Problem 71 - College Algebra Essentials

Question

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, you will be graphing the union of the solution sets of two inequalities. Graph the union of \(y>\frac{3}{2} x-2\) and \(y<4\)

Step-by-Step Solution

Answer

The union of the two inequalities \(y>\frac{3}{2} x-2\) and \(y<4\) is the entire area underneath the line \(y=4\) and the entire area above the line \(y=\frac{3}{2}x-2\) on the graph. It includes any overlapped area between these two areas.

Step 1: Graph the inequality \(y>\frac{3}{2} x-2\)

To graph this inequality, begin by graphing the line \(y=\frac{3}{2} x-2\). This line should be dashed because the inequality is '>' (greater than), not '≥' (greater than or equal to). Then, shade the area above the line because \(y>\frac{3}{2} x-2\) represents all the points above the line.

Step 2: Graph the inequality \(y<4\)

To graph this inequality, begin by graphing the line \(y=4\). This line should be dashed because the inequality is '<' (less than), not '≤' (less than or equal to). Then, shade the area below the line because \(y<4\) represents all the points below the line.

Step 3: Find the union of the solution sets

The union represents all the areas that are shaded for at least one of the inequalities. It's basically the combination of both shaded areas for the inequalities. Overlapping shaded areas represent solutions that satisfy both inequalities.

Step 4: Answer

The union of the two inequalities \(y>\frac{3}{2} x-2\) and \(y<4\) is the entire area underneath the line \(y=4\) and the entire area above the line \(y=\frac{3}{2} x-2\) on the graph. It would include any overlap between these two areas.

Key Concepts

Solution Sets of Inequalities

Understanding the solution sets of inequalities is crucial when it comes to analyzing relationships within an algebraic context. An inequality, such as \(y > \frac{3}{2} x - 2\), specifies a relationship between two expressions where they are not equal, but one is greater or less than the other. The 'solution set' is the collection of all possible values that satisfy the inequality.
When graphing an inequality, you start with the related equation (the boundary) and then determine which side of this boundary is included in the solution set. For instance, \(y > \frac{3}{2} x - 2\) involves graphing the line \(y = \frac{3}{2} x - 2\) with a dashed line to indicate that the points on the line are not part of the solution. The shading above the line shows all the points that make the inequality true. Graphically, any point within the shaded region is part of the solution set, and you can pick any value from this area, and it will satisfy the inequality.
When dealing with a system of inequalities, each inequality will have its own solution set. The way these sets interact is fundamental in determining the overall solution to the system.

Intersection of Solution Sets

Understanding Intersection

The intersection of solution sets occurs when you have two or more inequalities, and you're looking for the set of points that satisfy all of them simultaneously. In the graphical method, the intersection is represented by the overlapping shaded regions of each inequality's solution set.
For example, if you were given two inequalities, \(y > \frac{3}{2} x - 2\) and \(y < x + 3\), their individual solution sets would be shaded on a graph. The common area where both shadings overlap would symbolize the intersection. Only the points within the overlapping region would make both inequalities true at the same time.
This concept can be likened to a Venn diagram, where the intersection reflects the commonality between different sets. Finding the intersection is like discovering the common ground that satisfies all criteria imposed by the various inequalities involved.

Union of Solution Sets

Graphing the Union

The union of solution sets represents all points that satisfy any of the inequalities in the system. Contrary to the intersection where all conditions must be met, the union includes areas that fulfill at least one of the inequalities. When graphing, this means you shade the solution set for each inequality and consider any shaded part as the solution.
With the given exercise, graphing the union involved two inequalities: \(y > \frac{3}{2} x - 2\) and \(y < 4\). After graphing both inequalities independently, the union included any points that were above the line \(y = \frac{3}{2} x - 2\) or below the line \(y = 4\). The result is a combined shaded area where either of the inequalities' conditions is fulfilled. Visually, it may look like two separate patches or could take on a more complex shape depending on the inequalities and where their solution sets lie in relation to each other.
It's similar to combining two groups, allowing anyone belonging to at least one group to be included in the overall count. The union is a powerful concept that widens the scope of possible solutions by accepting any that qualify under the given inequalities.