Consider The System Of Inequalities And Its Graph.${ \begin{array}{l} y \leq -0.75x \ y \leq 3x - 2 \end{array} }$In Which Section Of The Graph Does The Actual Solution To The System Lie?1. Section 1 2. Section 2 3. Section 3 4. Section 4

Introduction

In mathematics, a system of inequalities is a set of two or more inequalities that are combined to form a single problem. These inequalities can be linear or non-linear, and they can be combined using various operations such as addition, subtraction, multiplication, and division. In this article, we will focus on a system of linear inequalities and its graph. We will analyze the system and determine in which section of the graph the actual solution to the system lies.

The System of Inequalities

The given system of inequalities is:

$$\begin{array}{l} y \leq -0.75x \\ y \leq 3x - 2 \end{array}$$

These two inequalities represent two lines on a graph. The first inequality, $y \leq -0.75x$, represents a line with a slope of -0.75 and a y-intercept of 0. The second inequality, $y \leq 3x - 2$, represents a line with a slope of 3 and a y-intercept of -2.

Graphing the Inequalities

To graph the inequalities, we need to find the points of intersection between the two lines. We can do this by setting the two equations equal to each other and solving for x.

$$-0.75x = 3x - 2$$

Solving for x, we get:

$$-3.75x = -2$$

$$x = \frac{2}{3.75}$$

$$x = \frac{4}{7.5}$$

$$x = \frac{8}{15}$$

Now that we have the x-coordinate of the point of intersection, we can find the y-coordinate by plugging the value of x into one of the original equations. Let's use the first equation:

$$y = -0.75x$$

$$y = -0.75 \times \frac{8}{15}$$

$$y = -\frac{6}{15}$$

$$y = -\frac{2}{5}$$

So, the point of intersection is $\left(\frac{8}{15}, -\frac{2}{5}\right)$.

Graphing the Inequalities on a Coordinate Plane

To graph the inequalities on a coordinate plane, we need to plot the two lines and shade the regions that satisfy the inequalities.

The first inequality, $y \leq -0.75x$, is a line with a slope of -0.75 and a y-intercept of 0. We can plot this line by starting at the y-intercept and moving down and to the left.

The second inequality, $y \leq 3x - 2$, is a line with a slope of 3 and a y-intercept of -2. We can plot this line by starting at the y-intercept and moving up and to the right.

Shading the Regions

To shade the regions that satisfy the inequalities, we need to determine which regions are above and below the lines.

For the first inequality, $y \leq -0.75x$, the region above the line is shaded. This is because the inequality is in the form $y \leq mx + b$, where $m$ is the slope and $b$ is the y-intercept. Since the slope is negative, the region above the line is the region that satisfies the inequality.

For the second inequality, $y \leq 3x - 2$, the region below the line is shaded. This is because the inequality is in the form $y \leq mx + b$, where $m$ is the slope and $b$ is the y-intercept. Since the slope is positive, the region below the line is the region that satisfies the inequality.

Determining the Solution Region

To determine the solution region, we need to find the intersection of the two shaded regions.

The first shaded region is above the line $y = -0.75x$, and the second shaded region is below the line $y = 3x - 2$. The intersection of these two regions is the region that satisfies both inequalities.

Conclusion

In conclusion, the system of inequalities is:

$$\begin{array}{l} y \leq -0.75x \\ y \leq 3x - 2 \end{array}$$

The graph of the system is a shaded region that satisfies both inequalities. The solution region is the intersection of the two shaded regions.

Answer

The actual solution to the system lies in Section 2.

Discussion

The system of inequalities is a common problem in mathematics, and it requires a deep understanding of linear inequalities and their graphs. The solution to the system lies in the intersection of the two shaded regions, and it requires careful analysis of the inequalities and their graphs.

Key Takeaways

• The system of inequalities is a set of two or more inequalities that are combined to form a single problem.
• The inequalities can be linear or non-linear, and they can be combined using various operations such as addition, subtraction, multiplication, and division.
• The graph of the system is a shaded region that satisfies both inequalities.
• The solution region is the intersection of the two shaded regions.

References

• [1] "Linear Inequalities and Their Graphs" by [Author's Name]
• [2] "Systems of Inequalities" by [Author's Name]

Additional Resources

• [1] Khan Academy: Linear Inequalities and Their Graphs
• [2] Mathway: Systems of Inequalities

Final Thoughts

Introduction

In our previous article, we discussed the system of inequalities and its graph. We analyzed the system and determined in which section of the graph the actual solution to the system lies. In this article, we will answer some frequently asked questions about the system of inequalities and its graph.

Q: What is a system of inequalities?

A system of inequalities is a set of two or more inequalities that are combined to form a single problem. These inequalities can be linear or non-linear, and they can be combined using various operations such as addition, subtraction, multiplication, and division.

Q: How do I graph a system of inequalities?

To graph a system of inequalities, you need to plot the two lines and shade the regions that satisfy the inequalities. The first inequality is plotted by starting at the y-intercept and moving down and to the left. The second inequality is plotted by starting at the y-intercept and moving up and to the right. The regions that satisfy the inequalities are shaded.

Q: How do I determine the solution region?

To determine the solution region, you need to find the intersection of the two shaded regions. The solution region is the region that satisfies both inequalities.

Q: What is the difference between a system of linear inequalities and a system of non-linear inequalities?

A system of linear inequalities is a set of linear inequalities that are combined to form a single problem. A system of non-linear inequalities is a set of non-linear inequalities that are combined to form a single problem. Non-linear inequalities are inequalities that are not in the form of a straight line.

Q: How do I solve a system of inequalities?

To solve a system of inequalities, you need to find the intersection of the two shaded regions. The solution region is the region that satisfies both inequalities.

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

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

• Plotting the wrong line
• Shading the wrong region
• Not finding the intersection of the two shaded regions

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

To check your work when graphing a system of inequalities, you need to:

• Verify that the lines are plotted correctly
• Verify that the regions are shaded correctly
• Verify that the solution region is the intersection of the two shaded regions

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

Some real-world applications of systems of inequalities include:

• Budgeting and financial planning
• Resource allocation and management
• Optimization problems

Conclusion

In conclusion, the system of inequalities is a fundamental concept in mathematics, and it requires a deep understanding of linear inequalities and their graphs. By understanding the system of inequalities, we can solve a wide range of problems in mathematics and other fields. We hope that this article has helped to answer some of your questions about the system of inequalities and its graph.

Additional Resources

• [1] Khan Academy: Linear Inequalities and Their Graphs
• [2] Mathway: Systems of Inequalities
• [3] Wolfram Alpha: Systems of Inequalities

Final Thoughts

The system of inequalities is a powerful tool for solving a wide range of problems in mathematics and other fields. By understanding the system of inequalities, we can optimize resources, make informed decisions, and solve complex problems. We hope that this article has helped to inspire you to learn more about the system of inequalities and its applications.