Problem 33 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution set for the inequalities exists and is an overlapped area from both equations.

Step 1: Plot the first inequality

Start by graphing \(x+2y \leq 4\). We can find the line \(x + 2y =4 \). This line will be the boundary line for this inequality. To graph it, simplify it to the form of \(y = mx + c\), rearrange to get \(y = -0.5x + 2\). This line intersects the y-axis at 2 and has negative slope. Hence the line goes down as \(x\) increases. We shade all the region below this line because the inequality is \(\leq\). That is, all the \(x\) and \(y\) points in this region satisfy this inequality.

Step 2: Plot the second inequality

Next, we graph \(y \geq x-3\). This line will be the boundary line for this inequality. To graph it, the form is already in \(y = mx + c \) format which is \(y = x - 3\). This line intersects the y-axis at -3 and has a gradient 1 (meaning it goes upwards as \(x\) increases). We shade all the region above the line because the inequality is \( \geq \). i.e., all the \(x\) and \(y\) points in this region satisfies this inequality.

Step 3: Find the Intersecting Region

Observe the graph. The solution to the system of inequalities will be the region where the shadings from the two inequalities overlap.

Key Concepts

Graphing Inequalities

To tackle the problem of graphing inequalities, the first step is to understand what an inequality tells us. Unlike simple linear equations, an inequality describes a range of values. Each inequality corresponds to a specific region on a graph.
When graphing inequalities, you start by turning the inequality into an equation. This is done to find the boundary line. For example, if you have the inequality \( x + 2y \leq 4 \), you start by graphing the line \( x + 2y = 4 \). This line is critical because it divides the plane into two halves. To plot this, convert it to slope-intercept form as \( y = -0.5x + 2 \). This tells you where the line intersects the y-axis and its slope.
Once the line is drawn, shading is the next step. The type of inequality (\( \leq \), \( \geq \)) indicates whether you shade above or below this line. For '\( \leq \)', shade below the line, encompassing all possible solutions. This methodology is universal for graphing linear inequalities.

Solution Set

The solution set of a system of inequalities consists of all the possible values of the variables that satisfy each inequality. In the context of this exercise, after graphing each inequality, finding the solution set involves identifying where these individual sets overlap.
First, graph both inequalities: \( x + 2y \leq 4 \) and \( y \geq x - 3 \). The solution set is where the shaded areas from each inequality graph intersect. This overlap is crucial because points in this area meet the conditions of both inequalities at once.
Check various points within the overlapping region by plugging them into the original inequalities to verify they work for both. These points represent solutions to the system. Essentially, the solution set is the graphical representation of all potential solutions that satisfies the given constraints.

Overlapping Regions

In systems of inequalities, the concept of overlapping regions is vital as it defines the feasible solution space. Each inequality divides the graph into two distinct zones. One zone contains solutions that satisfy the inequality, while the other doesn't.
Once both inequalities in a system are graphed, the overall solution lies in the area where these suitable regions intersect. This overlapping region is shaded doubly, indicating that these are the satisfying values for every inequality involved in the system.
Identifying overlapping regions helps visualize and solve real-world problems by showing how different constraints or conditions can coexist. It guides decisions by marking only those solutions permissible by all conditions, which is critical in optimization and constraint satisfaction fields.