Problem 40 - Elementary Algebra Within Reach

Question

How do you determine the vertices of the solution region for a system of linear inequalities?

Step-by-Step Solution

Answer

The vertices of the solution region for a system of linear inequalities are determined by first graphing each inequality and identifying the overlapping solution region. The intersection points within this region are the vertices.

Step 1: Graph the Inequalities

Start by graphing each inequality in the system, shading the region which is the solution to the inequality. If the inequality includes an 'equal to' sign (\(\le\) or \(\ge\)), draw a solid line. If it doesn't (\(<\) or \(>\)), draw a dashed line.

Step 2: Identify the Solution Region

The next step is to identify the solution region for the system. This will be the region where the shaded areas of all the inequalities overlap.

Step 3: Determine the Vertices

Lastly, identify all the points where two or more lines intersect within the solution region. These points are the vertices of the solution region.

Key Concepts

Graphing Inequalities

When graphing inequalities, you'll need to represent each inequality on a coordinate plane. It's important to note how you draw the boundary lines:

• If the inequality includes a "less than or equal to" (\(\leq\)) or "greater than or equal to" (\(\geq\)) sign, use a solid line. This means points on the line itself are included in the solution set.
• If the inequality is strictly "less than" (\(<\)) or "greater than" (\(>\)), use a dashed line. This indicates that points on the line are not included in the solution set.

After plotting the line(s), choose a test point not on the line (often the origin \((0,0)\), unless it falls on one of the lines) to determine which side of the line should be shaded. Substitute this point into the inequality:

• If the inequality holds true, shade the region containing that point.
• If not, shade the opposite side.

Shading Solution Regions

Once the boundary lines are drawn, the next step is shading the solution regions. This shading helps visualize which areas satisfy each inequality.

• Pick a section of the graph on either side of the line and test a point in that section.
• If your test point satisfies the inequality, shade that region to indicate it's part of the solution.
• If it doesn’t, shade the opposite region.

For a system of inequalities, each inequality will have its own shaded region. The solution to the system is the area where all these shaded regions overlap. This overlapping region represents all the points that satisfy all inequalities in the system.

Identifying Vertices

The vertices of the solution region are pivotal points where the boundary lines intersect. To identify these vertices:

• Look at where the lines of the inequalities intersect within the shaded solution region.
• The points of intersection are crucial because they are potential solutions where the inequalities meet.
• Check each vertex to ensure it falls within the overlapping shaded area, and that it conforms to all the given inequalities.

Vertices are important because, in optimization problems, these points often represent maximum or minimum values.