Which Ordered Pair Is A Solution To $-5x + 3y \ \textgreater \ 12$?A. (3, 9) B. (-5, 5) C. (3, 6) D. (-2, 5) E. (2, 8) F. (6, 0)

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Introduction

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Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving the linear inequality $-5x + 3y > 12$ and determine which ordered pair is a solution to this inequality.

Understanding Linear Inequalities

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A linear inequality is an inequality that can be written in the form $ax + by > c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables. The inequality $-5x + 3y > 12$ is a linear inequality, and we need to find the ordered pair that satisfies this inequality.

Graphing the Inequality

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To solve the inequality $-5x + 3y > 12$, we can graph the related equation $-5x + 3y = 12$ on a coordinate plane. The graph of this equation is a line, and we can use this line to determine the region that satisfies the inequality.

Finding the Solution Region

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To find the solution region, we need to determine which side of the line $-5x + 3y = 12$ satisfies the inequality $-5x + 3y > 12$. We can do this by choosing a point on one side of the line and checking if it satisfies the inequality.

Choosing a Test Point

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Let's choose the point $(0, 0)$ as a test point. We can plug this point into the inequality $-5x + 3y > 12$ to see if it satisfies the inequality.

Evaluating the Test Point

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Plugging in the point $(0, 0)$ into the inequality $-5x + 3y > 12$, we get:

$-5(0) + 3(0) > 12$

$0 > 12$

This is a false statement, so the point $(0, 0)$ does not satisfy the inequality.

Determining the Solution Region

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Since the point $(0, 0)$ does not satisfy the inequality, we know that the solution region lies on the side of the line $-5x + 3y = 12$ that is not containing the point $(0, 0)$. We can choose another point on the other side of the line and check if it satisfies the inequality.

Choosing Another Test Point

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Let's choose the point $(2, 8)$ as another test point. We can plug this point into the inequality $-5x + 3y > 12$ to see if it satisfies the inequality.

Evaluating the Test Point

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Plugging in the point $(2, 8)$ into the inequality $-5x + 3y > 12$, we get:

$-5(2) + 3(8) > 12$

$-10 + 24 > 12$

$14 > 12$

This is a true statement, so the point $(2, 8)$ satisfies the inequality.

Determining the Solution Region

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Since the point $(2, 8)$ satisfies the inequality, we know that the solution region lies on the side of the line $-5x + 3y = 12$ that contains the point $(2, 8)$. We can graph the solution region on a coordinate plane.

Graphing the Solution Region

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The solution region is the area above the line $-5x + 3y = 12$ and below the line $-5x + 3y = 12$.

Finding the Ordered Pair

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We need to find the ordered pair that lies in the solution region. Let's examine the answer choices and see which one lies in the solution region.

Examining the Answer Choices

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The answer choices are:

A. (3, 9) B. (-5, 5) C. (3, 6) D. (-2, 5) E. (2, 8) F. (6, 0)

We can plug each of these points into the inequality $-5x + 3y > 12$ to see if it satisfies the inequality.

Evaluating the Answer Choices

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Plugging in the point (3, 9) into the inequality $-5x + 3y > 12$, we get:

$-5(3) + 3(9) > 12$

$-15 + 27 > 12$

$12 > 12$

This is a false statement, so the point (3, 9) does not satisfy the inequality.

Plugging in the point (-5, 5) into the inequality $-5x + 3y > 12$, we get:

$-5(-5) + 3(5) > 12$

$25 + 15 > 12$

$40 > 12$

This is a true statement, so the point (-5, 5) satisfies the inequality.

Plugging in the point (3, 6) into the inequality $-5x + 3y > 12$, we get:

$-5(3) + 3(6) > 12$

$-15 + 18 > 12$

$3 > 12$

This is a false statement, so the point (3, 6) does not satisfy the inequality.

Plugging in the point (-2, 5) into the inequality $-5x + 3y > 12$, we get:

$-5(-2) + 3(5) > 12$

$10 + 15 > 12$

$25 > 12$

This is a true statement, so the point (-2, 5) satisfies the inequality.

Plugging in the point (2, 8) into the inequality $-5x + 3y > 12$, we get:

$-5(2) + 3(8) > 12$

$-10 + 24 > 12$

$14 > 12$

This is a true statement, so the point (2, 8) satisfies the inequality.

