Which Point Is A Solution To The Linear Inequality Y \textless − 1 2 X + 2 Y \ \textless \ -\frac{1}{2}x + 2 Y \textless − 2 1 X + 2 ?A. (2, 3) B. (2, 1) C. (3, -2) D. (-1, 3)

Mar 13, 2025 by ADMIN 183 views

Solving Linear Inequalities: A Step-by-Step Guide

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Introduction

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 $y \ \textless \ -\frac{1}{2}x + 2$ . We will break down the solution process into manageable steps and provide a clear explanation of each step.

Understanding Linear Inequalities

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. In this case, the linear inequality is $y \ \textless \ -\frac{1}{2}x + 2$ . To solve this inequality, we need to isolate the variable $y$ .

Isolating the Variable y

To isolate the variable $y$ , we need to get all the terms involving $y$ on one side of the inequality. In this case, we can do this by subtracting $-\frac{1}{2}x + 2$ from both sides of the inequality. This gives us:

y - (-\frac{1}{2}x + 2) < 0

Simplifying the left-hand side of the inequality, we get:

y + \frac{1}{2}x - 2 < 0

Finding the Boundary Line

The boundary line is the line that separates the region where the inequality is true from the region where the inequality is false. To find the boundary line, we need to set the inequality equal to zero and solve for $y$ . In this case, we can do this by setting $y + \frac{1}{2}x - 2 = 0$ and solving for $y$ . This gives us:

y = -\frac{1}{2}x + 2

Graphing the Boundary Line

To graph the boundary line, we need to plot the equation $y = -\frac{1}{2}x + 2$ on a coordinate plane. The boundary line is a straight line with a slope of $-\frac{1}{2}$ and a y-intercept of $2$ .

Finding the Solution Region

The solution region is the region where the inequality is true. To find the solution region, we need to test a point in the region to see if it satisfies the inequality. If the point satisfies the inequality, then the entire region is a solution to the inequality.

Testing Points

To test a point, we need to substitute the coordinates of the point into the inequality and see if it is true. Let's test the point $(2, 3)$ . Substituting the coordinates of the point into the inequality, we get:

3 + \frac{1}{2}(2) - 2 < 0

Simplifying the left-hand side of the inequality, we get:

3 + 1 - 2 < 0

2 < 0

Since $2$ is not less than $0$ , the point $(2, 3)$ does not satisfy the inequality. Therefore, the point $(2, 3)$ is not a solution to the inequality.

Testing Other Points

Let's test the other points in the list. Substituting the coordinates of the point $(2, 1)$ into the inequality, we get:

1 + \frac{1}{2}(2) - 2 < 0

Simplifying the left-hand side of the inequality, we get:

1 + 1 - 2 < 0

0 < 0

Since $0$ is not less than $0$ , the point $(2, 1)$ does not satisfy the inequality. Therefore, the point $(2, 1)$ is not a solution to the inequality.

Substituting the coordinates of the point $(3, -2)$ into the inequality, we get:

-2 + \frac{1}{2}(3) - 2 < 0

Simplifying the left-hand side of the inequality, we get:

-2 + \frac{3}{2} - 2 < 0

-2 + 1.5 - 2 < 0

-2.5 < 0

Since $-2.5$ is less than $0$ , the point $(3, -2)$ satisfies the inequality. Therefore, the point $(3, -2)$ is a solution to the inequality.

Substituting the coordinates of the point $(-1, 3)$ into the inequality, we get:

3 + \frac{1}{2}(-1) - 2 < 0

Simplifying the left-hand side of the inequality, we get:

3 - 0.5 - 2 < 0

0.5 < 0

Since $0.5$ is not less than $0$ , the point $(-1, 3)$ does not satisfy the inequality. Therefore, the point $(-1, 3)$ is not a solution to the inequality.

Conclusion

In this article, we solved the linear inequality $y \ \textless \ -\frac{1}{2}x + 2$ . We broke down the solution process into manageable steps and provided a clear explanation of each step. We found the boundary line, graphed the boundary line, and tested points to find the solution region. We concluded that the point $(3, -2)$ is a solution to the inequality.

Final Answer

The final answer is $\boxed{(3, -2)}$ .

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Introduction

In our previous article, we solved the linear inequality $y \ \textless \ -\frac{1}{2}x + 2$ . We broke down the solution process into manageable steps and provided a clear explanation of each step. In this article, we will answer some frequently asked questions about solving linear inequalities.

Q&A

Q: What is a linear inequality?

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 solve a linear inequality?

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

Q: What is the boundary line?

A: The boundary line is the line that separates the region where the inequality is true from the region where the inequality is false. To find the boundary line, you need to set the inequality equal to zero and solve for $y$ .

Q: How do I graph the boundary line?

A: To graph the boundary line, you need to plot the equation $y = mx + b$ on a coordinate plane, where $m$ is the slope and $b$ is the y-intercept.

Q: How do I find the solution region?

A: To find the solution region, you need to test a point in the region to see if it satisfies the inequality. If the point satisfies the inequality, then the entire region is a solution to the inequality.

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

A: A linear equation is an equation that can be written in the form $ax + by = c$ , where $a$ , $b$ , and $c$ are constants, and $x$ and $y$ are variables. 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: Can I use the same method to solve a linear inequality as I would to solve a linear equation?

A: No, you cannot use the same method to solve a linear inequality as you would to solve a linear equation. When solving a linear inequality, you need to isolate the variable on one side of the inequality, whereas when solving a linear equation, you need to isolate the variable on both sides of the equation.

Q: What is the importance of solving linear inequalities?

A: Solving linear inequalities is an important skill in mathematics, as it allows you to model real-world problems and make decisions based on the results. For example, you can use linear inequalities to determine the maximum or minimum value of a function, or to find the solution to a system of linear equations.

Conclusion

In this article, we answered some frequently asked questions about solving linear inequalities. We provided a clear explanation of each question and provided examples to illustrate the concepts. We hope that this article has been helpful in clarifying any confusion you may have had about solving linear inequalities.

Final Answer

The final answer is $\boxed{There is no final answer, as this is a Q&A article.}$