Problem 18 - Precalculus: Functions and Graphs

Question

Sketch the graph of the system of Inequalities. $$\left\\{\begin{array}{r}2 x+3 y \geq 6 \\\0 \leq x \leq 5 \\\0 \leq y \leq 4\end{array}\right.$$

Step-by-Step Solution

Answer

The feasible region is a polygon bounded by the line \(2x + 3y = 6\) and the rectangular region defined by \(0 \leq x \leq 5\) and \(0 \leq y \leq 4\).

Step 1: Graph the Boundary Line for the Inequality

Start by graphing the boundary line of the inequality \(2x + 3y = 6\). This line can be found by converting the inequality to equality. To graph this line, find the x-intercept and y-intercept.

Step 2: Find the Intercepts

To find the x-intercept, set \(y = 0\) and solve for \(x\):\[2x + 3(0) = 6 \Rightarrow x = 3\].To find the y-intercept, set \(x = 0\) and solve for \(y\):\[2(0) + 3y = 6 \Rightarrow y = 2\].Plot these points (3,0) and (0,2) and draw a straight line through them.

Step 3: Shade the Half-Plane for the Inequality

Since the inequality is \(2x + 3y \geq 6\), shade the region above the line. This includes the line itself because of the \(\geq\) sign.

Step 4: Graph the Constraints for x and y

The inequalities \(0 \leq x \leq 5\) and \(0 \leq y \leq 4\) define a rectangle. Draw vertical lines at \(x = 0\) and \(x = 5\), and horizontal lines at \(y = 0\) and \(y = 4\).

Step 5: Determine the Feasible Region

The solution to the system of inequalities is the region that satisfies all inequalities simultaneously. This is the intersection of the shaded region from Step 3 and the rectangle defined by \(0 \leq x \leq 5\) and \(0 \leq y \leq 4\). Shade this overlapping area to represent the feasible region.

Key Concepts

Boundary Lines

In graphing inequalities, the concept of boundary lines is critical for understanding which areas of the graph are part of the solution. A boundary line comes from the equation that represents the equality from your inequality.
For example, in the inequality \(2x + 3y \geq 6\), the boundary line is formed by making it an equality: \(2x + 3y = 6\). This line divides the coordinate plane into two parts - one that satisfies the inequality and the other that does not.
Boundary lines can be either solid or dashed. If the inequality allows the values on the line (\(\geq\) or \(\leq\)), draw a solid line. If the inequality strictly disallows values on the line (\(>\) or \(<\)), draw a dashed line. Here, since \(2x + 3y \geq 6\) includes the equality, the boundary line is solid.

X-Intercept

Finding the x-intercept is an essential step when graphing a boundary line. The x-intercept is the point where the graph of the line crosses the x-axis.

• To find this point, set \(y = 0\) in your line equation and solve for \(x\).
• In our example, by substituting \(y = 0\) in \(2x + 3y = 6\), we solve \(2x + 3(0) = 6\) to find \(x = 3\).

So, the x-intercept is at the point \((3,0)\). This process helps in graphing the boundary line accurately, as knowing one intercept automatically defines one point through which the line will pass.

Y-Intercept

The y-intercept is another critical intersection point of the graph. It is where the line crosses the y-axis.

• To identify this point, set \(x = 0\) in your line equation and solve for \(y\).
• Taking our example of bounds, with \(x = 0\), the equation \(2(0) + 3y = 6\) simplifies to \(3y = 6\), leading to \(y = 2\).

Thus, the y-intercept is \((0,2)\). Along with the x-intercept, this point ensures that you can accurately draw the boundary line, which is foundational in representing the regions of a system of inequalities.

Feasible Region

The feasible region is where the solution to a system of inequalities lies. It represents the area that satisfies all inequalities simultaneously. To locate the feasible region, follow these steps:

• First, identify the half-plane that satisfies each inequality. For \(2x + 3y \geq 6\), this is the region above or including the line since the inequality sign is \(\geq\).
• Next, consider all constraints given for \(x\) and \(y\), such as \(0 \leq x \leq 5\) and \(0 \leq y \leq 4\). These create a rectangle bounded by these values on the axes.
• The feasible region is found in the overlap of these areas: the shaded half-plane meeting the rectangle.

Shading this overlap identifies the set of solutions that satisfy all conditions. Recognizing feasible regions is key to tackling real-world optimization problems using linear programming.