Graphing Inequalities Understanding \(y \leq 3x+1\) And \(y \geq -x+2\)

Introduction

In the realm of mathematics, particularly in algebra and coordinate geometry, understanding how to graph inequalities is crucial. This article delves into the intricacies of graphing inequalities, focusing on the specific system defined by ${y \leq 3x+1}$ and ${y \geq -x+2}$. We will explore various aspects of these inequalities, including their boundary lines, slopes, and solutions. By the end of this comprehensive guide, you will have a clear understanding of how to analyze and interpret graphs of inequalities. The main goal is to determine which statements about the graph of these inequalities are true. Let’s embark on this mathematical journey and unravel the complexities of graphing inequalities together. This article aims to enhance your understanding of linear inequalities and their graphical representations, providing a solid foundation for more advanced mathematical concepts.

Understanding the Inequalities

To effectively analyze the given inequalities, ${y \leq 3x+1}$ and ${y \geq -x+2}$, it is essential to break them down into their fundamental components. Each inequality represents a region in the coordinate plane, bounded by a line. The first inequality, ${y \leq 3x+1}$, indicates that the region includes all points where the y-coordinate is less than or equal to the value of ${3x+1}$. This boundary is a straight line with a slope of 3 and a y-intercept of 1. The "less than or equal to" sign implies that the line itself is part of the solution, hence it will be a solid line on the graph.

Conversely, the second inequality, ${y \geq -x+2}$, represents the region where the y-coordinate is greater than or equal to ${-x+2}$. This boundary line has a slope of -1 and a y-intercept of 2. Similar to the first inequality, the "greater than or equal to" sign means the boundary line is included in the solution set, making it a solid line on the graph. The solution to the system of inequalities is the region where these two individual regions overlap. This overlapping region satisfies both inequalities simultaneously. Understanding these individual components is crucial for accurately graphing the inequalities and identifying the solution set. This involves not only plotting the lines but also determining which side of the line satisfies the inequality. Let’s proceed by examining the boundary lines in more detail, focusing on their slopes and intercepts, as these elements dictate the orientation and position of the lines on the coordinate plane.

Analyzing the Boundary Lines

The boundary lines of the inequalities play a crucial role in determining the solution set. Let’s dissect the characteristics of these lines to gain a deeper understanding. For the inequality ${y \leq 3x+1}$, the boundary line is represented by the equation ${y = 3x+1}$. This equation is in slope-intercept form, ${y = mx + b}$, where ${m}$ represents the slope and ${b}$ represents the y-intercept. In this case, the slope ${m}$ is 3, indicating that for every one unit increase in ${x}$, ${y}$ increases by 3 units. The y-intercept ${b}$ is 1, meaning the line crosses the y-axis at the point (0, 1). This positive slope signifies that the line ascends from left to right.

Turning our attention to the second inequality, ${y \geq -x+2}$, the boundary line is described by the equation ${y = -x+2}$. Again, this is in slope-intercept form. Here, the slope ${m}$ is -1, which means that for every one unit increase in ${x}$, ${y}$ decreases by 1 unit. The y-intercept ${b}$ is 2, indicating the line intersects the y-axis at the point (0, 2). The negative slope implies that this line descends from left to right. Knowing the slopes and y-intercepts allows us to accurately plot these lines on a graph. The slope provides the direction and steepness of the line, while the y-intercept gives us a fixed point to start plotting. We can use these characteristics to draw the lines and subsequently determine the regions that satisfy the inequalities. Understanding the slopes and intercepts is paramount in visualizing the behavior of the lines and how they interact to form the solution region.

Determining the Solution Region

Once we have established the boundary lines, the next step is to determine the solution region for each inequality. This involves identifying which side of the line satisfies the inequality. For the inequality ${y \leq 3x+1}$, we are looking for all points where the y-coordinate is less than or equal to ${3x+1}$. To find this region, we can test a point that is not on the line, such as the origin (0, 0). Plugging these coordinates into the inequality, we get ${0 \leq 3(0)+1}$, which simplifies to ${0 \leq 1}$. This statement is true, so the region including the origin satisfies the inequality. Therefore, we shade the region below the line ${y = 3x+1}$, as this region contains all the points where ${y}$ is less than or equal to ${3x+1}$.

