Which Is The Graph Of The Linear Inequality X − 2 Y ≥ − 12 X - 2y \geq -12 X − 2 Y ≥ − 12 ?

Mar 10, 2025 by ADMIN 92 views

**Graphing Linear Inequalities: Understanding the Basics**

Introduction

Graphing linear inequalities is a fundamental concept in mathematics, particularly in algebra and geometry. It involves representing the solution set of an inequality on a coordinate plane. In this article, we will focus on graphing the linear inequality $x - 2y \geq -12$ . We will explore the concept of linear inequalities, understand the properties of the given inequality, and learn how to graph it on a coordinate plane.

Understanding Linear Inequalities

A linear inequality is an inequality that can be written in the form $ax + by \geq c$ , where $a$ , $b$ , and $c$ are constants, and $x$ and $y$ are variables. The inequality $x - 2y \geq -12$ is a linear inequality, where $a = 1$ , $b = -2$ , and $c = -12$ .

Properties of the Inequality

To graph the inequality $x - 2y \geq -12$ , we need to understand its properties. The inequality can be rewritten as $y \leq \frac{1}{2}x + 6$ . This form makes it easier to identify the slope and the y-intercept of the line.

Slope: The slope of the line is $\frac{1}{2}$ . This means that for every unit increase in $x$ , the value of $y$ increases by $\frac{1}{2}$ unit.
Y-intercept: The y-intercept of the line is $6$ . This means that when $x = 0$ , the value of $y$ is $6$ .

Graphing the Inequality

To graph the inequality $x - 2y \geq -12$ , we need to graph the line $y = \frac{1}{2}x + 6$ and shade the region on one side of the line.

Graphing the Line: To graph the line $y = \frac{1}{2}x + 6$ , we can use the slope-intercept form of a line, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. In this case, $m = \frac{1}{2}$ and $b = 6$ . We can plot the y-intercept at $(0, 6)$ and use the slope to find other points on the line.
Shading the Region: To shade the region on one side of the line, we need to determine which side of the line satisfies the inequality. Since the inequality is $x - 2y \geq -12$ , we need to find the region where $y \leq \frac{1}{2}x + 6$ . This means that we need to shade the region below the line.

Graphing the Inequality on a Coordinate Plane

To graph the inequality $x - 2y \geq -12$ on a coordinate plane, we need to use a coordinate grid with $x$ and $y$ axes. We can graph the line $y = \frac{1}{2}x + 6$ and shade the region below the line to represent the solution set of the inequality.

Conclusion

Graphing linear inequalities is an important concept in mathematics, particularly in algebra and geometry. In this article, we learned how to graph the linear inequality $x - 2y \geq -12$ by understanding its properties, graphing the line, and shading the region on one side of the line. We also learned how to graph the inequality on a coordinate plane using a coordinate grid with $x$ and $y$ axes. By following these steps, we can graph any linear inequality and understand its solution set.

Example Problems

Graph the linear inequality $2x + 3y \geq 12$ .
Graph the linear inequality $x - 4y \geq 8$ .
Graph the linear inequality $3x + 2y \geq 10$ .

Solutions

To graph the linear inequality $2x + 3y \geq 12$ , we need to graph the line $y = -\frac{2}{3}x + 4$ and shade the region below the line.
To graph the linear inequality $x - 4y \geq 8$ , we need to graph the line $y = \frac{1}{4}x - 2$ and shade the region above the line.
To graph the linear inequality $3x + 2y \geq 10$ , we need to graph the line $y = -\frac{3}{2}x + 5$ and shade the region below the line.

Tips and Tricks

When graphing a linear inequality, make sure to graph the line and shade the region on one side of the line.
Use a coordinate grid with $x$ and $y$ axes to graph the inequality on a coordinate plane.
Make sure to label the axes and the line to make it easier to understand the graph.

Real-World Applications

Graphing linear inequalities has many real-world applications, such as:
- Finding the solution set of a system of linear inequalities.
- Graphing the profit or cost function of a business.
- Finding the maximum or minimum value of a function.

Conclusion

Introduction

Graphing linear inequalities is a fundamental concept in mathematics, particularly in algebra and geometry. In our previous article, we learned how to graph the linear inequality $x - 2y \geq -12$ by understanding its properties, graphing the line, and shading the region on one side of the line. In this article, we will answer some frequently asked questions about graphing linear inequalities.

Q&A

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

A: The main difference between graphing a linear equation and graphing a linear inequality is that a linear equation represents an equal relationship between two expressions, while a linear inequality represents a greater than or less than relationship between two expressions.

Q: How do I determine which side of the line to shade when graphing a linear inequality?

A: To determine which side of the line to shade, you need to look at the inequality sign. If the inequality sign is greater than or equal to (≥), you need to shade the region above the line. If the inequality sign is less than or equal to (≤), you need to shade the region below the line.

Q: Can I graph a linear inequality with a negative slope?

A: Yes, you can graph a linear inequality with a negative slope. To graph a linear inequality with a negative slope, you need to follow the same steps as graphing a linear inequality with a positive slope, but you need to use a negative slope.

Q: How do I graph a linear inequality with a fractional slope?

A: To graph a linear inequality with a fractional slope, you need to follow the same steps as graphing a linear inequality with a positive slope, but you need to use a fractional slope. You can use the slope-intercept form of a line, which is y = mx + b, where m is the slope and b is the y-intercept.

Q: Can I graph a linear inequality with a vertical line?

A: Yes, you can graph a linear inequality with a vertical line. To graph a linear inequality with a vertical line, you need to follow the same steps as graphing a linear inequality with a horizontal line, but you need to use a vertical line.

Q: How do I graph a system of linear inequalities?

A: To graph a system of linear inequalities, you need to graph each inequality separately and find the intersection of the two graphs. The solution set of the system is the region where the two graphs intersect.

Q: Can I use a graphing calculator to graph a linear inequality?

A: Yes, you can use a graphing calculator to graph a linear inequality. Graphing calculators can graph linear inequalities and find the solution set of the inequality.

Conclusion

Graphing linear inequalities is an important concept in mathematics, particularly in algebra and geometry. In this article, we answered some frequently asked questions about graphing linear inequalities. We hope that this article has helped you to understand how to graph linear inequalities and has provided you with the tools you need to solve problems involving linear inequalities.

Tips and Tricks

When graphing a linear inequality, make sure to graph the line and shade the region on one side of the line.
Use a coordinate grid with x and y axes to graph the inequality on a coordinate plane.
Make sure to label the axes and the line to make it easier to understand the graph.
Use a graphing calculator to graph a linear inequality and find the solution set of the inequality.

Real-World Applications

Graphing linear inequalities has many real-world applications, such as:
- Finding the solution set of a system of linear inequalities.
- Graphing the profit or cost function of a business.
- Finding the maximum or minimum value of a function.