Determine The Relationship Between The Point $(1,-5)$ And The Given System Of Inequalities.$ \begin{array}{l} y \leq 3x + 2 \\ y \ \textgreater \ -2x - 3 \end{array} $Explain Your Answer Both Algebraically And Graphically.
Introduction
In mathematics, understanding the relationship between a point and a system of inequalities is crucial in various fields, including algebra, geometry, and optimization. A system of inequalities is a set of mathematical statements that involve one or more variables and are connected by the words "and," "or," or "not." In this article, we will explore the relationship between the point $(1,-5)$ and the given system of inequalities.
The System of Inequalities
The given system of inequalities is:
\begin{array}{l} y \leq 3x + 2 \ y \ \textgreater \ -2x - 3 \end{array}
To understand the relationship between the point and the system, we need to analyze each inequality separately.
Analyzing the First Inequality
The first inequality is . This inequality represents a line with a slope of 3 and a y-intercept of 2. The symbol indicates that the line is included in the solution set. In other words, any point on or below the line satisfies the inequality.
Analyzing the Second Inequality
The second inequality is . This inequality represents a line with a slope of -2 and a y-intercept of -3. The symbol indicates that the line is not included in the solution set. In other words, any point above the line satisfies the inequality.
Graphing the System of Inequalities
To visualize the relationship between the point and the system, we need to graph the two inequalities on a coordinate plane.
Graphing the First Inequality
To graph the first inequality, we need to draw the line and shade the region below it.
import matplotlib.pyplot as plt
import numpy as np
# Define the line
x = np.linspace(-10, 10, 400)
y = 3*x + 2
# Create the plot
plt.plot(x, y, label='y = 3x + 2')
plt.fill_between(x, y, color='blue', alpha=0.2)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of the First Inequality')
plt.legend()
plt.grid(True)
plt.show()
Graphing the Second Inequality
To graph the second inequality, we need to draw the line and shade the region above it.
import matplotlib.pyplot as plt
import numpy as np
# Define the line
x = np.linspace(-10, 10, 400)
y = -2*x - 3
# Create the plot
plt.plot(x, y, label='y = -2x - 3')
plt.fill_between(x, y, color='red', alpha=0.2)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of the Second Inequality')
plt.legend()
plt.grid(True)
plt.show()
Determining the Relationship Between the Point and the System
Now that we have graphed the system of inequalities, we can determine the relationship between the point $(1,-5)$ and the system.
To do this, we need to check if the point satisfies both inequalities.
Checking the First Inequality
Substituting the coordinates of the point into the first inequality, we get:
Simplifying the inequality, we get:
Since the inequality is true, the point satisfies the first inequality.
Checking the Second Inequality
Substituting the coordinates of the point into the second inequality, we get:
Simplifying the inequality, we get:
Since the inequality is false, the point does not satisfy the second inequality.
Conclusion
In conclusion, the point $(1,-5)$ satisfies the first inequality but does not satisfy the second inequality. Therefore, the point is included in the solution set of the first inequality but is not included in the solution set of the second inequality.
Algebraic Explanation
The algebraic explanation for the relationship between the point and the system is as follows:
The point $(1,-5)$ satisfies the first inequality if and only if:
Simplifying the inequality, we get:
Since the inequality is true, the point satisfies the first inequality.
The point $(1,-5)$ does not satisfy the second inequality if and only if:
Simplifying the inequality, we get:
Since the inequality is false, the point does not satisfy the second inequality.
Graphical Explanation
The graphical explanation for the relationship between the point and the system is as follows:
The point $(1,-5)$ is included in the solution set of the first inequality because it lies below the line .
The point $(1,-5)$ is not included in the solution set of the second inequality because it lies below the line .
Final Answer
The final answer is that the point $(1,-5)$ satisfies the first inequality but does not satisfy the second inequality. Therefore, the point is included in the solution set of the first inequality but is not included in the solution set of the second inequality.
Introduction
In our previous article, we explored the relationship between a point and a system of inequalities. We analyzed the given system of inequalities and determined the relationship between the point $(1,-5)$ and the system. In this article, we will answer some frequently asked questions related to the topic.
Q1: What is a system of inequalities?
A system of inequalities is a set of mathematical statements that involve one or more variables and are connected by the words "and," "or," or "not." Each inequality in the system represents a region on a coordinate plane, and the solution set of the system is the intersection of these regions.
Q2: How do I graph a system of inequalities?
To graph a system of inequalities, you need to graph each inequality separately and then find the intersection of the regions. You can use a coordinate plane and a ruler to draw the lines and shade the regions.
Q3: What is the difference between a linear inequality and a nonlinear inequality?
A linear inequality is an inequality that can be written in the form or , where , , and are constants. A nonlinear inequality is an inequality that cannot be written in this form.
Q4: How do I determine the solution set of a system of inequalities?
To determine the solution set of a system of inequalities, you need to find the intersection of the regions represented by each inequality. You can use a coordinate plane and a ruler to draw the lines and shade the regions.
Q5: Can a point be included in the solution set of a system of inequalities?
Yes, a point can be included in the solution set of a system of inequalities if it lies in the intersection of the regions represented by each inequality.
Q6: Can a point be excluded from the solution set of a system of inequalities?
Yes, a point can be excluded from the solution set of a system of inequalities if it lies outside the intersection of the regions represented by each inequality.
Q7: How do I check if a point satisfies a linear inequality?
To check if a point satisfies a linear inequality, you need to substitute the coordinates of the point into the inequality and simplify. If the inequality is true, the point satisfies the inequality.
Q8: How do I check if a point satisfies a nonlinear inequality?
To check if a point satisfies a nonlinear inequality, you need to substitute the coordinates of the point into the inequality and simplify. If the inequality is true, the point satisfies the inequality.
Q9: Can a point satisfy both a linear inequality and a nonlinear inequality?
Yes, a point can satisfy both a linear inequality and a nonlinear inequality if it lies in the intersection of the regions represented by each inequality.
Q10: Can a point satisfy both a linear inequality and a nonlinear inequality if the nonlinear inequality is not included in the solution set of the linear inequality?
No, a point cannot satisfy both a linear inequality and a nonlinear inequality if the nonlinear inequality is not included in the solution set of the linear inequality.
Conclusion
In conclusion, determining the relationship between a point and a system of inequalities is a crucial concept in mathematics. By understanding the solution set of a system of inequalities, you can determine the relationship between a point and the system. We hope that this Q&A article has helped you to better understand the concept.
Final Answer
The final answer is that a point can satisfy both a linear inequality and a nonlinear inequality if it lies in the intersection of the regions represented by each inequality. However, a point cannot satisfy both a linear inequality and a nonlinear inequality if the nonlinear inequality is not included in the solution set of the linear inequality.