Graphing Solutions To Systems Of Inequalities A Step-by-Step Guide

In mathematics, particularly in algebra, understanding how to graph the solutions to systems of inequalities is a crucial skill. It allows us to visualize the set of points that satisfy multiple inequality conditions simultaneously. This article will delve into the process of graphing solutions for a system of inequalities in the coordinate plane, using the specific example:

$ \begin{aligned} 3 y & \ \textgreater \ 2 x+12 \ 2 x+y & \leq-5 \end{aligned} $

We will explore each step in detail, ensuring a comprehensive understanding of the techniques involved.

Understanding Linear Inequalities

Before we tackle the system of inequalities, it's essential to grasp the concept of linear inequalities. A linear inequality is similar to a linear equation but uses inequality symbols such as greater than ($>$), less than ($<$), greater than or equal to ($\geq$), or less than or equal to ($\leq$). When graphed on a coordinate plane, a linear inequality represents a region rather than a single line. This region includes all the points that satisfy the inequality. The boundary line of this region is determined by the corresponding linear equation (i.e., replacing the inequality symbol with an equal sign).

To effectively graph linear inequalities, several key concepts must be understood. First, the slope-intercept form of a linear equation, represented as $y = mx + b$, plays a crucial role. Here, 'm' denotes the slope, indicating the line's steepness and direction, while 'b' represents the y-intercept, the point where the line crosses the y-axis. Transforming inequalities into slope-intercept form simplifies the graphing process by providing a clear understanding of the line's orientation and position on the coordinate plane. For instance, an inequality like $2x + y \leq -5$ can be rearranged into $y \leq -2x - 5$, immediately revealing a slope of -2 and a y-intercept of -5. This transformation not only aids in accurately plotting the boundary line but also in determining the shaded region that satisfies the inequality. When the inequality includes '$>$' or '$<$', the boundary line is dashed to indicate that points on the line are not part of the solution set, whereas '$\geq$' or '$\leq$' use a solid line to include the boundary. Secondly, determining which side of the line to shade is vital for representing the inequality correctly. This is often achieved by selecting a test point not on the line, such as (0,0), and substituting its coordinates into the inequality. If the inequality holds true, the region containing the test point is shaded; otherwise, the opposite region is shaded. For example, after graphing the line for $y \leq -2x - 5$, testing (0,0) gives $0 \leq -5$, which is false, indicating that the region not containing (0,0) should be shaded. This step ensures that the graphical representation accurately reflects the solution set of the inequality.

Step-by-Step Solution

Let's graph the solution to the given system of inequalities step-by-step:

1. Isolate y in Each Inequality

The first step is to rewrite each inequality to isolate 'y' on one side. This transformation puts the inequalities into a form that is easier to graph, specifically the slope-intercept form ($y = mx + b$), which provides immediate insight into the slope and y-intercept of the boundary line. For the first inequality, $3y > 2x + 12$, dividing both sides by 3 isolates 'y', resulting in $y > \frac{2}{3}x + 4$. This form clearly shows that the boundary line has a slope of $\frac{2}{3}$ and a y-intercept of 4, which are crucial parameters for graphing. Similarly, for the second inequality, $2x + y \leq -5$, isolating 'y' involves subtracting 2x from both sides, leading to $y \leq -2x - 5$. This reveals a slope of -2 and a y-intercept of -5 for the second boundary line. The transformation to slope-intercept form not only simplifies the graphing process but also aids in understanding the relationship between the inequalities and their graphical representations, making it a cornerstone of solving systems of inequalities.

• Inequality 1: $3 y > 2 x+12$ Divide both sides by 3: $y > \frac{2}{3} x+4$

• Inequality 2: $2 x+y \leq-5$ Subtract 2x from both sides: $y \leq-2 x-5$

2. Graph the Boundary Lines

Next, we graph the boundary lines for each inequality. The boundary line is the line obtained by replacing the inequality symbol with an equal sign. For the inequality $y > \frac{2}{3}x + 4$, the boundary line is $y = \frac{2}{3}x + 4$. This line has a slope of $\frac{2}{3}$ and a y-intercept of 4. Since the inequality is 'greater than' (rather than 'greater than or equal to'), we draw the line as a dashed line to indicate that points on the line are not included in the solution. To plot this line, we start at the y-intercept (0,4) and use the slope to find another point. A slope of $\frac{2}{3}$ means that for every 3 units we move to the right along the x-axis, we move 2 units up along the y-axis. This leads us to the point (3,6), allowing us to draw the dashed line accurately.

