Problem 64 - A Graphical Approach to Precalculus with Limits

Question

Graph the solution set of each system of inequalities by hand. $$\begin{array}{l}-2

Step-by-Step Solution

Answer

The solution is the rectangular region \(-2 < x < 3\) and \(-1 \leq y \leq 5\), below the dashed line \(2x + y = 6\).

Step 1: Graph the Inequality for x

First, let's graph the inequality \(-2 < x < 3\). This represents a vertical strip between the vertical lines \(x = -2\) and \(x = 3\), where neither line is included in the solution set, so they should be drawn as dashed lines.

Step 2: Graph the Inequality for y

Next, graph the inequality \(-1 \leq y \leq 5\). This represents a horizontal strip between \(y = -1\) and \(y = 5\), where both lines are included in the solution set, so they should be drawn as solid lines.

Step 3: Determine the Intersection of x and y Regions

The solution set for the first two inequalities \(-2 < x < 3\) and \(-1 \leq y \leq 5\) is the rectangular region bounded by these lines in the xy-plane.

Step 4: Graph the Inequality for 2x + y < 6

Now, graph the inequality \(2x + y < 6\). First, rewrite it as an equation: \(2x + y = 6\), which is a line. To find points for this line, set \(x = 0\) and solve for \(y\), then set \(y = 0\) and solve for \(x\). This gives points \((0, 6)\) and \((3, 0)\). Draw this line with a dashed pattern because the inequality is strict and does not include the line itself. Shade below the line since it's \(2x + y < 6\).

Step 5: Identify the Feasible Region

Combine the solutions from previous steps. The feasible region where all conditions overlap is the area within the rectangle from the intersection of the first two inequalities that also lies below the line and not on it from the third inequality.

Key Concepts

Solution Sets

In mathematics, a solution set encompasses all possible solutions that satisfy a given equation or inequality. When dealing with inequalities, the solution set refers to a range or region of values that satisfy these inequalities. In the provided exercise, the inequalities include both variable constraints and conditions represented visually on a graph.

For example, the inequality \(-2 < x < 3\) outlines a range for the variable \(x\). This range forms a vertical strip on a graph, meaning \(x\) can take any value between -2 and 3, but not exactly -2 or 3, as they are part of a dashed line instead of solid.

• Similarly, ##-1 \leq y \leq 5 refers to the range for \(y\). Here, \(y\) includes the endpoints because the inequality is non-strict, thus forming a solid horizontal line along -1 and 5.
• The last inequality, \(2x + y < 6\), introduces another layer by using both \(x\) and \(y\) to define a linear relationship. These solution sets, when plotted, help visualize where all conditions in the system are satisfied simultaneously.

System of Inequalities

A system of inequalities consists of multiple inequalities that are solved together to find common solutions. This exercise gives three different inequalities that form the system, each defining a separate condition or limit involving the variables \(x\) and \(y\).

Graphing a system of inequalities involves plotting each inequality on a coordinate plane and determining where their solution sets overlap.

• The inequality \(-2 < x < 3\) and ##-1 \leq y \leq 5 form a bounded rectangular area on the graph.
• The third inequality, \(2x + y < 6\), adds a linear boundary. By graphing the line \(2x + y = 6\) and shading the area below it helps us define the part of the rectangle that satisfies all inequalities together.

Understanding a system of inequalities is key to identifying scenarios or conditions where multiple constraints are at play and need to be met simultaneously. This is crucial in fields like optimization and decision-making.

Feasible Region

The feasible region is a concept used to identify the area on a graph where all solutions to a system of inequalities can be found. It represents the overlap of the solution sets of all inequalities in the system.

In the exercise mentioned, the feasible region is determined by the intersection of the constraints given:

• It's the area bounded by the vertical lines defined by \(-2 < x < 3\) and horizontal lines where \(-1 \leq y \leq 5\).
• Moreover, it needs to be below the dashed line \(2x + y = 6\).

Together, these conditions define a specific region on the graph. This region reflects all possible combinations of \(x\) and \(y\) that satisfy the set of inequalities presented in the system.

When graphically represented, the feasible region is often shaded to depict where the solutions lie. It provides a visual tool for decisions involving practical problems where similar constraints are often present.