2 < 4$, which is true. For the second inequality, we have $3 + 4 > 3$, which simplifies to $7 > 3$. This is also true.

Conclusion

Based on our analysis, we have found that the ordered pair $(3, 4)$ satisfies both inequalities. Therefore, the correct answer is A. $(3, 4)$.

Why is this the correct answer?

This is the correct answer because it satisfies both inequalities. The first inequality $2x - y < 4$ is satisfied when $2(3) - 4 < 4$, which simplifies to $2 < 4$. The second inequality $x + y > 3$ is satisfied when $3 + 4 > 3$, which simplifies to $7 > 3$. Therefore, the ordered pair $(3, 4)$ satisfies both inequalities and is the correct answer.

What if we had chosen a different answer?

If we had chosen a different answer, we would have had to plug it into both inequalities to see if it satisfied them. If it did not satisfy both inequalities, then it would not be the correct answer. For example, if we had chosen the answer $(1, -3)$, we would have had to plug it into both inequalities to see if it satisfied them. For the first inequality, we would have had $2(1) - (-3) < 4$, which simplifies to $5 < 4$. This is not true, so $(1, -3)$ is not the correct answer.

Conclusion

In conclusion, the correct answer is A. $(3, 4)$. This is because it satisfies both inequalities, and none of the other answer choices do.

Why is this important?

This is important because it shows how to solve systems of inequalities. Systems of inequalities are used in many real-world applications, such as finance, economics, and engineering. By understanding how to solve systems of inequalities, we can make better decisions and solve problems more efficiently.

What are some common applications of systems of inequalities?

Some common applications of systems of inequalities include:

• Finance: Systems of inequalities are used to model financial transactions and make investment decisions.
• Economics: Systems of inequalities are used to model economic systems and make predictions about economic trends.
• Engineering: Systems of inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

In conclusion, systems of inequalities are an important tool in many fields, including finance, economics, and engineering. By understanding how to solve systems of inequalities, we can make better decisions and solve problems more efficiently.

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 through a common variable or variables.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you need to find the values of the variables that satisfy all the inequalities in the system. You can do this by graphing the inequalities on a coordinate plane and finding the intersection of the two graphs.

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

A: A system of equations is a set of two or more equations that are related to each other through a common variable or variables. A system of inequalities, on the other hand, is a set of two or more inequalities that are related to each other through a common variable or variables.

Q: How do I graph a system of inequalities?

A: To graph a system of inequalities, you need to graph each inequality separately on a coordinate plane. You can use a graphing calculator or draw the graphs by hand.

Q: What is the intersection of two graphs?

A: The intersection of two graphs is the point or points where the two graphs meet.

Q: How do I find the intersection of two graphs?

A: To find the intersection of two graphs, you need to set the two equations equal to each other and solve for the variable.

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

A: The solution to a system of inequalities is the set of all possible values of the variables that satisfy all the inequalities in the system.

Q: How do I check if a solution is correct?

A: To check if a solution is correct, you need to plug the values of the variables into each inequality and check if the inequality is true.

Q: What if I have a system of three or more inequalities?

A: If you have a system of three or more inequalities, you can use the same method as before to solve the system. However, you may need to use a graphing calculator or a computer program to help you solve the system.

Q: Can I use a graphing calculator to solve a system of inequalities?

A: Yes, you can use a graphing calculator to solve a system of inequalities. Graphing calculators can help you graph the inequalities and find the intersection of the two graphs.

Q: What are some common applications of systems of inequalities?

A: Some common applications of systems of inequalities include:

• Finance: Systems of inequalities are used to model financial transactions and make investment decisions.
• Economics: Systems of inequalities are used to model economic systems and make predictions about economic trends.
• Engineering: Systems of inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.

Q: Why is it important to understand systems of inequalities?

A: Understanding systems of inequalities is important because it can help you make better decisions and solve problems more efficiently. Systems of inequalities are used in many real-world applications, and being able to solve them can give you a competitive edge in your career.

Q: Can I use systems of inequalities to solve real-world problems?

A: Yes, you can use systems of inequalities to solve real-world problems. Systems of inequalities can be used to model a wide range of problems, from finance and economics to engineering and science.

Q: What are some tips for solving systems of inequalities?

A: Some tips for solving systems of inequalities include:

• Start by graphing the inequalities on a coordinate plane.
• Find the intersection of the two graphs.
• Check if the solution is correct by plugging the values of the variables into each inequality.
• Use a graphing calculator or a computer program to help you solve the system.
• Practice, practice, practice! The more you practice solving systems of inequalities, the more comfortable you will become with the process.