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

Feb 26, 2025 by ADMIN 76 views

**Graphing Linear Inequalities: A Step-by-Step Guide**

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Introduction

Linear inequalities are a fundamental concept in mathematics, and graphing them is an essential skill for students and professionals alike. In this article, we will focus on graphing the linear inequality $2x - 3y < 12$ . We will break down the process into manageable steps, providing a clear and concise 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. The inequality $2x - 3y < 12$ is a linear inequality, where $a = 2$ , $b = -3$ , and $c = 12$ .

Graphing Linear Inequalities

To graph a linear inequality, we need to find the boundary line and then determine the region that satisfies the inequality. The boundary line is the line that divides the region into two parts: the region that satisfies the inequality and the region that does not.

Finding the Boundary Line

The boundary line for the inequality $2x - 3y < 12$ is the line $2x - 3y = 12$ . To find the boundary line, we can rewrite the inequality as an equation by replacing the inequality symbol 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 $b = -4$ .

import numpy as np
import matplotlib.pyplot as plt

# Define the slope and y-intercept
m = 2/3
b = -4

# Generate x values
x = np.linspace(-10, 10, 400)

# Calculate y values
y = m * x + b

# Plot the boundary line
plt.plot(x, y, label='Boundary Line')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Boundary Line')
plt.legend()
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Determining the Region

To determine the region that satisfies the inequality, we need to test a point in each region. Let's test the point $(0, 0)$ , which is in the region that does not satisfy the inequality.

# Test the point (0, 0)
x_test = 0
y_test = 0

# Calculate the value of the inequality at the test point
value = 2 * x_test - 3 * y_test

# Print the result
print(f'The value of the inequality at the test point (0, 0) is {value}')

Since the value of the inequality at the test point $(0, 0)$ is greater than $12$ , we know that the region that does not satisfy the inequality is the region above the boundary line.

Graphing the Linear Inequality

Now that we have determined the region that satisfies the inequality, we can graph the linear inequality.

import numpy as np
import matplotlib.pyplot as plt

# Define the slope and y-intercept
m = 2/3
b = -4

# Generate x values
x = np.linspace(-10, 10, 400)

# Calculate y values
y = m * x + b

# Plot the boundary line
plt.plot(x, y, label='Boundary Line')

# Plot the region that satisfies the inequality
plt.fill_between(x, y, color='blue', alpha=0.2)

# Add labels and title
plt.xlabel('x')
plt.ylabel('y')
plt.title('Linear Inequality')
plt.legend()
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Conclusion

In this article, we have graphed the linear inequality $2x - 3y < 12$ . We have broken down the process into manageable steps, providing a clear and concise explanation of each step. We have found the boundary line, graphed the boundary line, determined the region that satisfies the inequality, and graphed the linear inequality.

Final Thoughts

Graphing linear inequalities is an essential skill for students and professionals alike. By following the steps outlined in this article, you can graph any linear inequality. Remember to find the boundary line, graph the boundary line, determine the region that satisfies the inequality, and graph the linear inequality.

Common Mistakes

When graphing linear inequalities, there are several common mistakes to avoid. These include:

Graphing the wrong boundary line: Make sure to graph the correct boundary line.
Determining the wrong region: Make sure to determine the correct region that satisfies the inequality.
Graphing the wrong linear inequality: Make sure to graph the correct linear inequality.

Real-World Applications

Graphing linear inequalities has many real-world applications. These include:

Optimization problems: Graphing linear inequalities can help solve optimization problems.
Budgeting: Graphing linear inequalities can help create a budget.
Scheduling: Graphing linear inequalities can help create a schedule.

Conclusion

In conclusion, graphing linear inequalities is an essential skill for students and professionals alike. By following the steps outlined in this article, you can graph any linear inequality. Remember to find the boundary line, graph the boundary line, determine the region that satisfies the inequality, and graph the linear inequality.

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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 graph a linear inequality?

A: To graph a linear inequality, you need to find the boundary line and then determine the region that satisfies the inequality. The boundary line is the line that divides the region into two parts: the region that satisfies the inequality and the region that does not.

Q: What is the boundary line?

A: The boundary line is the line that divides the region into two parts: the region that satisfies the inequality and the region that does not. It is the line that is equal to 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 symbol with an equal sign.

Q: What is the region that satisfies the inequality?

