Graph The Following System Of Inequalities.$\[ \begin{align*} y & \leq \frac{1}{3} X - 2 \\ x & \ \textless \ 4 \end{align*} \\]

Introduction

Graphing a system of inequalities involves finding the solution set that satisfies both inequalities in the system. In this article, we will focus on graphing the system of inequalities given by $y \leq \frac{1}{3} x - 2$ and $x < 4$. We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding the Inequalities

Before we can graph the system of inequalities, we need to understand the individual inequalities. The first inequality is $y \leq \frac{1}{3} x - 2$. This is a linear inequality in the slope-intercept form, where the slope is $\frac{1}{3}$ and the y-intercept is $-2$. The inequality indicates that the solution set lies on or below the line defined by the equation $y = \frac{1}{3} x - 2$.

The second inequality is $x < 4$. This is a linear inequality in the form of $x < a$, where $a$ is the boundary value. In this case, the boundary value is $4$, and the inequality indicates that the solution set lies to the left of the line $x = 4$.

Graphing the First Inequality

To graph the first inequality, we need to graph the line defined by the equation $y = \frac{1}{3} x - 2$. We can do this by plotting two points on the line and drawing a line through them. Let's choose the points $(0, -2)$ and $(3, -1)$.

import matplotlib.pyplot as plt
import numpy as np

# Define the x-values
x = np.array([0, 3])

# Define the y-values
y = np.array([-2, -1])

# Plot the points
plt.scatter(x, y)

# Plot the line
plt.plot(x, y, 'r-')

# Set the title and labels
plt.title('Graph of y = (1/3)x - 2')
plt.xlabel('x')
plt.ylabel('y')

# Display the plot
plt.show()

Graphing the Second Inequality

To graph the second inequality, we need to graph the line defined by the equation $x = 4$. We can do this by plotting a single point on the line and drawing a vertical line through it. Let's choose the point $(4, 0)$.

import matplotlib.pyplot as plt
import numpy as np

# Define the x-value
x = np.array([4])

# Plot the point
plt.scatter(x, np.array([0]))

# Plot the vertical line
plt.axvline(x=4, color='r')

# Set the title and labels
plt.title('Graph of x < 4')
plt.xlabel('x')
plt.ylabel('y')

# Display the plot
plt.show()

Graphing the System of Inequalities

Now that we have graphed the individual inequalities, we can graph the system of inequalities by shading the region that satisfies both inequalities. The solution set lies on or below the line defined by the equation $y = \frac{1}{3} x - 2$ and to the left of the line $x = 4$.

import matplotlib.pyplot as plt
import numpy as np

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

# Define the y-values
y1 = (1/3)*x - 2
y2 = np.zeros_like(x)

# Plot the lines
plt.plot(x, y1, 'r-')
plt.plot(x, y2, 'r-')

# Plot the vertical line
plt.axvline(x=4, color='r')

# Shade the region
plt.fill_between(x, y1, color='lightblue')

# Set the title and labels
plt.title('Graph of the System of Inequalities')
plt.xlabel('x')
plt.ylabel('y')

# Display the plot
plt.show()

Conclusion

Introduction

Graphing a system of inequalities can be a challenging task, but with the right guidance, it can be made easier. In this article, we will provide a Q&A guide to help you understand the process of graphing a system of inequalities.

Q: What is a system of inequalities?

A: A system of inequalities is a set of two or more inequalities that must be satisfied simultaneously. In other words, the solution set must lie in the region that satisfies all the inequalities in the system.

Q: How do I graph a system of inequalities?

A: To graph a system of inequalities, you need to graph each inequality separately and then shade the region that satisfies both inequalities. You can use the following steps:

1. Graph the first inequality by plotting two points on the line and drawing a line through them.
2. Graph the second inequality by plotting a single point on the line and drawing a vertical line through it.
3. Shade the region that satisfies both inequalities.

Q: What if the inequalities are not linear?

A: If the inequalities are not linear, you can still graph them by finding the boundary lines and then shading the region that satisfies both inequalities. You can use the following steps:

1. Find the boundary lines by solving the inequalities.
2. Graph the boundary lines by plotting two points on each line and drawing a line through them.
3. Shade the region that satisfies both inequalities.

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

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

Q: What if the system of inequalities has multiple solutions?

A: If the system of inequalities has multiple solutions, you need to shade the region that satisfies all the inequalities. You can use the following steps:

1. Graph each inequality separately.
2. Shade the region that satisfies all the inequalities.

Q: How do I check my graph for accuracy?

A: To check your graph for accuracy, you need to make sure that the shading is correct and that the boundary lines are correctly graphed. You can use the following steps:

1. Check the inequality signs to make sure that the shading is correct.
2. Check the boundary lines to make sure that they are correctly graphed.
3. Check the region that satisfies both inequalities to make sure that it is correctly shaded.

Q: What are some common mistakes to avoid when graphing a system of inequalities?

A: Some common mistakes to avoid when graphing a system of inequalities include:

• Graphing the wrong boundary lines.
• Shading the wrong region.
• Not checking the inequality signs.
• Not checking the boundary lines.

Conclusion

Graphing a system of inequalities can be a challenging task, but with the right guidance, it can be made easier. By following the steps outlined in this article, you can graph a system of inequalities with confidence. Remember to check your graph for accuracy and to avoid common mistakes.