Question

Use a graphing calculator or a computer to graph the system of inequalities. Give the coordinates of each vertex of the solution region. \(x+y \leq 11\) \(5 x-3 y \leq 15\) \(x \geq 0\)

Short answer

The vertices of the solution region are at the coordinates (0,0), (3,0) and (0,11).

Step-by-step solution

Graphing Each Inequality

Plot each of the inequalities on a graph individually. The inequality \(x+y \leq 11\) is a straight line with y-intercept at (0,11) and x-intercept at (11,0). The inequality \(5x-3y \leq 15\) is also a straight line, which passes through the points (0,-5) and (3,0). The inequality \(x \geq 0\) represents the portion of the graph to the right of the y-axis.

Shading the Solution Area

For each inequality, shade the region that satisfies the inequality. For \(x+y \leq 11\), the solution region is everything below the line. For \(5x-3y \leq 15\), the solution region is above the line. And for \(x \geq 0\), the solution region is to the right of the y-axis. The common shaded area represents the solution for all these inequalities.

Finding the Vertices

The vertices of the solution region are the intersection points of the lines. You can find these by either visually inspecting the graph or by solving the equations of the lines. The vertices are (0,0), (3,0) and (0,11).

Key concepts

Graphing Calculator

Mastering the use of a graphing calculator is a pivotal skill for students tackling systems of inequalities. This tool enables learners to visually analyze and solve complex problems that might be time-consuming if done by hand. A graphing calculator simplifies the process by allowing users to input equations directly, and it generates the corresponding graphs promptly.

For the given exercise, each inequality can be keyed into a graphing calculator, set to graph mode. This direct approach not just reflects the equations visually, but also assists in identifying the solution region efficiently. Most advanced graphing calculators also offer features to trace along the graph and provide coordinates of key points such as vertices, intercepts, and intersections, which are often essential in forming the solution set.

When instructing students on using graphing calculators, it's helpful to point out the importance of correctly entering inequalities with attention to detail. A misplaced sign or coefficient can lead to a completely different graph and thus an incorrect solution. Always encourage double-checking the input against the original problem.

Solution Region

The solution region is a fundamental concept when graphing systems of inequalities. This region represents the set of all possible solutions that satisfy every inequality within the system. In graphical terms, it's where the shaded areas of all the inequalities overlap on the graph.

Students often find shading the correct region challenging. A practical tip is to pick a test point, preferably the origin (0,0) if it's not on any of the lines, to determine which side of the line to shade. If the inequality holds true for the test point, the area towards the test point is shaded.

In our exercise, the final solution region is a convex polygon, sometimes also referred to as the 'feasible region'. Identifying this region correctly is crucial, as it visually encapsulates all potential solutions. Students should be familiar with the fact that the coordinates of the vertices of this polygon may represent optimal solutions, particularly in linear programming scenarios.

Inequality Graphing

The art of inequality graphing involves representing inequalities on a coordinate plane. Unlike equations that equate to a line, inequalities include all points on one side of the line as part of their solution set. Students must understand how to graph the associated line and then determine which side of the line forms the solution set.

For example, with the inequality \( x+y \leq 11 \), one would graph the line \( x+y = 11\) and then shade below the line since all the points below it satisfy the 'less than or equal to' condition. It's analogous to saying, 'the sum of x and y is no greater than 11.'

In classroom practice, a solid educational approach includes tasks that require students to

Analyze the Slope and Intercept

This step is done before shading, analyzing graphically why the slope affects the angle of the line and how the intercept positions the line. Regular exercises should help students internalize how changes in the coefficients of x and y affect the steepness and orientation of the line on the graph. Moreover, becoming proficient in

Identifying Boundary Lines

These are the lines that form the 'border' of the solution set. Remember to address the distinction between solid lines for inequalities including equality (\( \leq \) or \( \geq \)) and dashed lines for strict inequalities (\( < \) or \( > \)), clarifying that the former include the points on the line while the latter do not. In conjunction, employing these strategies in classroom instruction will lead to a comprehensive understanding of inequality graphing.