Problem 43 - Thinking Mathematically

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, in Exercises 43-44, 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 solution is a region which is below the line \(y = 4\) and above the line \(y = \frac{3}{2}x -2\). In other words, it's a region that satisfies either of the inequalities, or both.

Step 1: Graph the First Inequality

Start by graphing the inequality \(y > \frac{3}{2}x -2\). To do this, first plot the line \(y = \frac{3}{2}x -2\). Remember, this line should be a dashed one, because the inequality is 'greater than', not 'greater than or equal to'. The line divides the graph in two. Then choose a test point not on the line. The point (0,0) is a good choice. Substituting \(x = 0\) and \(y = 0\) into the inequality gives \(0 > -2\), which is true. So (0,0) is in the solution set, and then the entire half-plane that contains (0,0) is shaded.

Step 2: Graph the Second Inequality

Now graph the inequality \(y < 4\). Again, begin by graphing the line \(y = 4\), which will be a dashed horizontal line running through point \(y = 4\). Take a test point below the line, for example (0,0), and substitute the coordinates into the inequality to see if it is true. \(0 < 4\) is true, meaning the area below the line should be shaded.

Step 3: Find the Union of the Inequalities

The union of the solution sets of the two inequalities is all the points that satisfy either of the inequalities. So, look for the regions where either of the shaded regions from step 1 or 2, or both, cover. This combined region is the solution to the system.

Key Concepts

Understanding Solution Sets of Inequalities

When we talk about the solution sets of inequalities, we're referring to finding all the possible values that satisfy a given inequality. Graphically, this means shading the region of the graph where the inequality holds true. In the context of the exercise given, we dealt with two inequalities. The solution set for each inequality is represented by a shaded area on a graph. For the inequality (y > \frac{3}{2}x -2), we start by graphing the corresponding line (y = \frac{3}{2}x -2), using a dashed line to indicate that points on the line are not included in the solution set.

To determine the correct region to shade, we selected a test point, like (0,0), and substituted it into the inequality. Since our test point satisfied the inequality \(0 > -2\), we shaded the entire half-plane above the line. This area represents all the points where \(y\) is greater than \(\frac{3}{2}x -2\).

It's critical for students to understand the concept of selecting a proper test point; it's often convenient to use (0,0) if it's not on the borderline, as it usually makes the math simpler. Remember, the solution set is a visual representation of all the (x, y) coordinate pairs that make the inequality true.

Union of Inequalities Explained

The union of inequalities comes into play when we have more than one inequality and we want to know the set of all points that satisfy at least one of the inequalities. Rather than looking for the overlap, as in an intersection, we are combining the shaded regions. In the exercise, after graphing the first inequality, we followed a similar process for \(y < 4\), using \(y = 4\) as our borderline and a test point to determine the region of shading.

The union is the combined area where either of the inequalities is true. So, if a point lies in the shaded area of the first inequality, the second inequality, or in both, it's part of the solution set for the union. Having a clear understanding of this concept is essential for correctly addressing more complex systems of inequalities. We simply look where the shading from both inequalities overlaps or exists side by side and consider this entire region part of the union set. This means that if a point satisfies either or both of the inequalities, it is included in the solution.

Graphing Linear Inequalities

The process of graphing linear inequalities is a foundational skill that helps in visualizing and solving these inequalities. Start by treating the inequality as if it were an equation to draw the boundary line. It's important to note whether the inequality is 'strict' (represented with a dashed line) or 'inclusive' (represented with a solid line). Strict inequalities, such as \(y > \frac{3}{2}x -2\) or \(y < 4\), mean that the values on the line are not part of the solution, hence a dashed line is used.

Once the boundary line is in place, choose a test point not on the line to determine which side of the line is part of the solution set. If the inequality is satisfied when you plug in the point, then that side of the line is shaded. In our exercise, the lines \(y = \frac{3}{2}x -2\) and \(y = 4\) were drawn dashed, and the regions where the inequalities held true were accordingly shaded. Giving students the skills to graph inequalities properly lays the groundwork for them to solve and understand systems of inequalities, which is a critical aspect of algebra.