For the second inequality, ${y \geq -x+2}$, we want to find all points where the y-coordinate is greater than or equal to ${-x+2}$. Again, we can test the origin (0, 0). Substituting these values into the inequality, we get ${0 \geq -(0)+2}$, which simplifies to ${0 \geq 2}$. This statement is false, so the region including the origin does not satisfy this inequality. Consequently, we shade the region above the line ${y = -x+2}$, as this region contains all the points where ${y}$ is greater than or equal to ${-x+2}$. The solution to the system of inequalities is the intersection of these two shaded regions. This overlapping area represents all the points that satisfy both inequalities simultaneously. It is crucial to accurately shade the correct regions to identify the solution set. Testing points is a reliable method to determine which side of the line to shade. Let’s move on to testing specific points to see if they are solutions to the system.

Testing Solution Points

To verify whether a particular point is a solution to the system of inequalities, we must check if the coordinates of the point satisfy both inequalities. Let's consider the point (1, 3) as suggested in the original problem. To check if this point is a solution, we substitute ${x = 1}$ and ${y = 3}$ into both inequalities.

For the first inequality, ${y \leq 3x+1}$, we have:

${ 3 \leq 3(1) + 1 }$ ${ 3 \leq 3 + 1 }$ ${ 3 \leq 4 }$

This statement is true, so the point (1, 3) satisfies the first inequality.

Now, let's check the second inequality, ${y \geq -x+2}$:

${ 3 \geq -(1) + 2 }$ ${ 3 \geq -1 + 2 }$ ${ 3 \geq 1 }$

This statement is also true, indicating that the point (1, 3) satisfies the second inequality as well. Since (1, 3) satisfies both inequalities, it is indeed a solution to the system. This method of testing points is a straightforward way to confirm whether a given point lies within the solution region. If a point fails to satisfy even one of the inequalities, it is not a solution to the system. By testing points, we gain a concrete understanding of what it means for a point to be a solution to a system of inequalities. It reinforces the concept that the solution region encompasses all points that meet the criteria set by the inequalities. Now, let's consolidate our findings and address the specific statements provided in the problem.

Evaluating the Statements

Now that we have a thorough understanding of the inequalities and their graphical representation, we can evaluate the statements provided in the original problem. Let's revisit the statements:

• A. The slope of one boundary line is 2.
• B. Both boundary lines are solid.
• C. A solution to the system is (1,3).

We have already analyzed the slopes of the boundary lines. The line ${y = 3x+1}$ has a slope of 3, and the line ${y = -x+2}$ has a slope of -1. Neither of these slopes is equal to 2. Therefore, statement A is false.

Next, we consider whether the boundary lines are solid. As discussed earlier, the inequalities ${y \leq 3x+1}$ and ${y \geq -x+2}$ include the "equal to" condition ( ${\leq}$ and ${\geq}$ ). This means that the boundary lines themselves are part of the solution set and are represented as solid lines on the graph. Thus, statement B is true.

Lastly, we tested the point (1, 3) and found that it satisfies both inequalities. Therefore, it is a solution to the system. This confirms that statement C is true.

By systematically evaluating each statement based on our analysis, we can accurately determine which statements are true and which are false. This process highlights the importance of understanding the underlying concepts of graphing inequalities, including slopes, boundary lines, and solution regions. Let's summarize the true statements to provide a clear conclusion to our analysis.

Conclusion

In summary, after thoroughly analyzing the system of inequalities ${y \leq 3x+1}$ and ${y \geq -x+2}$, we have determined the following:

• The slope of one boundary line is not 2. Statement A is false.
• Both boundary lines are solid because the inequalities include the "equal to" condition. Statement B is true.
• The point (1, 3) is indeed a solution to the system. Statement C is true.

Therefore, the true statements about the graph of ${y \leq 3x+1}$ and ${y \geq -x+2}$ are that both boundary lines are solid, and (1, 3) is a solution to the system. This exploration underscores the significance of understanding linear inequalities and their graphical representations. By breaking down the problem into smaller parts—analyzing boundary lines, determining solution regions, and testing points—we can effectively solve complex mathematical problems. Graphing inequalities is a fundamental skill in algebra, and mastering it opens the door to more advanced mathematical concepts. We hope this comprehensive guide has provided you with a clear and thorough understanding of graphing inequalities and has enhanced your problem-solving abilities in mathematics. Remember, practice is key to mastering these concepts, so continue to explore and solve various inequality problems to solidify your understanding.