Which Is The Graph Of The Linear Inequality $2x - 3y \ \textless \ 12$?

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Introduction

Linear inequalities are mathematical expressions that contain a linear function and an inequality sign. They are used to describe a set of points that satisfy a certain condition. In this article, we will focus on the linear inequality $2x - 3y \ \textless \ 12$ and determine its graph.

Understanding Linear Inequalities

A linear inequality is a mathematical expression that contains a linear function and an inequality sign. The general form of a linear inequality is $ax + by \ \textless \ c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables. The inequality sign can be either less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥).

Graphing Linear Inequalities

To graph a linear inequality, we need to find the boundary line and determine the direction of the inequality. The boundary line is the line that separates the region where the inequality is true from the region where it is false. The direction of the inequality is determined by the inequality sign.

Finding the Boundary Line

The boundary line for the linear inequality $2x - 3y \ \textless \ 12$ is the line $2x - 3y = 12$. To find the equation of the boundary line, we can rewrite the inequality as an equation by replacing the inequality sign with an equal sign.

Graphing the Boundary Line

To graph the boundary line, we can use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope of the boundary line is $m = \frac{2}{3}$, and the y-intercept is $b = 4$.

Determining the Direction of the Inequality

The direction of the inequality is determined by the inequality sign. Since the inequality sign is less than (<), the region where the inequality is true is below the boundary line.

Graphing the Linear Inequality

To graph the linear inequality $2x - 3y \ \textless \ 12$, we need to graph the boundary line and shade the region below it. The graph of the linear inequality is a half-plane that lies below the boundary line.

Conclusion

In conclusion, the graph of the linear inequality $2x - 3y \ \textless \ 12$ is a half-plane that lies below the boundary line. The boundary line is the line $2x - 3y = 12$, and the direction of the inequality is determined by the inequality sign.

Example

Let's consider an example to illustrate the graph of the linear inequality. Suppose we want to graph the linear inequality $2x - 3y \ \textless \ 12$.

Step 1: Find the Boundary Line

The boundary line for the linear inequality $2x - 3y \ \textless \ 12$ is the line $2x - 3y = 12$.

Step 2: Graph the Boundary Line

To graph the boundary line, we can use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope of the boundary line is $m = \frac{2}{3}$, and the y-intercept is $b = 4$.

Step 3: Determine the Direction of the Inequality

The direction of the inequality is determined by the inequality sign. Since the inequality sign is less than (<), the region where the inequality is true is below the boundary line.

Step 4: Graph the Linear Inequality

To graph the linear inequality $2x - 3y \ \textless \ 12$, we need to graph the boundary line and shade the region below it. The graph of the linear inequality is a half-plane that lies below the boundary line.

Final Answer

The final answer is a half-plane that lies below the boundary line $2x - 3y = 12$.

Introduction

Graphing linear inequalities can be a challenging task, especially for those who are new to the concept. In this article, we will answer some of the most frequently asked questions about graphing linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is a mathematical expression that contains a linear function and an inequality sign. The general form of a linear inequality is $ax + by \ \textless \ c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to find the boundary line and determine the direction of the inequality. The boundary line is the line that separates the region where the inequality is true from the region where it is false. The direction of the inequality is determined by the inequality sign.

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 it is false. It is the line that is equal to the value of the inequality.

Q: How do I find the boundary line?

A: To find the boundary line, you can rewrite the inequality as an equation by replacing the inequality sign with an equal sign.

Q: What is the direction of the inequality?

A: The direction of the inequality is determined by the inequality sign. If the inequality sign is less than (<), the region where the inequality is true is below the boundary line. If the inequality sign is greater than (>), the region where the inequality is true is above the boundary line.

Q: How do I graph the linear inequality?

A: To graph the linear inequality, you need to graph the boundary line and shade the region where the inequality is true.

Q: What is the graph of a linear inequality?

A: The graph of a linear inequality is a half-plane that lies below or above the boundary line, depending on the direction of the inequality.

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

A: Yes, you can graph a linear inequality with a negative slope. The graph will be a half-plane that lies below or above the boundary line, depending on the direction of the inequality.

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

A: Yes, you can graph a linear inequality with a zero slope. The graph will be a half-plane that lies below or above the boundary line, depending on the direction of the inequality.

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

A: Yes, you can graph a linear inequality with a fractional slope. The graph will be a half-plane that lies below or above the boundary line, depending on the direction of the inequality.

Q: Can I graph a linear inequality with a negative y-intercept?

A: Yes, you can graph a linear inequality with a negative y-intercept. The graph will be a half-plane that lies below or above the boundary line, depending on the direction of the inequality.

Q: Can I graph a linear inequality with a fractional y-intercept?

A: Yes, you can graph a linear inequality with a fractional y-intercept. The graph will be a half-plane that lies below or above the boundary line, depending on the direction of the inequality.

Q: How do I determine the direction of the inequality?

A: To determine the direction of the inequality, you need to look at the inequality sign. If the inequality sign is less than (<), the region where the inequality is true is below the boundary line. If the inequality sign is greater than (>), the region where the inequality is true is above the boundary line.

Q: Can I graph a linear inequality with a negative x-intercept?

A: Yes, you can graph a linear inequality with a negative x-intercept. The graph will be a half-plane that lies below or above the boundary line, depending on the direction of the inequality.

Q: Can I graph a linear inequality with a fractional x-intercept?

A: Yes, you can graph a linear inequality with a fractional x-intercept. The graph will be a half-plane that lies below or above the boundary line, depending on the direction of the inequality.

Conclusion

In conclusion, graphing linear inequalities can be a challenging task, but with the right tools and techniques, it can be done easily. By following the steps outlined in this article, you can graph linear inequalities with ease.

Example

Let's consider an example to illustrate the graph of a linear inequality. Suppose we want to graph the linear inequality $2x - 3y \ \textless \ 12$.

Step 1: Find the Boundary Line

The boundary line for the linear inequality $2x - 3y \ \textless \ 12$ is the line $2x - 3y = 12$.

Step 2: Graph the Boundary Line

To graph the boundary line, we can use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope of the boundary line is $m = \frac{2}{3}$, and the y-intercept is $b = 4$.

Step 3: Determine the Direction of the Inequality

The direction of the inequality is determined by the inequality sign. Since the inequality sign is less than (<), the region where the inequality is true is below the boundary line.

Step 4: Graph the Linear Inequality

To graph the linear inequality $2x - 3y \ \textless \ 12$, we need to graph the boundary line and shade the region below it. The graph of the linear inequality is a half-plane that lies below the boundary line.

Final Answer

The final answer is a half-plane that lies below the boundary line $2x - 3y = 12$.