Which Is The Graph Of The System X + 3 Y \textgreater − 3 X + 3y \ \textgreater \ -3 X + 3 Y \textgreater − 3 And Y \textless 1 2 X + 1 Y \ \textless \ \frac{1}{2}x + 1 Y \textless 2 1 X + 1 ?

Mar 7, 2025 by ADMIN 199 views

**Graphing Systems of Inequalities: A Step-by-Step Guide**

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Introduction

Graphing systems of inequalities is a fundamental concept in mathematics, particularly in algebra and geometry. It involves representing a set of inequalities as a graph on a coordinate plane. In this article, we will explore how to graph the system of inequalities $x + 3y > -3$ and $y < \frac{1}{2}x + 1$ . We will break down the process into manageable steps, making it easier to understand and visualize the graph.

Understanding the Inequalities

Before we dive into graphing, let's understand the two inequalities:

$x + 3y > -3$ : This inequality represents a linear inequality in two variables, $x$ and $y$ . The inequality is greater than, indicating that the region above the line is shaded.
$y < \frac{1}{2}x + 1$ : This inequality also represents a linear inequality in two variables, $x$ and $y$ . The inequality is less than, indicating that the region below the line is shaded.

Graphing the First Inequality

To graph the first inequality, $x + 3y > -3$ , we need to find the boundary line. The boundary line is the line that divides the region into two parts: the region above the line and the region below the line.

Finding the x-intercept: To find the x-intercept, we set $y = 0$ and solve for $x$ . This gives us $x + 3(0) > -3$ , which simplifies to $x > -3$ . So, the x-intercept is $(-3, 0)$ .
Finding the y-intercept: To find the y-intercept, we set $x = 0$ and solve for $y$ . This gives us $0 + 3y > -3$ , which simplifies to $3y > -3$ . Dividing both sides by 3, we get $y > -1$ . So, the y-intercept is $(0, -1)$ .
Slope: The slope of the boundary line is the coefficient of $x$ in the inequality, which is 1. Since the inequality is greater than, the region above the line is shaded.

Graphing the Second Inequality

To graph the second inequality, $y < \frac{1}{2}x + 1$ , we need to find the boundary line. The boundary line is the line that divides the region into two parts: the region above the line and the region below the line.

Finding the x-intercept: To find the x-intercept, we set $y = 0$ and solve for $x$ . This gives us $0 < \frac{1}{2}x + 1$ , which simplifies to $\frac{1}{2}x > -1$ . Multiplying both sides by 2, we get $x > -2$ . So, the x-intercept is $(-2, 0)$ .
Finding the y-intercept: To find the y-intercept, we set $x = 0$ and solve for $y$ . This gives us $y < \frac{1}{2}(0) + 1$ , which simplifies to $y < 1$ . So, the y-intercept is $(0, 1)$ .
Slope: The slope of the boundary line is the coefficient of $x$ in the inequality, which is $\frac{1}{2}$ . Since the inequality is less than, the region below the line is shaded.

Graphing the System of Inequalities

To graph the system of inequalities, we need to find the intersection of the two regions. The intersection of the two regions is the region that satisfies both inequalities.

Finding the intersection: To find the intersection, we need to find the point where the two boundary lines intersect. We can do this by solving the system of equations formed by the two boundary lines.
Solving the system of equations: The system of equations is:

\begin{align*} x + 3y &= -3 \\ y &= \frac{1}{2}x + 1 \end{align*}

Substituting the expression for $y$ from the second equation into the first equation, we get:

\begin{align*} x + 3\left(\frac{1}{2}x + 1\right) &= -3 \\ x + \frac{3}{2}x + 3 &= -3 \\ \frac{5}{2}x &= -6 \\ x &= -\frac{12}{5} \end{align*}

Substituting the value of $x$ into the second equation, we get:

\begin{align*} y &= \frac{1}{2}\left(-\frac{12}{5}\right) + 1 \\ y &= -\frac{6}{5} + 1 \\ y &= \frac{1}{5} \end{align*}

So, the point of intersection is $\left(-\frac{12}{5}, \frac{1}{5}\right)$ .

Conclusion

In this article, we graphed the system of inequalities $x + 3y > -3$ and $y < \frac{1}{2}x + 1$ . We found the boundary lines for each inequality and shaded the regions above and below the lines. We then found the intersection of the two regions, which is the region that satisfies both inequalities. The graph of the system of inequalities is a shaded region that represents the solution to the system.

