Identifying Regions Satisfying Inequalities X + Y < 5 And 1 ≤ Y < 6

In mathematics, inequalities play a crucial role in defining ranges and constraints for variables. When dealing with two variables, such as x and y, inequalities can describe regions on a coordinate plane. These regions represent sets of points that satisfy the given conditions. In this comprehensive guide, we will delve into the process of identifying regions that satisfy a system of inequalities, specifically focusing on the inequalities x + y < 5 and 1 ≤ y < 6. Our primary goal is to provide a clear and step-by-step approach, enabling readers to confidently tackle similar problems and gain a deeper understanding of inequality solutions. We will not only break down the mathematical concepts but also emphasize the practical applications of these skills in various fields, ensuring a well-rounded learning experience.

To accurately identify the region that satisfies the given inequalities, we must first meticulously analyze each inequality individually. Let's begin with the first inequality, x + y < 5. This inequality represents a region on the coordinate plane that lies below the line x + y = 5. To visualize this, we can rewrite the equation in slope-intercept form (y = mx + b) as y < -x + 5. This form makes it clear that the line has a slope of -1 and a y-intercept of 5. All points below this line will satisfy the inequality. The line itself is not included in the solution set because of the "less than" (<) symbol, rather than "less than or equal to" (≤). Hence, the line will be drawn as a dashed line to indicate that it is not part of the solution. This understanding is crucial for accurately shading the correct region on the graph and visualizing the set of points that meet the condition x + y < 5. Next, let’s consider the second inequality, 1 ≤ y < 6. This compound inequality constrains the values of y to lie between 1 and 6, inclusive of 1 but exclusive of 6. It defines a horizontal strip on the coordinate plane. The lower bound, y ≥ 1, includes all points on or above the horizontal line y = 1. The upper bound, y < 6, includes all points below the horizontal line y = 6. Therefore, the solution set for this inequality is the area between these two horizontal lines. The line y = 1 will be drawn as a solid line because the inequality includes the equals sign (≤), indicating that points on this line are part of the solution. Conversely, the line y = 6 will be drawn as a dashed line because the inequality does not include the equals sign (<), meaning that points on this line are not part of the solution. By carefully analyzing both inequalities, we can start to visualize the overlapping region that satisfies both conditions simultaneously. This region will be the intersection of the area below the line y = -x + 5 and the horizontal strip between y = 1 and y = 6.

Graphing inequalities is a critical step in visualizing the solution set. To graph the first inequality, x + y < 5, we first graph the boundary line x + y = 5. As we discussed earlier, this line can be rewritten in slope-intercept form as y = -x + 5, making it easier to plot. We start by plotting the y-intercept at (0, 5) and then use the slope of -1 to find additional points. For example, moving one unit to the right and one unit down gives us the point (1, 4), and so on. Since the inequality is strict (i.e., x + y is strictly less than 5), we draw the line as a dashed line to indicate that points on the line are not included in the solution. To determine which side of the line to shade, we can test a point, such as (0, 0), in the inequality. Substituting x = 0 and y = 0 into x + y < 5 yields 0 < 5, which is true. Therefore, we shade the region that includes the origin, which is the area below the dashed line. Now, let's graph the second inequality, 1 ≤ y < 6. This inequality represents a horizontal strip between the lines y = 1 and y = 6. The line y = 1 is a solid line because the inequality includes y being equal to 1, meaning that points on this line are part of the solution. The line y = 6 is a dashed line because the inequality only includes y being less than 6, not equal to 6. This means that points on the line y = 6 are not included in the solution. We shade the region between these two horizontal lines, capturing all points where the y-coordinate is between 1 and 6. By graphing both inequalities on the same coordinate plane, we can visually identify the overlapping region. This overlapping region represents the set of points that simultaneously satisfy both inequalities. It is the area where the shaded regions of both inequalities intersect. The vertices of this region are particularly important as they define the boundaries of the solution set. Identifying these vertices will help us precisely define the region that satisfies both inequalities.