A: The region that satisfies the inequality is the region that is below the boundary line. This is the region where the inequality is true.

Q: How do I determine the region that satisfies the inequality?

A: To determine the region that satisfies the inequality, you need to test a point in each region. If the point satisfies the inequality, then the region is the region that satisfies the inequality.

Q: What is the significance of the boundary line?

A: The boundary line is significant because it divides the region into two parts: the region that satisfies the inequality and the region that does not. It is the line that is equal to 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 process is the same as graphing a linear inequality with a positive slope.

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

A: Yes, you can graph a linear inequality with a zero slope. The process is the same as graphing a linear inequality with a positive slope.

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

A: Yes, you can graph a linear inequality with a vertical line. The process is the same as graphing a linear inequality with a positive slope.

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

A: Yes, you can graph a linear inequality with a horizontal line. The process is the same as graphing a linear inequality with a positive slope.

Q: What are some real-world applications of graphing linear inequalities?

A: Some real-world applications of graphing linear inequalities include:

Optimization problems: Graphing linear inequalities can help solve optimization problems.
Budgeting: Graphing linear inequalities can help create a budget.
Scheduling: Graphing linear inequalities can help create a schedule.

Q: How do I use graphing linear inequalities in real-world applications?

A: To use graphing linear inequalities in real-world applications, you need to identify the variables and the constraints. Then, you can graph the linear inequality and use it to solve the problem.

Q: What are some common mistakes to avoid when graphing linear inequalities?

A: Some common mistakes to avoid when graphing linear inequalities include:

Graphing the wrong boundary line: Make sure to graph the correct boundary line.
Determining the wrong region: Make sure to determine the correct region that satisfies the inequality.
Graphing the wrong linear inequality: Make sure to graph the correct linear inequality.

Q: How do I avoid common mistakes when graphing linear inequalities?

A: To avoid common mistakes when graphing linear inequalities, you need to carefully read the problem and identify the variables and the constraints. Then, you can graph the linear inequality and use it to solve the problem.

Q: What are some tips for graphing linear inequalities?

A: Some tips for graphing linear inequalities include:

Read the problem carefully: Make sure to read the problem carefully and identify the variables and the constraints.
Graph the boundary line correctly: Make sure to graph the correct boundary line.
Determine the correct region: Make sure to determine the correct region that satisfies the inequality.
Graph the correct linear inequality: Make sure to graph the correct linear inequality.

Q: How do I graph linear inequalities with multiple variables?

A: To graph linear inequalities with multiple variables, you need to identify the variables and the constraints. Then, you can graph the linear inequality and use it to solve the problem.

Q: What are some real-world applications of graphing linear inequalities with multiple variables?

A: Some real-world applications of graphing linear inequalities with multiple variables include:

Optimization problems: Graphing linear inequalities with multiple variables can help solve optimization problems.
Budgeting: Graphing linear inequalities with multiple variables can help create a budget.
Scheduling: Graphing linear inequalities with multiple variables can help create a schedule.

Q: How do I use graphing linear inequalities with multiple variables in real-world applications?

A: To use graphing linear inequalities with multiple variables in real-world applications, you need to identify the variables and the constraints. Then, you can graph the linear inequality and use it to solve the problem.

Q: What are some common mistakes to avoid when graphing linear inequalities with multiple variables?

A: Some common mistakes to avoid when graphing linear inequalities with multiple variables include:

Graphing the wrong boundary line: Make sure to graph the correct boundary line.
Determining the wrong region: Make sure to determine the correct region that satisfies the inequality.
Graphing the wrong linear inequality: Make sure to graph the correct linear inequality.

Q: How do I avoid common mistakes when graphing linear inequalities with multiple variables?

A: To avoid common mistakes when graphing linear inequalities with multiple variables, you need to carefully read the problem and identify the variables and the constraints. Then, you can graph the linear inequality and use it to solve the problem.

Q: What are some tips for graphing linear inequalities with multiple variables?

A: Some tips for graphing linear inequalities with multiple variables include:

Read the problem carefully: Make sure to read the problem carefully and identify the variables and the constraints.
Graph the boundary line correctly: Make sure to graph the correct boundary line.
Determine the correct region: Make sure to determine the correct region that satisfies the inequality.
Graph the correct linear inequality: Make sure to graph the correct linear inequality.