Problem 38 - Introductory Algebra for College Students

Question

Graph the solution set of each system of linear inequalities. If the system has no solutions, state this and explain why. $$\left\\{\begin{array}{l}x-y \leq 3 \\\x+y \leq 3 \\\x \geq-2\end{array}\right.$$

Step-by-Step Solution

Answer

The solution set to the system of inequalities is the region that is shaded in all three graphs, which is the region that satisfies all inequalities. If there is a point that lies in the region of intersection and satisfies all inequalities, then the solution set is correct. However, if there is no such region, then the system of inequalities is said to have no solutions.

Step 1: Graph each inequality

Each inequality will represent a half-plane. Start with the first inequality \(x - y \leq 3\). To graph it easily, write it in slope-intercept form: \(y \geq x - 3\). The boundary line is \(y = x - 3\), this is a straight line with slope 1 and y-intercept -3. Represent this line in a graph and shade the half-plane above the line, since \(y\) is greater or equal. Repeat this process with the following inequalities \(x + y \leq 3\) and \(x \geq -2\).

Step 2: Find the intersection of all inequalities

The solution set of the system of inequalities is the intersection of all the individual solution sets. Thus, you need to find the region that is shaded in all graphs. This is the area that is a solution to all three inequalities.

Step 3: Validate the solution set

You must make sure that the area obtained in the previous step is the correct solution. To check, pick a point from the intersection area and substitute its coordinates into each inequality to verify that all inequalities are satisfied. If the point lies in the region of intersection, then it is confirmed that the solution set found is correct.

Key Concepts

Solution Set

In mathematics, when we discuss the "solution set" of inequalities, we are referring to all the possible values—that is, pairs of numbers \(x, y\)—that satisfy every inequality in the system. The solution set encompasses the region on a graph where all conditions imposed by the inequalities are true simultaneously. For example, the system given involves three inequalities:

• \(x - y \leq 3\)
• \(x + y \leq 3\)
• \(x \geq -2\)

These inequalities define boundaries on a graph, and the solution set is the overlapping region that fulfills all these inequalities simultaneously. If there is no overlapping region, meaning no common set of \(x, y\) satisfies all, the solution set is empty. Finding this overlapping region is central to solving systems of linear inequalities.
It is helpful to use the graphical method because it allows for a visual representation, making it easier to identify the solution set.

Graphing

Graphing is a vital skill in handling systems of linear inequalities. By plotting them on a graph, each inequality forms a boundary line, and you consider the area "allowed" by that inequality. For this exercise, start by converting inequalities into equations to graph the boundary lines:

• \(y = x - 3\) for the inequality \(x - y \leq 3\) simplifies to \(y \geq x - 3\), instructing you to shade above this line.
• \(y = -x + 3\) for \(x + y \leq 3\) tells you to shade below.
• \(x = -2\) designates a vertical boundary, shading to the right.

Each of these inequalities determines a half-plane, and their graphical representation helps identify where all conditions overlap. It can be tricky to keep in mind which side of the line to shade. For an inequality of type \(y \leq mx + b\), if in standard y-style, shade beneath the line; shade above for \(y \geq mx + b\).
Accurate graphing allows us to visually determine the valid region where all inequalities intersect, helping simplify analysis significantly.

Intersection of Inequalities

The "intersection of inequalities" is where we find harmony between competing mathematical statements. Once each inequality is graphically represented, the next task is to determine the area common to all these inequalities. This intersection is the portion of the graph that represents the solution set. We do this by identifying regions that are shaded by more than one inequality.

• The intersection is not just any overlapping area, but one where all individual conditions are met.
• Typically, the final solution set is a polygonal region, but it could sometimes be a line or a point.
• Validating solutions involves picking a test point from this intersected region and ensuring all original inequalities embrace it.

Whenever you have more than one inequality, combining them into a single, coherent graphical solution demands attention to each boundary’s half-plane and their mutual overlaps. It's a balancing act, carefully shading and evaluating, to find where all inequalities genuinely align.