Problem 44 - Intermediate Algebra

Question

Use a graphing calculator to solve each system. See Using Your Calculator: Solving Systems of Inequalities. $$\left\\{\begin{array}{l}3 x+y<-2 \\\y>3(1-x)\end{array}\right.$$

Step-by-Step Solution

Answer

Graph both inequalities and find the overlapping region.

Step 1: Set up the inequalities

We have the system of inequalities: 1. \(3x + y < -2\)2. \(y > 3(1 - x)\). First, understand that these represent two regions on a graph, and we need to find the overlapping area where both inequalities are true.

Step 2: Graph the first inequality

Rewrite the first inequality as \(y < -3x - 2\). On your graphing calculator, enter this inequality. This will display as a shaded region below the line \(y = -3x - 2\). Make sure to visualize that the line itself is dashed because the inequality is strict (it does not include the line).

Step 3: Graph the second inequality

Rewrite the second inequality as \(y > 3 - 3x\). Enter this inequality on the graphing calculator as well. This will display as a shaded region above the line \(y = 3 - 3x\). The line is dashed because the inequality is strict.

Step 4: Identify the overlapping region

On the graphing calculator, locate the region where the shaded areas from both inequalities overlap. This region represents the solution to the system of inequalities. Ensure both shading patterns are correctly input to see where they intersect.

Step 5: Check specific points

To verify your solution, choose a few test points within the overlapping region and substitute them back into both original inequalities to ensure they are satisfied by both.

Key Concepts

Systems of Inequalities

A system of inequalities involves having more than one inequality at the same time. Each inequality describes a specific area or region on a graph. In our case, we have two inequalities:

• \(3x + y < -2\)
• \(y > 3(1 - x)\)

These inequalities might look familiar as they are similar to linear equations, but instead of a single line, they create shaded regions on a graph. Solving such systems means finding all possible values of \(x\) and \(y\) that satisfy both inequalities simultaneously. Essentially, we're seeking the overlapping area where both conditions are met on the graph.
This forms the basis of finding solutions to systems of inequalities.

Graphical Representation

Graphical representation is a method of showcasing these inequalities on a coordinate plane. Using a graphing calculator is helpful as it allows you to enter each inequality and visually see the region it represents. For each inequality:

• The first inequality \(3x + y < -2\) is represented by shading below the line \(y = -3x - 2\).
• The second inequality \(y > 3 - 3x\) is represented by shading above the line \(y = 3 - 3x\).

It's crucial to notice how these lines are dashed. This indicates that the solutions do not include the lines themselves. The graphical representation helps us easily see which values might work and where the inequalities intersect.

Overlapping Regions

The overlapping regions on the graph are key to solving a system of inequalities. After plotting each inequality, look for where the shaded areas intersect or overlap. This overlapping area is where the solution to both inequalities exists. If you've used your graphing calculator correctly, these regions will appear clearly. If the regions are not overlapping, there is no solution that satisfies both given inequalities simultaneously. Confirm that the graphing calculator is showing correct shading so you can reliably find this intersection.

Test Points Verification

To make sure that the overlapping region you identified truly belongs to the system of inequalities, checking some test points is a good practice. Pick points from within the region where both shadings overlap:

• Choose a point like \((0, 0)\) or another within the intersection area.
• Substitute this point back into both original inequalities \(3x + y < -2\) and \(y > 3 (1-x)\).
• If the test points satisfy both inequalities, then you have correctly identified an overlapping area.

Test points help reinforce that your graphical representation and the solution identified from it are accurate.