Graphing Linear Inequality (1/2)x - 2y > -6 A Comprehensive Guide

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Navigating the realm of linear inequalities can initially seem daunting, but with a structured approach, it becomes a manageable task. This guide delves into the intricacies of graphing the linear inequality 12x−2y>−6{\frac{1}{2}x - 2y > -6}, providing a step-by-step explanation to enhance understanding and proficiency. Linear inequalities, unlike linear equations, represent a range of solutions rather than a single point. The inequality 12x−2y>−6{\frac{1}{2}x - 2y > -6} is a classic example, where we seek to identify all the coordinate pairs (x, y) that satisfy the condition. Graphing these inequalities is a visual method that illuminates the solution set, making it easier to comprehend and apply in various contexts. Before we dive into the specifics, it’s essential to grasp the foundational concepts. A linear inequality is characterized by variables raised to the first power, combined with inequality symbols such as >, <, ≥, or ≤. The graph of a linear inequality is a region in the coordinate plane, bounded by a line. This line is crucial as it delineates the boundary between solutions that satisfy the inequality and those that do not. The inequality 12x−2y>−6{\frac{1}{2}x - 2y > -6} is a perfect example to illustrate these principles. Our journey begins with transforming this inequality into a more manageable form, which will pave the way for accurate graphing. We will explore each step meticulously, ensuring a solid understanding of the underlying concepts and techniques. By the end of this guide, you'll be equipped to graph not just this inequality, but any linear inequality with confidence and precision.

Step 1: Transforming the Inequality into Slope-Intercept Form

The first crucial step in graphing any linear inequality, including 12x−2y>−6{\frac{1}{2}x - 2y > -6}, is to transform it into the slope-intercept form. The slope-intercept form, represented as y = mx + b, offers a clear understanding of the line's slope (m) and y-intercept (b), which are essential for graphing. This form not only simplifies the graphing process but also provides valuable insights into the behavior of the line. For our inequality, 12x−2y>−6{\frac{1}{2}x - 2y > -6}, we need to isolate y on one side. This involves a series of algebraic manipulations, each carefully executed to maintain the integrity of the inequality. The initial step is to eliminate the x term from the left side. We achieve this by subtracting 12x{\frac{1}{2}x} from both sides of the inequality. This operation yields the modified inequality: * -2y > -\frac1}{2}x - 6*. Now, our focus shifts to isolating y. Since y is multiplied by -2, we must divide both sides of the inequality by -2. However, a critical rule in dealing with inequalities comes into play here when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is because multiplying or dividing by a negative number flips the number line, changing the relative order of the values. Applying this rule, we divide both sides by -2 and reverse the inequality sign, transforming the inequality into: *y < \frac{1{4}x + 3*. This is the slope-intercept form of our inequality. We can now clearly see that the slope (m) is 14{\frac{1}{4}} and the y-intercept (b) is 3. This transformation is a pivotal step, as it lays the groundwork for accurately graphing the inequality. The slope and y-intercept provide the necessary information to plot the boundary line, which is the next step in our graphing journey. Understanding the significance of the slope-intercept form and the rule for reversing the inequality sign when dividing by a negative number is paramount for mastering the graphing of linear inequalities. With the inequality now in slope-intercept form, we are well-prepared to proceed to the next stage: plotting the boundary line.

Step 2: Plotting the Boundary Line

With the inequality 12x−2y>−6{\frac{1}{2}x - 2y > -6} transformed into the slope-intercept form y < \frac1}{4}x + 3*, the next step is to plot the boundary line. This line serves as the visual separator between the solutions that satisfy the inequality and those that do not. The equation of the boundary line is obtained by replacing the inequality sign with an equals sign, resulting in y = \frac{1}{4}x + 3. This linear equation represents the line we need to plot on the coordinate plane. To plot the line, we need at least two points. The slope-intercept form provides us with one point directly the y-intercept. The y-intercept is the point where the line crosses the y-axis, and in our equation, it is 3. This gives us the point (0, 3). The slope, ${\frac{14}}$, provides us with the information needed to find another point. The slope represents the change in y for every unit change in x. A slope of 14{\frac{1}{4}} means that for every 4 units we move to the right along the x-axis, we move 1 unit up along the y-axis. Starting from the y-intercept (0, 3), we move 4 units to the right and 1 unit up, which gives us the point (4, 4). Now that we have two points, (0, 3) and (4, 4), we can draw the line. However, there's a crucial detail we need to consider the type of line we draw. Since our original inequality is *y < \frac{1{4}x + 3, which uses a strict inequality (<), the boundary line is not included in the solution set. This means we draw a dashed line to indicate that the points on the line do not satisfy the inequality. If the inequality had been y ≤ \frac{1}{4}x + 3, we would have drawn a solid line to indicate that the points on the line are included in the solution set. Plotting the boundary line accurately is paramount, as it forms the basis for identifying the solution region. The dashed line in our case clearly signifies that the solutions lie on one side of the line, but not on the line itself. With the boundary line plotted, we are now ready to determine which side of the line represents the solution region. This involves the crucial step of testing a point.