The key to solving systems of inequalities graphically lies in identifying the region where the individual solutions overlap. This overlapping region, known as the region of intersection, represents the set of points that simultaneously satisfy all the inequalities in the system. In our case, we have two inequalities: x + y < 5 and 1 ≤ y < 6. We have already graphed each of these inequalities, so now we need to find where their shaded regions intersect. Visual inspection of the graph is crucial here. The region of intersection is the area where the shading from both inequalities overlaps. It is bounded by the lines y = 1, y = 6, and x + y = 5. This region is a polygon, and its shape is determined by the points where the boundary lines intersect. To precisely define this region, we need to find the coordinates of the vertices, which are the corner points of the polygon. These vertices are the points where the lines intersect. The vertices are found by solving the equations of the intersecting lines simultaneously. For example, one vertex is the intersection of the lines y = 1 and x + y = 5. Substituting y = 1 into the second equation gives us x + 1 = 5, which simplifies to x = 4. Thus, one vertex is at the point (4, 1). Similarly, we need to find the intersection point of the lines y = 6 and x + y = 5. Substituting y = 6 into the second equation gives us x + 6 = 5, which simplifies to x = -1. However, this point (-1, 6) is not part of our solution because the inequality y < 6 does not include the line y = 6. Therefore, this intersection point is not a vertex of our region. The final vertex we need to find is the intersection of the lines y = 6 and the y-axis. Since the inequality 1 ≤ y < 6 bounds the y values, we consider only the line y=1 as lower bound and the intersection of y=1 and x + y = 5, which we already found to be (4,1). To fully describe the solution region, we must state the coordinates of the vertices and emphasize that the region includes all points within the polygon formed by these vertices, but excludes the portion along the dashed line y = 6. This precise identification of the region of intersection is the ultimate goal of graphically solving systems of inequalities.

Inequalities are not just abstract mathematical concepts; they have a wide range of practical applications in various fields, making them an essential tool for problem-solving and decision-making. One significant application is in optimization problems, where inequalities are used to define constraints or limitations. For instance, in business and economics, companies often need to maximize profits or minimize costs subject to certain constraints, such as budget limits, resource availability, or production capacity. These constraints can be expressed as inequalities, and the feasible region—the region satisfying all constraints—is the area within which the optimal solution lies. By using techniques like linear programming, businesses can find the best possible solution within this feasible region, whether it's maximizing production output or minimizing operational expenses. In engineering, inequalities are crucial for designing structures and systems that meet specific performance criteria. For example, when designing a bridge, engineers must ensure that the structure can withstand certain loads without exceeding stress limits. These load-bearing requirements can be expressed as inequalities, and the design must fall within the solution set to guarantee safety and stability. Similarly, in control systems, inequalities are used to ensure that systems operate within desired ranges, such as maintaining a certain temperature or pressure level. Computer science also leverages inequalities in algorithm design and analysis. For example, the efficiency of an algorithm can be expressed using inequalities, allowing computer scientists to compare different algorithms and determine which one performs best under specific conditions. In data analysis and machine learning, inequalities are used to define decision boundaries and classify data points into different categories. For instance, in a spam filter, inequalities can be used to classify emails as either spam or not spam based on certain criteria, such as the frequency of specific words or phrases. In everyday life, inequalities help us make informed decisions. Whether it's budgeting our finances, planning our time, or making health-related choices, we often encounter constraints and limitations that can be expressed as inequalities. Understanding how to work with inequalities allows us to make better decisions and optimize our outcomes. The ability to solve inequalities and interpret their solutions is a valuable skill that extends far beyond the classroom. It empowers us to tackle real-world challenges, make informed decisions, and achieve our goals more effectively.

In conclusion, the process of identifying regions that satisfy inequalities, such as x + y < 5 and 1 ≤ y < 6, involves a combination of analytical and graphical techniques. We began by carefully breaking down each inequality, understanding that x + y < 5 represents the region below the line x + y = 5, and 1 ≤ y < 6 defines a horizontal strip between the lines y = 1 and y = 6. Graphing these inequalities allowed us to visualize the solution sets and identify the overlapping region, which represents the set of points that satisfy both conditions simultaneously. The region of intersection, a polygon bounded by the lines y = 1, y = 6, and x + y = 5, was precisely defined by finding the coordinates of its vertices. These vertices, found by solving the equations of the intersecting lines, provide the corner points of the solution region. Moreover, we explored the practical applications of inequalities in various fields, including optimization problems, engineering design, computer science, and everyday decision-making. These real-world examples highlight the importance of understanding inequalities as a powerful tool for problem-solving and achieving optimal outcomes. By mastering the techniques discussed in this guide, readers can confidently tackle similar problems, gaining a deeper appreciation for the role of inequalities in mathematics and beyond. The ability to interpret and solve inequalities is a valuable skill that empowers individuals to make informed decisions and navigate complex situations effectively. As we continue to encounter constraints and limitations in our daily lives and professional endeavors, a solid understanding of inequalities will undoubtedly prove to be an asset in achieving our goals and optimizing our outcomes. This guide has provided a comprehensive foundation for understanding inequalities, and further exploration and practice will solidify these skills, enabling readers to apply them successfully in various contexts.