Graphing Inequalities Unveiling The Solution For -2x - 3y < 6
In the realm of mathematics, inequalities play a crucial role in defining relationships between variables and sketching regions on a coordinate plane. This article delves into the process of identifying the graph that accurately represents the inequality -2x - 3y < 6. We will explore the steps involved in converting the inequality into a more manageable form, plotting the boundary line, and determining the shaded region that satisfies the given condition. Understanding these concepts is fundamental for comprehending linear inequalities and their graphical representations.
Demystifying Linear Inequalities
Before we dive into the specific inequality at hand, let's first establish a firm grasp on the fundamentals of linear inequalities. A linear inequality, at its core, is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). These inequalities, when plotted on a coordinate plane, define a region rather than a single line, as in the case of linear equations. This region encompasses all the points whose coordinates satisfy the inequality.
Linear inequalities are ubiquitous in various real-world applications, from optimizing resource allocation in business to modeling constraints in engineering problems. Their ability to represent a range of possibilities makes them an indispensable tool for decision-making and problem-solving. In the context of graphing, linear inequalities provide a visual representation of the solution set, offering a clear understanding of the values that satisfy the given conditions.
Grasping the fundamentals of linear inequalities is essential for accurately interpreting and representing them graphically. The inequality symbol dictates the type of boundary line (dashed or solid) and the region to be shaded. A dashed line indicates that the points on the line are not included in the solution set, while a solid line signifies their inclusion. The shading then visually represents the area containing all the points that satisfy the inequality. For instance, an inequality like y > mx + b would be represented by a dashed line with the region above it shaded, indicating that all points above the line satisfy the inequality.
Transforming the Inequality into Slope-Intercept Form
The first step in graphing the inequality -2x - 3y < 6 is to transform it into the slope-intercept form, which is given by y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. This form provides valuable insights into the line's orientation and position on the coordinate plane, making it easier to visualize and graph the inequality.
To achieve this transformation, we need to isolate 'y' on one side of the inequality. Let's begin by adding 2x to both sides of the inequality: -3y < 2x + 6. Next, we divide both sides by -3. However, a crucial point to remember is that dividing or multiplying an inequality by a negative number reverses the inequality sign. Therefore, we obtain y > (-2/3)x - 2. This transformed inequality is now in slope-intercept form, with a slope of -2/3 and a y-intercept of -2.
Converting the inequality to slope-intercept form not only simplifies the graphing process but also provides a clear understanding of the relationship between 'x' and 'y'. The slope, -2/3, indicates that for every 3 units we move horizontally, the line descends by 2 units. The y-intercept, -2, tells us that the line crosses the y-axis at the point (0, -2). This information is invaluable for accurately plotting the boundary line and subsequently determining the shaded region.
Plotting the Boundary Line
Now that we have the inequality in slope-intercept form, y > (-2/3)x - 2, we can proceed to plot the boundary line on the coordinate plane. The boundary line is the graphical representation of the equation y = (-2/3)x - 2, which serves as the dividing line between the regions that satisfy the inequality and those that do not. The nature of the inequality symbol determines whether the boundary line is solid or dashed.
Since our inequality uses the 'greater than' symbol (>), the boundary line will be dashed. This indicates that the points on the line itself are not included in the solution set. To plot the line, we can start with the y-intercept, which is -2. This gives us the point (0, -2) on the line. Then, using the slope of -2/3, we can find other points on the line. For instance, moving 3 units to the right and 2 units down from (0, -2) gives us the point (3, -4). Similarly, moving 3 units to the left and 2 units up from (0, -2) gives us the point (-3, 0).
Connecting these points with a dashed line gives us the boundary line for the inequality. The dashed line visually represents the solutions to the equation y = (-2/3)x - 2, but it's crucial to remember that these points do not satisfy the original inequality y > (-2/3)x - 2. The dashed line serves as a visual separator, guiding us in identifying the region that contains the solutions to the inequality.
Determining the Shaded Region
The final step in graphing the inequality -2x - 3y < 6 is to determine the shaded region, which represents all the points whose coordinates satisfy the inequality. To do this, we can use a test point. A test point is any point that is not on the boundary line. We substitute the coordinates of the test point into the original inequality and check if the inequality holds true.
A convenient test point to use is the origin (0, 0), as it simplifies calculations. Substituting x = 0 and y = 0 into the inequality y > (-2/3)x - 2, we get 0 > (-2/3)(0) - 2, which simplifies to 0 > -2. This inequality is true, indicating that the origin (0, 0) is part of the solution set. Therefore, we shade the region that contains the origin.
If the test point had not satisfied the inequality, we would have shaded the region on the opposite side of the boundary line. Shading the appropriate region visually represents the solution set of the inequality, providing a clear understanding of all the points that satisfy the given condition. In this case, since the origin satisfied the inequality, we shade the region above the dashed line, indicating that all points in this region have coordinates that make the inequality y > (-2/3)x - 2 true.
Conclusion: The Graphical Representation of -2x - 3y < 6
In conclusion, the graph corresponding to the inequality -2x - 3y < 6 is a region bounded by a dashed line with a slope of -2/3 and a y-intercept of -2. The region above the dashed line is shaded, indicating that all points in this region satisfy the inequality. This graphical representation provides a visual understanding of the solution set, making it easier to identify points that meet the given condition.
Understanding the steps involved in graphing linear inequalities is essential for solving various mathematical problems and real-world applications. By transforming the inequality into slope-intercept form, plotting the boundary line, and determining the shaded region, we can effectively represent the solution set graphically. This skill is invaluable for comprehending mathematical relationships and making informed decisions based on graphical interpretations.
