Problem 39 - Precalculus with Limits

Question

In Exercises 33-46, sketch the graph (and label the vertices) of the solution set of the system of inequalities. $$\left\\{\begin{array}{c}{-3 x+2 y<6} \\ {x-4 y>-2} \\ {2 x+y<3}\end{array}\right.$$

Step-by-Step Solution

Answer

The graph consists of three lines, each representing a linear inequality. The vertices can be determined by finding where these lines intersect. The solution set is the region where the shaded areas of all three inequalities overlap, representing a region where all inequalities hold true.

Step 1: Graphing the Inequalities

Each inequality can be transformed into an equation by replacing the inequality symbol with an equals sign. This, done for all three, gives: -3x + 2y = 6, x - 4y = -2, and 2x + y = 3. Plotting these equations individually will result in three lines on the graph.

Step 2: Identifying the Solution Areas

Each inequality divides the whole plane into two regions: one that satisfies, and one that doesn't. Therefore, each of the equations when replaced the equals sign with an inequality sign, it's crucial to perform a simple test, such as using the origin (0,0), to see which region to shade. The shaded regions are where the inequalities hold true.

Step 3: Finding the Solution Set

Given the shading from each inequality, the solution set to the system of inequalities is the region where the shaded areas of all three overlap. This is the area where all inequalities hold, and hence, is the solution to the system of inequalities.

Key Concepts

Sketching Inequality Graphs

Understanding how to sketch graphs of inequalities is a fundamental skill in precalculus and algebra. To begin with, consider the boundary lines by replacing the inequality sign with an equals sign. For example, the inequality (-3x + 2y < 6) becomes the line (-3x + 2y = 6). This line represents all the points that are the exact solution to the equation. However, because we have an inequality, we are interested in the area on one side of this line.

To determine which side of the line to shade, you can use a test point that is not on the line, often the origin (0,0) if it's not on the boundary line. If the test point satisfies the inequality, you shade the side of the line where the test point lies; otherwise, you shade the opposite side. Remember to draw dashed lines for '<' or '>' inequalities to indicate that points on the line are not included in the solution set, and solid lines for 'eq' or 'eq' inequalities to include those points.

The process needs to be repeated for each inequality in the system. Then, the intersection of the shaded areas will identify the solution set that satisfies all the inequalities simultaneously.

• Convert inequalities to equations
• Graph each line on a coordinate plane
• Use a test point to determine which side to shade
• Shade the appropriate region for each inequality
• Identify the intersection area for the system's solution set

Solution Set of Inequalities

Comprehending and locating the solution set of a system of inequalities is a vital concept in precalculus. The 'solution set' refers to the collection of all possible solutions (points) that satisfy all inequalities simultaneously. It’s a region on the graph and not just a line or a single point.

You find it by first graphing each inequality as if it were a line, then determining where to shade for each individual inequality, and finally, looking for where these shaded areas overlap. This common overlapped region is your solution set. This region could be bounded, meaning it's a finite area, or unbounded, extending infinitely in one or more directions.
In our example with inequalities (-3x + 2y < 6), (x - 4y > -2), and (2x + y < 3), the solution set is the overlapped region after correctly shading the regions for each inequality.
In practice, the solution set gives us a visual representation of all the coordinates (x,y) that can satisfy a given system of inequalities, which is incredibly helpful for solving real-life problems like optimization and resource allocation.

• The solution set is where all shaded regions overlap.
• It includes all points that satisfy all the inequalities.
• This set may be bounded or unbounded.

Graphical Solutions in Precalculus

In precalculus, graphical solutions are a powerful means to solve systems of inequalities. They provide a visual and intuitive way to understand and solve complex problems. When working graphically, you should always keep in mind the scale and axes limits to ensure accurate representations of the solution set.

Graphical solutions involve plotting each inequality on a set of axes and identifying the feasible region. Not only does this method help visualize the solution, but it also allows you to analyze how changing an inequality will affect the solution set. This can be particularly poignant when discussing real-life scenarios in economics or engineering, where alterations in constraints could lead to adjusting resources or strategies.
Understanding graphical solutions can also form a basis for learning more advanced topics, such as linear programming, which often relies on similar principles to identify optimal solutions. It's essential to learn to draw precise and clear graphs, as inaccuracies can lead to a misunderstanding of the solution set.

• Plot inequalities on a graph to examine their solutions.
• Useful for understanding the impact of varying constraints.
• Good foundation for more advanced mathematical concepts.
• Accuracy in graphing is critical for correct solutions.