Problem 48 - College Algebra

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&3 x+y \leq 6\\\&x \geq-2\\\&y \leq 4\end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the set of points that fall within the overlapping region of the graphs of the individual inequalities. If there's no overlapping area, the system has no solution. Mathematical formulas are not necessary for the short answer in this case.

Step 1: Analyze and Graph each Inequality Individually

The system is composed of three inequalities: \(3x + y \leq 6\), \(x \geq -2\), and \(y \leq 4\). To create a graph for each inequality: \n\n1. For \(3x + y \leq 6\), rearrange it for y: \(y \leq -3x + 6\). This is a line with a slope of -3 and y-intercept of 6. Choose points below the line due to the ‘less than or equal to' sign.\n\n2. For \(x \geq -2\), this means x has to be greater than or equal to -2. Draw a vertical line at \(x = -2\) on the graph, and pick points to the right (or on) this line.\n\n3. For \(y \leq 4\), this suggests y-values need to be less than or equal to 4. Draw a horizontal line at \(y = 4\) on the graph, and pick points below (or on) this line.

Step 2: Combine the Graphs

Now, it's time to merge all three inequalities on a single graph. This implies the solution set will consist of the region that satisfies all three inequalities simultaneously. In other words, the solution to the given system of inequalities is the intersection of the regions identified in Step 1. The area of overlap is the region which satisfies all three inequalities.

Step 3: Identify the Solution Set

The final step is to observe the area that satisfies all the inequalities. This overlapping region represents the set of all possible (x, y) points that satisfy the system of inequalities. It is the solution set. If there is no overlapping region, it indicates that the system has no solution.

Key Concepts

Graphing Inequalities

When tackling systems of inequalities, a good start is to graph each inequality separately on the coordinate plane. By doing so, it is easier to visualize where these inequalities hold true.

• For each linear inequality, convert it to the form of a straight line first. For instance, for the inequality \(3x + y \leq 6\), rearrange to \(y \leq -3x + 6\) for easy graphing.
• The line \(y = -3x + 6\) would be solid, showing that points on the line are included (thanks to the 'less than or equal to').
• Shade the region below the line because of the 'less than' aspect.
• Repeat this process for the other inequalities: For \(x \geq -2\), shade to the right of the vertical dashed line at \(x = -2\); for \(y \leq 4\), shade below the dashed line at \(y = 4\).

By following these steps, each inequality is transformed into a graph segment on the overall graph. This creates a visual breakdown of divisible regions in the coordinate plane.

Intersection of Regions

Once each inequality is graphically represented in its individual form, the next major step involves identifying where these shaded regions cross or overlap each other. This is called the intersection of regions, and it is critical for finding solutions within systems of inequalities.

• The intersection is the shaded area common to all the individual graphs you've drawn. This is the piece of the graph that meets the criteria set by every inequality simultaneously.
• For our example: Any point within the intersection satisfies \(3x + y \leq 6\), is to the right of \(x = -2\), and below \(y = 4\).
• The intersection is typically a closed or bounded region unless the lines are parallel and never intersect.

Always double-check the lighter shaded regions to ensure you’ve included only the points that hold true for every inequality involved.

Solution Set of Inequalities

The final piece of solving a system of inequalities is identifying and recording the solution set. This entails pinpointing the exact area or regions that serve as solutions to the inequalities combined.

• The solution set is essentially the collection of points that lie within the overlapping shaded region, as discussed in the context of intersection of regions.
• An example solution set might include specific points or a more generalized description of the shaded region, especially useful for inequalities involving large and continuous areas.
• If the shaded areas do not overlap at all, it implies that there is no solution to the system, signifying impossibility or non-existence of common (x, y) pairs.

Mapping out the solution set verifies if a legitimate solution exists and clarifies where these solutions can be found within the coordinate plane.