Final Answer

The final answer is: $\boxed{\left(-\frac{12}{5}, \frac{1}{5}\right)}$

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Q: What is the difference between a system of equations and a system of inequalities?

A: A system of equations is a set of equations that are solved simultaneously to find the solution. A system of inequalities, on the other hand, is a set of inequalities that are solved simultaneously to find the solution. In a system of inequalities, the solution is a region on the coordinate plane, rather than a single point.

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 find the intersection of the two regions. You can use the following steps:

Graph the boundary line for each inequality.
Shade the region above the line for the first inequality and the region below the line for the second inequality.
Find the intersection of the two regions by solving the system of equations formed by the two boundary lines.

Q: What is the significance of the boundary line in a system of inequalities?

A: The boundary line in a system of inequalities is the line that divides the region into two parts: the region above the line and the region below the line. The boundary line is the line that represents the equation that is equal to the inequality.

Q: How do I find the intersection of two regions in a system of inequalities?

A: To find the intersection of two regions in a system of inequalities, you need to solve the system of equations formed by the two boundary lines. You can use the following steps:

Write the system of equations formed by the two boundary lines.
Solve the system of equations using substitution or elimination.
Find the point of intersection by substituting the value of one variable into the other equation.

Q: What is the solution to a system of inequalities?

A: The solution to a system of inequalities is the region on the coordinate plane that satisfies both inequalities. The solution is a shaded region that represents the set of all points that satisfy both inequalities.

Q: How do I determine the solution to a system of inequalities?

A: To determine the solution to a system of inequalities, you need to graph each inequality separately and then find the intersection of the two regions. You can use the following steps:

Graph the boundary line for each inequality.
Shade the region above the line for the first inequality and the region below the line for the second inequality.
Find the intersection of the two regions by solving the system of equations formed by the two boundary lines.

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 boundary line incorrectly.
Shading the region incorrectly.
Failing to find the intersection of the two regions.
Not solving the system of equations correctly.

Q: How do I check my work when graphing a system of inequalities?

A: To check your work when graphing a system of inequalities, you can use the following steps:

Graph the boundary line for each inequality.
Shade the region above the line for the first inequality and the region below the line for the second inequality.
Find the intersection of the two regions by solving the system of equations formed by the two boundary lines.
Check that the solution is a shaded region that represents the set of all points that satisfy both inequalities.

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

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

Modeling population growth and decline.
Modeling the spread of disease.
Modeling the growth of a company.
Modeling the spread of a rumor.

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

A: To use graphing systems of inequalities in real-world applications, you can use the following steps:

Identify the variables and the relationships between them.
Write the system of inequalities that represents the relationships.
Graph the system of inequalities to find the solution.
Use the solution to make predictions or decisions.

Q: What are some common challenges when graphing systems of inequalities?

A: Some common challenges when graphing systems of inequalities include:

Graphing the boundary line incorrectly.
Shading the region incorrectly.
Failing to find the intersection of the two regions.
Not solving the system of equations correctly.

Q: How do I overcome common challenges when graphing systems of inequalities?

A: To overcome common challenges when graphing systems of inequalities, you can use the following steps:

Double-check your work when graphing the boundary line.
Make sure to shade the region correctly.
Find the intersection of the two regions by solving the system of equations formed by the two boundary lines.
Check that the solution is a shaded region that represents the set of all points that satisfy both inequalities.

Q: What are some tips for graphing systems of inequalities?

A: Some tips for graphing systems of inequalities include:

Use a ruler to draw the boundary line.
Use a pencil to shade the region.
Make sure to label the axes and the boundary line.
Check your work carefully to avoid mistakes.

Q: How do I use technology to graph systems of inequalities?

A: To use technology to graph systems of inequalities, you can use the following steps:

Use a graphing calculator to graph the system of inequalities.
Use a computer program to graph the system of inequalities.
Use a online graphing tool to graph the system of inequalities.

Q: What are some online resources for graphing systems of inequalities?

A: Some online resources for graphing systems of inequalities include:

Khan Academy: Graphing Systems of Inequalities
Mathway: Graphing Systems of Inequalities
Wolfram Alpha: Graphing Systems of Inequalities