Problem 38 - Algebra and Trigonometry

Question

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{array}{c}{y < x+6} \\ {3 x+2 y \geq 12} \\ {x-2 y \leq 2}\end{array}\right. $$

Step-by-Step Solution

Answer

Identify and graph intersecting boundary lines for vertices; solution set is bounded if contained.

Step 1: Graph the Boundary Lines

Graph the boundary lines for each inequality. The first inequality is a dotted line since it is a strict inequality \(y = x + 6\). The second line \(3x + 2y = 12\) and the third line \(x - 2y = 2\) are solid lines since the inequalities include equality.

Step 2: Determine Inequality Regions

For \(y < x + 6\), shade below the line since it is a 'less than' inequality. For \(3x + 2y \geq 12\), shade above the line, and for \(x - 2y \leq 2\), shade below the line. The solution region is where all shaded areas overlap.

Step 3: Finding the Intersection Points

To find the vertices of the solution region, find intersections of the boundary lines: \(3x + 2y = 12\) with \(x - 2y = 2\), \(3x + 2y = 12\) with \(y = x + 6\), and \(x - 2y = 2\) with \(y = x + 6\). Solve each pair to find precise points.

Step 4: Check Boundary-Line Solutions

Solve the system of linear equations for each pair. For example, solving \(3x + 2y = 12\) and \(x - 2y = 2\), we find \(x = 2, y = 0\). Apply similar steps to other line pairs to find other vertices.

Step 5: Evaluate Boundedness

The solution set is defined by all regions overlapping from all shaded areas. If the area doesn't extend infinitely in any direction (i.e., it is enclosed), it is bounded. Check the vertices and sketch to confirm.

Key Concepts

Graphing Inequalities

When dealing with systems of inequalities, the first step is to graph each individual inequality. Think of each inequality like a different rule that draws a line on the graph indicating a boundary. This boundary helps us visualize the solutions that satisfy the inequality.

• A boundary line could be dotted or solid. If the inequality is strict, like '<' or '>', use a dotted line to show that points on the line are not included in the solution. For example, the inequality \(y < x+6\) would be graphed as a dotted line indicating \(y = x+6\).
• For inequalities with '≈' or '≤', use a solid line because points on the line are considered part of the solution, like the line \(3x + 2y = 12\) from this system.

Once the lines are drawn, the next task is to shade the appropriate area that fulfills the inequality. This shaded region will represent all the solutions for that inequality. For instance:

• If the inequality is '<', like with \(y < x+6\), we shade below the line.
• For '≥', like \(3x + 2y \geq 12\), we shade above the line.

Understanding this process is key to visualizing the solutions intersected by these conditions, helping you see where they overlap.

Intersection Points

Intersection points are crucial for identifying where solutions of different inequalities come together. They represent the vertices of the region that satisfies all inequalities. To find these points, you solve the equations formed by the boundary lines. Here’s how you find them:

• Pick two boundary lines at a time and set them as equality equations. For \(3x + 2y = 12\) and \(x - 2y = 2\), solving these gives us the point (2, 0).
• Repeat this process for other combinations of lines, like \(x - 2y = 2\) with \(y = x + 6\), and \(3x + 2y = 12\) with \(y = x + 6\).

By solving these systems of equations, you determine the precise coordinates of each intersection point. These coordinates are the vertices of the region defined by the inequalities. Make sure to check each vertex by substituting back into the original inequalities to ensure they all satisfy the conditions. Knowing how to do this makes the system of inequalities more manageable and less intimidating.

Bounded Solution

A bounded solution in terms of a system of inequalities means that the feasible region—the area where all inequalities overlap—is completely enclosed. It doesn't go off infinitely in any direction. Observing whether a solution is bounded involves both visual and analytical checks. Here’s how you can evaluate:

• After graphing all inequalities and finding intersection points, examine the shaded region.
• If it looks like an enclosed shape such as a triangle or polygon, your solution is bounded. In this exercise, if the vertices result in a closed triangle, then it implies boundedness.
• If the shaded region continues outwards without stopping, extends infinitely in all or some directions, your solution is unbounded.

A bounded solution provides a finite feasible region where each constraint meets. This is crucial, especially in real-world scenarios where solutions need to be within a specific area or limit. Identifying this helps determine if a system's solutions fit the given practical conditions.