For the second inequality, $y \leq -2x - 5$, the boundary line is $y = -2x - 5$. This line has a slope of -2 and a y-intercept of -5. Because the inequality includes 'less than or equal to', we draw a solid line to indicate that points on the line are part of the solution set. Starting from the y-intercept (0,-5), we use the slope of -2 to find another point. A slope of -2, which can be seen as $\frac{-2}{1}$, means that for every 1 unit we move to the right, we move 2 units down. This leads us to the point (1,-7), allowing us to draw the solid line precisely. Graphing these boundary lines correctly is a critical step in visualizing the solution set for the system of inequalities.

• For $y > \frac{2}{3} x+4$, graph the dashed line $y=\frac{2}{3} x+4$.
• For $y \leq-2 x-5$, graph the solid line $y=-2 x-5$.

3. Shade the Appropriate Regions

The next crucial step in graphing the solutions to a system of inequalities is shading the appropriate regions on the coordinate plane. This process involves determining which side of each boundary line represents the solutions that satisfy the respective inequality. To accomplish this, we employ a method using test points. The most commonly used test point is the origin (0,0), provided it does not lie on the boundary line itself. Substituting the coordinates of the test point into the inequality reveals whether the region containing the test point is part of the solution set.

For the first inequality, $y > \frac{2}{3}x + 4$, substituting (0,0) yields $0 > \frac{2}{3}(0) + 4$, which simplifies to $0 > 4$. This statement is false, indicating that the region containing the origin does not satisfy the inequality. Therefore, we shade the region above the dashed line $y = \frac{2}{3}x + 4$, representing all points where the y-coordinate is greater than $\frac{2}{3}x + 4$. This shading effectively visualizes the solutions for the first inequality.

Moving to the second inequality, $y \leq -2x - 5$, we again substitute the test point (0,0), resulting in $0 \leq -2(0) - 5$, which simplifies to $0 \leq -5$. This statement is also false, implying that the region containing the origin is not part of the solution set for this inequality. Consequently, we shade the region below the solid line $y = -2x - 5$, encompassing all points where the y-coordinate is less than or equal to $-2x - 5$. This shading visually represents the solutions for the second inequality.

The accurate determination of the shaded regions is paramount in solving systems of inequalities graphically. Shading the correct side of each boundary line ensures that the graphical representation precisely reflects the solutions that satisfy each inequality, setting the stage for identifying the overall solution set for the system.

• For $y > \frac{2}{3} x+4$, test the point (0,0): $0 > \frac{2}{3}(0) + 4$ is false, so shade the region above the dashed line.
• For $y \leq-2 x-5$, test the point (0,0): $0 \leq-2(0)-5$ is false, so shade the region below the solid line.

4. Identify the Solution Region

The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap. This overlapping region represents the set of all points that satisfy both inequalities simultaneously. In the coordinate plane, the overlapping region is the area where the shadings from the individual inequalities intersect. Every point within this region, and along any solid boundary lines forming its edges, constitutes a solution to the system.

Visually, the solution region is the area that has been shaded twice—once for each inequality. This area may be a bounded polygon, an unbounded region extending to infinity, or even an empty set if the inequalities have no solutions in common. The accurate identification of this region is crucial, as it provides a clear graphical representation of the solution set. Points within this region, when their coordinates are substituted into the original inequalities, will satisfy all conditions, confirming their inclusion in the solution set.

The boundaries of the solution region are defined by the boundary lines of the inequalities. Where these lines are solid, the points on the lines are included in the solution. However, where the lines are dashed, the points on the lines are excluded. The points of intersection of the boundary lines, known as vertices, are particularly significant. If the solution region is a bounded polygon, the vertices represent the extreme points of the solution set and are critical in optimization problems. By carefully examining the graph and identifying the overlapping shaded region, we can effectively visualize and communicate the solution to the system of inequalities.

This region represents the solution to the system of inequalities. Any point in this region satisfies both inequalities.

Conclusion

Graphing the solution to a system of inequalities involves several key steps: isolating 'y' in each inequality, graphing the boundary lines (dashed or solid as appropriate), shading the correct regions based on test points, and identifying the overlapping region as the solution. This process provides a visual representation of the solution set, making it easier to understand and interpret the solutions to systems of inequalities. By following these steps carefully, you can accurately graph the solutions to any system of linear inequalities in the coordinate plane.

This skill is foundational in various mathematical applications, including linear programming, optimization problems, and understanding constraints in real-world scenarios. Mastering the graphical representation of inequality solutions opens doors to more advanced mathematical concepts and practical problem-solving.