Graphing Inequalities Y ≤ -7/4x + 10 And Y > 5 A Comprehensive Guide

Introduction: Understanding the Power of Graphing Inequalities

In the realm of mathematics, graphing inequalities serves as a powerful tool for visualizing solutions sets and understanding relationships between variables. Unlike equations that represent specific points or lines, inequalities define regions on a coordinate plane, encompassing a range of possible values. This article delves into the process of graphing linear inequalities, specifically focusing on the example of $y \leq -\frac{7}{4}x + 10$ and $y > 5$. We will break down each step, from understanding the inequality symbols to shading the appropriate regions, ensuring a comprehensive understanding of this fundamental concept.

Graphing inequalities is not merely an abstract mathematical exercise; it has practical applications in various fields, including economics, computer science, and engineering. For instance, in economics, inequalities can represent budget constraints or production possibilities, while in computer science, they can define feasible regions for optimization problems. By mastering the art of graphing inequalities, you gain a valuable skill that extends beyond the classroom.

Before we dive into the specifics of our example, let's establish a solid foundation by revisiting the basics of linear inequalities and their graphical representation. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), $\leq$ (less than or equal to), and $\geq$ (greater than or equal to). The graph of a linear inequality is a region on the coordinate plane, bounded by a line, that represents all the points that satisfy the inequality. The line itself may or may not be included in the solution set, depending on the inequality symbol used. For instance, an inequality with a < or > symbol will have a dashed boundary line, indicating that the points on the line are not part of the solution, while an inequality with a $\leq$ or $\geq$ symbol will have a solid boundary line, indicating that the points on the line are included.

Part 1: Graphing $y \leq -\frac{7}{4}x + 10$

Step 1: Transforming the Inequality into Slope-Intercept Form

The first step in graphing the inequality $y \leq -\frac{7}{4}x + 10$ is to recognize that it is already presented in slope-intercept form, which is $y = mx + b$, where m represents the slope and b represents the y-intercept. This form is incredibly useful because it directly provides the two key pieces of information needed to graph a line: the slope and the y-intercept. In our case, the inequality $y \leq -\frac{7}{4}x + 10$ has a slope of $-\frac{7}{4}$ and a y-intercept of 10. Understanding the slope-intercept form is crucial for efficiently graphing linear equations and inequalities.

The slope, $-\frac{7}{4}$, tells us how steep the line is and its direction. A negative slope indicates that the line slopes downward from left to right. The numerical value of the slope, $\frac{7}{4}$, represents the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. In this case, for every 4 units we move to the right along the x-axis, we move 7 units down along the y-axis. The y-intercept, 10, tells us where the line crosses the y-axis. It is the point (0, 10) on the coordinate plane. These two pieces of information, the slope and the y-intercept, are the foundation for accurately graphing the line that bounds our inequality. Recognizing and utilizing the slope-intercept form simplifies the graphing process significantly.

Step 2: Graphing the Boundary Line

Now that we've identified the slope and y-intercept, we can proceed to graph the boundary line. To do this, we'll first treat the inequality as an equation: $y = -\frac{7}{4}x + 10$. We start by plotting the y-intercept, which is the point (0, 10). This is where the line intersects the y-axis. Next, we use the slope to find another point on the line. Recall that the slope is $-\frac{7}{4}$, which means for every 4 units we move to the right, we move 7 units down. Starting from the y-intercept (0, 10), we move 4 units to the right and 7 units down, landing us at the point (4, 3). We can plot this point as well.

With two points on the line, we can now draw the line. However, before we draw the line, we need to consider the inequality symbol. Since our inequality is $y \leq -\frac{7}{4}x + 10$, it includes the "equal to" part ($\leq$). This means that the points on the line are part of the solution set, and we should draw a solid line to represent this. If the inequality were $y < -\frac{7}{4}x + 10$, we would draw a dashed line to indicate that the points on the line are not included in the solution set. The choice between a solid and dashed line is crucial for accurately representing the inequality's solution.

The solid line serves as the boundary between the region where the inequality is true and the region where it is false. It's a visual representation of the boundary condition defined by the inequality. By drawing a solid line, we're emphasizing that the points on this line satisfy the condition $y = -\frac{7}{4}x + 10$ and are therefore included in the solution set of the inequality $y \leq -\frac{7}{4}x + 10$. This distinction is important for a complete and accurate graphical representation of the inequality.

Step 3: Shading the Correct Region

After graphing the boundary line, the next step is to determine which region of the coordinate plane represents the solution set for the inequality. This is achieved by shading the appropriate region. Since our inequality is $y \leq -\frac{7}{4}x + 10$, we are looking for all the points (x, y) where the y-coordinate is less than or equal to the value of $-\frac{7}{4}x + 10$. This corresponds to the region below the line.

To confirm this, we can use a test point. A test point is any point that is not on the line. A common and convenient test point is (0, 0), as long as the line does not pass through the origin. Let's substitute (0, 0) into the inequality: $0 \leq -\frac{7}{4}(0) + 10$. This simplifies to $0 \leq 10$, which is a true statement. Since the test point (0, 0) satisfies the inequality, the region containing (0, 0) is part of the solution set. This confirms that we should shade the region below the line.

The act of shading visually represents the infinite set of solutions to the inequality. Every point in the shaded region has coordinates (x, y) that make the inequality $y \leq -\frac{7}{4}x + 10$ true. Conversely, every point in the unshaded region does not satisfy the inequality. The shading provides a clear and intuitive way to understand the range of possible solutions. It transforms an abstract mathematical statement into a concrete visual representation, making it easier to grasp the concept of inequalities and their solutions. The choice of shading the region below the line is directly tied to the "less than or equal to" ($\leq$) symbol in the inequality.

