Problem 36 - College Algebra Essentials

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l}x \leq 3 \\\y \leq-1\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of inequalities is any point in the region that is to the left of \(x = 3\), below \(y = -1\), and including the lines \(x = 3\) and \(y = -1\).

Step 1: Graph the first inequality

Plot the inequality \(x \leq 3\) on a graph. This inequality tells that x can be any value that is less than or equal to 3. For this, draw a solid vertical line at \(x = 3\). The solid line is used because \(x\) is less than or equal to 3, not just less than 3, and so, it is included in the solution. The area to the left of this line, including the line itself, will be your feasible region for this inequality.

Step 2: Graph the second inequality

Graph the inequality \(y \leq -1\) on the same graph. This inequality states that \(y\) can be any value less than or equal to -1. To graph this inequality, draw a solid horizontal line on \(y = -1\). The line is solid because the values on the line are a part of the solution. The area below this line, including the line itself, is your feasible region for this inequality.

Step 3: Find the overlapping feasible region

Note where the two feasible regions intersect. This region satisfies both inequalities, which means it is the solution to the system of inequalities. In this case, it will be the region below and to the left of the intersection of the lines. The intersection is the point (3, -1). Therefore, the solution to the system is any point in the region that is to the left of \(x = 3\), below \(y = -1\), and including the lines \(x = 3\) and \(y = -1\).

Key Concepts

Graphing Inequalities

When graphing inequalities like \(x \leq 3\) and \(y \leq -1\), it's essential to understand the types of lines used and the regions they describe. For inequalities that include the equal sign (\(\leq\) or \(\geq\)), we use a solid line. This means all the points on that line are part of the solution as well. To graph \(x \leq 3\), draw a vertical line at \(x = 3\). Everything to the left of this line, including the line itself, makes up the solution for this inequality.

For \(y \leq -1\), draw a horizontal line at \(y = -1\). Similar to the first inequality, include the line in the solution, highlighting everything below it. Because of these lines, the graph starts to show areas that satisfy the inequalities. This helps us find the solution set when these regions overlap.

Solution Set

The solution set of a system of inequalities consists of all the points that satisfy each inequality simultaneously. Once we've graphed each inequality, identify where their regions overlap. This intersection is the solution set, meaning any point in this area satisfies all inequalities in the system.

By graphing \(x \leq 3\) and \(y \leq -1\), and identifying the shared area, we determine the solution set. The solution set is the region that is simultaneously to the left of the line \(x = 3\) and below the line \(y = -1\). Every point in this shared region is a part of the solution.

Coordinate Plane

The coordinate plane is a two-dimensional space formed by two perpendicular number lines, the x-axis (horizontal) and y-axis (vertical). It's essential for graphing inequalities, as it provides a visual representation of the solution set. Each point on the plane is described by an ordered pair \((x, y)\).

For our inequalities \(x \leq 3\) and \(y \leq -1\), using the coordinate plane allows us to see how these inequalities interact. The plane helps in understanding which points satisfy both inequalities by outlining the overlapping region. Graphing on the coordinate plane makes it easier to identify and verify the solution set effectively.

Feasible Region

The feasible region is the area on the graph that represents all possible solutions to a system of inequalities. It's found by looking at the overlap of the shaded regions that each inequality describes. In our case, the feasible region tells us where the conditions \(x \leq 3\) and \(y \leq -1\) are both satisfied.

As you graph each inequality individually, watch for their intersection. The feasible region for this system of inequalities lies to the left of the line \(x = 3\) and below the line \(y = -1\). These combined conditions create a rectangle that extends infinitely in the negative x and y directions, including the boundary lines themselves. The feasible region helps visualize the solution set, making abstract algebraic concepts clearer and more tangible.