Problem 71 - Algebra and Trigonometry

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 71–72, 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 graph of the solution set for these inequalities will be a region that is above the line \(y=\frac{3}{2}x-2\) and below the line \(y=4\). This region signifies the points that satisfy either one or both of the inequalities.

Step 1: Graph first inequality

The first inequality to graph is \(y>\frac{3}{2} x-2\). This is a linear inequality. To graph this, the line \(y=\frac{3}{2}x-2\) is plotted first. It has a slope of 3/2, hence it rises 3 units for every 2 units it runs. Also, it intersects the y-axis at -2. Next, because 'y' is greater than \(\frac{3}{2} x-2\), every point above this line (excluding the line) will be included to represent this inequality in the graph.

Step 2: Graph the second inequality

The second inequality to graph is \(y<4\). This is a horizontal line and it's drawn on the graph at the level where y equals 4. Because 'y' is less than 4, every point below this line (excluding the line) will be included to represent this inequality.

Step 3: Finding the union

The union of these two inequalities will be the combination of the shaded areas representing both inequalities. The final graph should represent all points that satisfy either one or both of the inequalities.

Key Concepts

Graphing Inequalities

When we are graphing inequalities, we need to identify all the points that satisfy a given condition. In this case, our focus is on linear inequalities, which involve graphs of lines.
To graph a linear inequality such as \(y > \frac{3}{2}x - 2\), we first plot the line that corresponds to the equation \(y = \frac{3}{2}x - 2\). This gives us a solid visual reference.

• The line itself is called the boundary line. For \(y > \frac{3}{2}x - 2\), we draw a dashed line because the values on the line are not included in the solution set. The inequality sign “>” indicates that we do not include the line itself, hence a dashed line is used instead of a solid one.
• Next, shade all the region above this line. The job of shading is to visually represent all the solutions that satisfy the inequality.

By employing these steps, you can graph any linear inequality and visually represent the set of solutions on a coordinate plane.

Linear Inequalities

Linear inequalities like \(y > \frac{3}{2}x - 2\) and \(y < 4\) are essential building blocks in algebra that represent vast sets of solutions.
Each inequality separates the coordinate plane into two regions.

• The boundary lines form the edges of these regions. With \(y > \frac{3}{2}x - 2\), the boundary line \(y = \frac{3}{2}x - 2\) tilts on the plane with a defined slope and intercept, dividing it into two parts. The area above it shows where solutions for the inequality exist.
• Similarly, \(y < 4\) represents solutions that lie below the horizontal boundary line \(y = 4\).

Understanding how each inequality defines regions on the plane will help you solve more complex systems in the future.

Union of Solution Sets

The union of solution sets is a concept that significantly differs from finding intersections. Instead of looking for common solutions that fit both inequalities, we combine solutions from each inequality. This concept is crucial when dealing with multiple inequalities in a system.

• To graph the union of \(y > \frac{3}{2}x - 2\) and \(y < 4\), you start by graphing each inequality separately as detailed in earlier steps.
• Create a combined graph by shading all regions from both inequalities. This results in a bigger, combined solution area on the graph.
• Visually, the union is the collective area that satisfies either inequality or both. It's like creating a larger pool of solutions by merging individual ones.

Understanding and graphing unions help in solving problems that require accounting for multiple conditions or constraints.