Part 2: Graphing $y > 5$

Step 1: Understanding the Inequality

The inequality $y > 5$ represents all points on the coordinate plane where the y-coordinate is greater than 5. Unlike the previous inequality, this one is simpler because it only involves the variable y and a constant. There is no x term, which means the boundary line will be a horizontal line. The absence of an x term simplifies the graphing process, as we only need to focus on the y-coordinate. However, it's crucial to understand that the concept of a boundary line and shading still applies, even though the line is horizontal.

Understanding the inequality is the key to correctly graphing it. The symbol ">" signifies "greater than," meaning that any y-value strictly larger than 5 will satisfy the inequality. This implies that the points on the line y = 5 itself are not part of the solution set. This distinction is important and will influence how we draw the boundary line. The inequality $y > 5$ defines a region of the coordinate plane that extends infinitely upwards from the horizontal line y = 5. Visualizing this region is the first step in accurately graphing the inequality.

Step 2: Graphing the Boundary Line

The boundary line for the inequality $y > 5$ is the horizontal line $y = 5$. To graph this line, we locate the point where y is equal to 5 on the y-axis. This point is (0, 5). Since the inequality symbol is ">" (greater than), which does not include "equal to," we will draw a dashed line through this point. A dashed line indicates that the points on the line are not included in the solution set. This is a crucial distinction, as it accurately represents the inequality's condition that y must be strictly greater than 5.

The dashed line serves as a visual cue that the boundary is not part of the solution. It's a way of communicating that while values very close to 5 are included in the solution, the value 5 itself is not. This is different from an inequality like $y \geq 5$, where the boundary line would be solid, indicating that points on the line are part of the solution. The choice of a dashed line is directly linked to the strict inequality symbol (">") and is essential for an accurate graphical representation.

The horizontal line $y = 5$ divides the coordinate plane into two regions: the region above the line where y > 5 and the region below the line where y < 5. Our goal is to identify and shade the region that satisfies the inequality $y > 5$. The dashed line acts as a clear visual separator between these two regions, guiding us in the next step of shading the correct area.

Step 3: Shading the Correct Region

Since the inequality is $y > 5$, we are looking for all points where the y-coordinate is greater than 5. This corresponds to the region above the dashed line $y = 5$. To confirm this, we can use a test point. A convenient test point is (0, 6), which is clearly above the line. Substituting this point into the inequality, we get $6 > 5$, which is a true statement. This confirms that the region above the line is the solution set.

Therefore, we shade the region above the dashed line $y = 5$ to represent all the points that satisfy the inequality $y > 5$. This shaded region extends infinitely upwards, indicating that any point with a y-coordinate greater than 5 is a solution. The shading visually represents the infinite set of solutions to the inequality. Every point in the shaded region has a y-coordinate greater than 5, and therefore satisfies the inequality. The unshaded region below the line represents points where the y-coordinate is less than or equal to 5, and these points do not satisfy the inequality. The shading provides a clear and intuitive way to understand the range of possible solutions for the inequality $y > 5$.

Part 3: Combining the Graphs

Identifying the Overlapping Region

To find the solution set for the system of inequalities:

$y \leq -\frac{7}{4}x + 10$

$y > 5$

we need to identify the region where the solutions of both inequalities overlap. This overlapping region represents all the points that satisfy both inequalities simultaneously. Graphically, this is the area where the shaded regions of the two inequalities intersect.

The process of identifying the overlapping region involves careful observation of the two graphs. We have already graphed each inequality separately, shading the region that represents its solutions. Now, we need to overlay these graphs and look for the area that is shaded in both. This area represents the common solution set, where both inequalities are true. The overlapping region is the visual representation of the intersection of the two solution sets, and it provides a clear and concise way to understand the combined solution of the system of inequalities. The boundaries of this region are defined by the boundary lines of the individual inequalities, and the type of line (solid or dashed) is crucial for accurately representing the inclusion or exclusion of the boundary in the solution set.

The Solution Set

The solution set is the region that is shaded in both graphs. This region is bounded by the solid line of $y \leq -\frac{7}{4}x + 10$ and the dashed line of $y > 5$. The solid line indicates that the points on that line are included in the solution set, while the dashed line indicates that the points on that line are not included. Therefore, the solution set includes all points within the overlapping region, including the points on the solid boundary line but excluding the points on the dashed boundary line.

The solution set represents the complete answer to the problem of solving the system of inequalities. It is the set of all points (x, y) that make both inequalities true. This set is infinite, as it encompasses a continuous region on the coordinate plane. The graphical representation of the solution set provides a powerful visual tool for understanding the combined effect of the two inequalities. It allows us to see the constraints imposed by each inequality and how they interact to define the feasible region. This understanding is crucial for various applications, such as optimization problems where we seek to find the best solution within a set of constraints defined by inequalities.

Conclusion: Mastering Graphing Inequalities

Graphing inequalities is a fundamental skill in mathematics with wide-ranging applications. By understanding the steps involved, from transforming inequalities into slope-intercept form to shading the correct regions, you can effectively visualize solution sets and solve systems of inequalities. The example of $y \leq -\frac{7}{4}x + 10$ and $y > 5$ provides a clear illustration of this process. Remember to pay close attention to the inequality symbols and whether the boundary lines should be solid or dashed. With practice, you can master the art of graphing inequalities and unlock their power in various mathematical and real-world contexts. This skill forms the bedrock for understanding more advanced mathematical concepts and is invaluable in fields that rely on mathematical modeling and problem-solving.