Problem 57 - College Algebra

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x-y \leq 2 \\ x>-2 \\ y \leq 3 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution set is the region on the coordinate plane that satisfies all three inequalities, represented by the overlay of the three half-planes.

Step 1: Draw each inequality

Start by drawing each inequality on the same coordinate plane. The inequality \(x-y \leq 2\) can be rewritten as \(y \geq x-2\), which represents an upward slanting half-plane. The line \(x = -2\) represents the vertical half-plane to the right of \(x = -2\). The line \(y = 3\) represents the horizontal half-plane below \(y = 3\). Each line is the boundary of its respective half-plane and should be included in the graph, as they are all 'less than or equal to' or 'greater than' type inequalities.

Step 2: Overlay the half-planes

Next, overlay the half-planes drawn in step 1, to find the intersection, which is the region that satisfies all inequalities.

Step 3: Identify the solution set

The solution set to the system of inequalities is the region that satisfies all the inequalities, which is the overlay of all three half-planes from step 2.

Key Concepts

graphing inequalities

Graphing inequalities involves representing the solutions of an inequality on a coordinate plane. For example, when dealing with a linear inequality like \(x - y \leq 2\), you first transform it into an equation, like \(y = x - 2\), to plot its boundary line. You then check the inequality sign to determine which half-plane to shade. If it's a "less than or equal to" (\(\leq\)) sign, shade below the line; if "greater than" (\(>\)), shade above. This shading reveals all the possible solutions to the inequality.

This method is very helpful because it gives a visual representation of solutions. By graphing each inequality in a system, you determine where all solutions overlap, which will become part of the complete solution set.

half-planes

A half-plane is essentially one side of the boundary defined by a linear equation. When working with inequalities, each inequality divides the coordinate plane into two parts: the half where the inequality holds true and the half where it doesn't. For instance, the inequality \(x - y \leq 2\) translates to the half-plane that lies above or on the line \(y = x - 2\).

Understanding half-planes is crucial in solving systems of inequalities. Each inequality contributes a half-plane, and the solution to the system is where these half-planes intersect or overlap. This shared section is where all inequalities are true simultaneously.

coordinate plane

The coordinate plane is a two-dimensional space where we can plot points, lines, and shapes to understand their mathematical relationships. It's defined by an x-axis (horizontal) and a y-axis (vertical), intersecting at a point called the origin (0,0).

On this plane, all our graphing takes place. Each point on the plane has coordinates (x, y) that tell us where it is in relation to the axes. By plotting inequalities on this plane, we represent the solutions visually, using lines and shaded regions to show where inequalities are satisfied.

The coordinate plane is a powerful tool for visualizing complex geometric and algebraic relationships, making it easier to analyze and solve problems.

solution set

The solution set is the collection of all values that satisfy the given system of inequalities. In the context of a graph, this is the region where all the shaded half-planes intersect. When you have a system like \(\begin{cases} x-y \leq 2 \ x > -2 \ y \leq 3 \end{cases}\), the solution set is that part of the coordinate plane where all these conditions are true.

This region may take different shapes depending on the inequalities. In some cases, there might even be no overlapping region, indicating no solution exists. By pinpointing this area, one can find all potential solutions that meet every condition imposed by the inequalities, providing a comprehensive answer to the system.