Plugging in the point (6, 0) into the inequality $-5x + 3y > 12$, we get:

$-5(6) + 3(0) > 12$

$-30 + 0 > 12$

$-30 > 12$

This is a false statement, so the point (6, 0) does not satisfy the inequality.

Determining the Correct Answer

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Based on the evaluations above, we can see that the points (-5, 5), (-2, 5), and (2, 8) satisfy the inequality. However, we need to choose only one answer, and the correct answer is the one that is most consistent with the solution region.

Conclusion

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In conclusion, the ordered pair that is a solution to the inequality $-5x + 3y > 12$ is (-5, 5). This is because the point (-5, 5) lies in the solution region, which is the area above the line $-5x + 3y = 12$ and below the line $-5x + 3y = 12$.

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Q: What is a linear inequality?

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A: A linear inequality is an inequality that can be written in the form $ax + by > c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables.

Q: How do I graph a linear inequality?

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A: To graph a linear inequality, you need to graph the related equation $ax + by = c$ on a coordinate plane. The graph of this equation is a line, and you can use this line to determine the region that satisfies the inequality.

Q: How do I find the solution region of a linear inequality?

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A: To find the solution region, you need to determine which side of the line $ax + by = c$ satisfies the inequality. You can do this by choosing a point on one side of the line and checking if it satisfies the inequality.

Q: What is a test point?

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A: A test point is a point on one side of the line $ax + by = c$ that you use to check if it satisfies the inequality.

Q: How do I choose a test point?

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A: You can choose any point on one side of the line $ax + by = c$ as a test point. It's best to choose a point that is easy to work with, such as the origin $(0, 0)$.

Q: What is the solution region of a linear inequality?

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A: The solution region of a linear inequality is the area on the coordinate plane that satisfies the inequality. It's the area above the line $ax + by = c$ and below the line $ax + by = c$.

Q: How do I determine the correct answer for a linear inequality?

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A: To determine the correct answer, you need to find the ordered pair that lies in the solution region. You can do this by plugging in the answer choices into the inequality and checking if they satisfy the inequality.

Q: What is the most important thing to remember when solving linear inequalities?

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A: The most important thing to remember is to always graph the related equation $ax + by = c$ on a coordinate plane and use it to determine the solution region.

Q: Can I use a calculator to solve linear inequalities?

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A: Yes, you can use a calculator to solve linear inequalities. However, it's always best to graph the related equation on a coordinate plane and use it to determine the solution region.

Q: How do I know if an ordered pair is a solution to a linear inequality?

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A: To know if an ordered pair is a solution to a linear inequality, you need to plug it into the inequality and check if it satisfies the inequality.

Q: Can I have multiple solutions to a linear inequality?

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A: Yes, you can have multiple solutions to a linear inequality. The solution region is the area on the coordinate plane that satisfies the inequality, and there can be multiple points in this region.

Q: How do I write a linear inequality in standard form?

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A: To write a linear inequality in standard form, you need to rewrite it in the form $ax + by > c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables.

Q: Can I use a linear inequality to solve a system of equations?

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A: Yes, you can use a linear inequality to solve a system of equations. You can use the inequality to determine the solution region and then find the points in this region that satisfy the system of equations.

Q: How do I use a linear inequality to solve a system of equations?

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A: To use a linear inequality to solve a system of equations, you need to graph the related equations on a coordinate plane and use the inequality to determine the solution region. Then, you can find the points in this region that satisfy the system of equations.

Q: Can I use a linear inequality to solve a system of inequalities?

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A: Yes, you can use a linear inequality to solve a system of inequalities. You can use the inequality to determine the solution region and then find the points in this region that satisfy the system of inequalities.

Q: How do I use a linear inequality to solve a system of inequalities?

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A: To use a linear inequality to solve a system of inequalities, you need to graph the related equations on a coordinate plane and use the inequality to determine the solution region. Then, you can find the points in this region that satisfy the system of inequalities.

Conclusion

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In conclusion, solving linear inequalities is an important skill in mathematics, and it's essential to understand the concepts and techniques involved. By following the steps outlined in this article, you can solve linear inequalities and determine the solution region. Remember to always graph the related equation on a coordinate plane and use it to determine the solution region.