Step 3: Shading the Solution Region

After plotting the dashed boundary line for the inequality y < 14{\frac{1}{4}}x + 3, the next crucial step is to shade the solution region. The solution region represents all the points (x, y) that satisfy the inequality. To determine which side of the line to shade, we use a simple yet effective technique: testing a point. The most common and often easiest point to test is the origin (0, 0), provided the boundary line does not pass through it. If the boundary line passed through the origin, we would need to choose a different test point. To test the origin, we substitute x = 0 and y = 0 into the inequality y < 14{\frac{1}{4}}x + 3. This gives us 0 < 14{\frac{1}{4}}(0) + 3, which simplifies to 0 < 3. This statement is true, meaning that the point (0, 0) does indeed satisfy the inequality. Therefore, the region that contains the origin is the solution region. We shade this side of the dashed line to visually represent all the solutions to the inequality. If, on the other hand, the test point had not satisfied the inequality, we would have shaded the opposite side of the line. The shaded region extends infinitely in the direction that satisfies the inequality, indicating that there are countless solutions. Each point within the shaded region, when its coordinates are substituted into the original inequality, will result in a true statement. The dashed boundary line and the shaded region together provide a complete graphical representation of the solution set for the inequality y < 14{\frac{1}{4}}x + 3. This visual representation is invaluable for understanding the range of solutions and how they relate to the inequality. Shading the correct region is critical for accurately portraying the solution set. The test point method provides a reliable way to determine which side of the line to shade, ensuring that the graph accurately reflects the solutions to the inequality. With the solution region shaded, we have successfully graphed the linear inequality 12x−2y>−6{\frac{1}{2}x - 2y > -6}, visually representing all its solutions.

Step 4: Interpreting the Graph

Once the graph of the linear inequality 12x−2y>−6{\frac{1}{2}x - 2y > -6} is complete, with the dashed boundary line and the shaded solution region, the final step is to interpret the graph. The graph serves as a visual representation of all the solutions to the inequality, providing valuable insights that go beyond the algebraic expression. The dashed line, y = 14{\frac{1}{4}}x + 3, acts as the boundary, but it is not included in the solution set. This is a critical distinction because it highlights that points on the line itself do not satisfy the original inequality, 12x−2y>−6{\frac{1}{2}x - 2y > -6}. The shaded region, which lies below the dashed line, represents all the points (x, y) that satisfy the inequality. This region extends infinitely in its direction, indicating an infinite number of solutions. Every point within this shaded region, when its coordinates are substituted into the original inequality, will yield a true statement. For instance, consider the point (0, 0), which we used as our test point. Substituting x = 0 and y = 0 into 12x−2y>−6{\frac{1}{2}x - 2y > -6} gives us 12{\frac{1}{2}}(0) - 2(0) > -6, which simplifies to 0 > -6. This is a true statement, confirming that (0, 0) is indeed a solution and lies within the shaded region. Conversely, any point outside the shaded region, such as (0, 5), will not satisfy the inequality. Substituting x = 0 and y = 5 into 12x−2y>−6{\frac{1}{2}x - 2y > -6} gives us 12{\frac{1}{2}}(0) - 2(5) > -6, which simplifies to -10 > -6. This is a false statement, demonstrating that (0, 5) is not a solution. The graph provides a clear and intuitive way to understand the solution set of the inequality. It allows us to quickly identify solutions and non-solutions, and it visually represents the infinite nature of the solution set. Interpreting the graph effectively involves understanding the meaning of the dashed line and the shaded region, and recognizing that every point within the shaded region is a solution to the inequality. With this understanding, the graph becomes a powerful tool for solving and analyzing linear inequalities. By understanding the visual representation of solutions, we complete the process of graphing linear inequalities, bridging the gap between algebraic expressions and their geometric interpretations.

Conclusion

In conclusion, graphing the linear inequality 12x−2y>−6{\frac{1}{2}x - 2y > -6} involves a systematic approach that combines algebraic manipulation with graphical representation. The process begins with transforming the inequality into the slope-intercept form, y < 14{\frac{1}{4}}x + 3, which provides the crucial information about the slope and y-intercept. This transformation sets the stage for plotting the boundary line, a dashed line in this case, to indicate that the points on the line are not included in the solution set. The next step involves shading the solution region, which is determined by testing a point, such as the origin (0, 0). If the test point satisfies the inequality, the region containing the point is shaded; otherwise, the opposite region is shaded. The shaded region represents all the points (x, y) that satisfy the inequality, providing a visual representation of the infinite solution set. Interpreting the graph is the final step, where the dashed line and the shaded region are understood in the context of the inequality. The dashed line marks the boundary, while the shaded region encompasses all the solutions. This graphical representation provides a powerful tool for understanding and analyzing linear inequalities. Mastering the graphing of linear inequalities is a fundamental skill in mathematics, with applications in various fields, including economics, engineering, and computer science. It allows for the visualization of solutions and provides a deeper understanding of the relationships between variables. The steps outlined in this guide provide a clear and concise method for graphing linear inequalities, empowering you to tackle more complex problems with confidence. By understanding the underlying principles and practicing the techniques, you can effectively graph and interpret linear inequalities, enhancing your mathematical proficiency and problem-solving abilities. This comprehensive guide serves as a valuable resource for anyone seeking to master the art of graphing linear inequalities, bridging the gap between algebraic concepts and their visual representations. With practice and a solid understanding of these steps, graphing linear inequalities becomes an accessible and valuable skill in your mathematical toolkit.