Which Ordered Pair Is A Solution To The Linear Inequality Below?$y \geq -2x + 7$A. $(-1, 5$\] B. $(3, 2$\] C. $(2, -8$\] D. $(-4, 0$\]

Introduction

Linear inequalities are a fundamental concept in mathematics, and they play a crucial role in various fields such as algebra, geometry, and calculus. In this article, we will focus on solving linear inequalities, specifically the inequality $y \geq -2x + 7$. We will explore the concept of linear inequalities, learn how to solve them, and apply this knowledge to find the solution to the given inequality.

What are Linear Inequalities?

A linear inequality is an inequality that involves a linear expression, which is an expression that can be written in the form $ax + b$, where $a$ and $b$ are constants, and $x$ is the variable. Linear inequalities can be written in the following forms:

• $ax + b > c$
• $ax + b < c$
• $ax + b \geq c$
• $ax + b \leq c$

where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Solving Linear Inequalities

To solve a linear inequality, we need to isolate the variable $x$ on one side of the inequality. We can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

For example, consider the inequality $2x + 3 > 5$. To solve this inequality, we can subtract 3 from both sides of the inequality, which gives us $2x > 2$. We can then divide both sides of the inequality by 2, which gives us $x > 1$.

Solving the Inequality $y \geq -2x + 7$

To solve the inequality $y \geq -2x + 7$, we can use the same steps as before. We can start by isolating the variable $x$ on one side of the inequality. We can do this by subtracting 7 from both sides of the inequality, which gives us $y - 7 \geq -2x$. We can then add $2x$ to both sides of the inequality, which gives us $y - 7 + 2x \geq 0$. We can then add 7 to both sides of the inequality, which gives us $y + 2x \geq 7$.

Finding the Solution to the Inequality

To find the solution to the inequality $y \geq -2x + 7$, we need to find the values of $x$ and $y$ that satisfy the inequality. We can do this by graphing the inequality on a coordinate plane.

The inequality $y \geq -2x + 7$ can be graphed as a line with a slope of -2 and a y-intercept of 7. The line has a negative slope, which means that it slopes downward from left to right. The inequality $y \geq -2x + 7$ is satisfied by all points on or above the line.

Evaluating the Answer Choices

Now that we have found the solution to the inequality $y \geq -2x + 7$, we can evaluate the answer choices to see which one is a solution to the inequality.

• A. $(-1, 5)$: To determine if this point is a solution to the inequality, we can plug in the values of $x$ and $y$ into the inequality. We get $5 \geq -2(-1) + 7$, which simplifies to $5 \geq 9$. This is not true, so the point $(-1, 5)$ is not a solution to the inequality.
• B. $(3, 2)$: To determine if this point is a solution to the inequality, we can plug in the values of $x$ and $y$ into the inequality. We get $2 \geq -2(3) + 7$, which simplifies to $2 \geq -6 + 7$. This is true, so the point $(3, 2)$ is a solution to the inequality.
• C. $(2, -8)$: To determine if this point is a solution to the inequality, we can plug in the values of $x$ and $y$ into the inequality. We get $-8 \geq -2(2) + 7$, which simplifies to $-8 \geq -4 + 7$. This is not true, so the point $(2, -8)$ is not a solution to the inequality.
• D. $(-4, 0)$: To determine if this point is a solution to the inequality, we can plug in the values of $x$ and $y$ into the inequality. We get $0 \geq -2(-4) + 7$, which simplifies to $0 \geq 8 + 7$. This is not true, so the point $(-4, 0)$ is not a solution to the inequality.

Conclusion

In this article, we have learned how to solve linear inequalities, specifically the inequality $y \geq -2x + 7$. We have also evaluated the answer choices to see which one is a solution to the inequality. The correct answer is B. $(3, 2)$.

Final Answer

The final answer is B. $(3, 2)$.

Introduction

Linear inequalities are a fundamental concept in mathematics, and they play a crucial role in various fields such as algebra, geometry, and calculus. In this article, we will provide a Q&A guide to help you understand linear inequalities and how to solve them.

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression, which is an expression that can be written in the form $ax + b$, where $a$ and $b$ are constants, and $x$ is the variable.

Q: What are the different types of linear inequalities?

A: There are four types of linear inequalities:

• $ax + b > c$
• $ax + b < c$
• $ax + b \geq c$
• $ax + b \leq c$

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable $x$ on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear equation is an equation that involves a linear expression, whereas a linear inequality is an inequality that involves a linear expression. In other words, a linear equation is an equation that can be written in the form $ax + b = c$, whereas a linear inequality is an inequality that can be written in one of the four forms listed above.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to graph the corresponding linear equation and then shade the region that satisfies the inequality. For example, if the inequality is $y \geq 2x + 3$, you would graph the line $y = 2x + 3$ and then shade the region above the line.

Q: What is the solution to a linear inequality?

A: The solution to a linear inequality is the set of all values of $x$ that satisfy the inequality. In other words, it is the set of all values of $x$ that make the inequality true.

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

A: To find the solution to a linear inequality, you need to isolate the variable $x$ on one side of the inequality and then determine the values of $x$ that satisfy the inequality. You can do this by graphing the inequality on a coordinate plane and then identifying the region that satisfies the inequality.

Q: What is the difference between a solution and a solution set?

A: A solution is a single value of $x$ that satisfies the inequality, whereas a solution set is the set of all values of $x$ that satisfy the inequality.

Q: How do I determine if a point is a solution to a linear inequality?

A: To determine if a point is a solution to a linear inequality, you need to plug in the values of $x$ and $y$ into the inequality and then determine if the inequality is true.

Q: What is the importance of linear inequalities in real-life applications?

A: Linear inequalities are used in a wide range of real-life applications, including finance, economics, and engineering. They are used to model and solve problems that involve constraints and limitations.

Conclusion

In this article, we have provided a Q&A guide to help you understand linear inequalities and how to solve them. We hope that this guide has been helpful in answering your questions and providing you with a better understanding of linear inequalities.

Final Answer

The final answer is that linear inequalities are an important concept in mathematics and have many real-life applications. They are used to model and solve problems that involve constraints and limitations, and they are an essential tool for anyone who wants to succeed in mathematics and other fields.