Problem 42 - Calculus Single Variable

Question

Sketch the given region. \(\\{(x, y): x+5 y \geq 4, x \leq 2, y \geq-8\\}\)

Step-by-Step Solution

Answer

The solution region is above and to the left of the line \(x + 5y = 4\), left of \(x = 2\), and above \(y = -8\).

Step 1: Understanding the Inequalities

First, we need to understand the three inequalities: 1. \(x + 5y \geq 4\): This describes a half-plane above or on the line \(x + 5y = 4\). 2. \(x \leq 2\): This describes the region to the left of the vertical line \(x = 2\).3. \(y \geq -8\): This describes the region above or on the horizontal line \(y = -8\).

Step 2: Graph the Line for \(x + 5y = 4\)

To graph \(x + 5y = 4\), find two points on the line by substituting values for \(x\) and \(y\). Let \(x = 0\), then: \[0 + 5y = 4 \Rightarrow y = \frac{4}{5}\]Let \(y = 0\), then: \[x + 0 = 4 \Rightarrow x = 4\]Plot these points (0, \(\frac{4}{5}\)) and (4, 0) and draw the line through them.

Step 3: Shade the Region for \(x + 5y \geq 4\)

Since the region \(x + 5y \geq 4\) is above the line, shade the half-plane that includes points like (0, 1) which satisfy the inequality. Test by substituting a sample point; (0,1) gives \(0 + 5\times1 = 5\), which is greater than 4.

Step 4: Graph the Vertical Line \(x = 2\)

Draw a vertical line where \(x = 2\) on the graph, which is a straight line parallel to the y-axis. Since the inequality is \(x \leq 2\), shade the region left of this line.

Step 5: Graph the Horizontal Line \(y = -8\)

Draw a horizontal line where \(y = -8\) on the graph, which is a straight line parallel to the x-axis. Since the inequality is \(y \geq -8\), shade the region above this line.

Step 6: Identify the Intersection of All Shaded Regions

The final step is to look for the area that satisfies all three conditions: above the line \(x + 5y = 4\), to the left of \(x = 2\), and above \(y = -8\). The intersection of these shaded regions is the solution region.

Key Concepts

Inequality Regions

In the realm of graphing, inequality regions help us define areas on a plane that satisfy a set of given conditions. Consider an inequality such as \(x + 5y \geq 4\). This tells us about the space on a graph above the line \(x + 5y = 4\). By rearranging the terms and treating the inequality as an equation, we can initially establish the boundaries that need to be considered.
Once we know the line, the inequality symbol (\(\geq\) in this case) guides us on which part of the plane to shade.

• If the inequality is \(\geq\), the region includes the line and points above it.
• If it is \(\leq\), it includes the line and points below it.

For each linear inequality, it’s crucial to determine these boundaries accurately by finding specific points and then testing which side of the line satisfies the inequality.

Coordinate Plane

The coordinate plane is a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis). This grid system allows us to locate points using ordered pairs \((x, y)\). Every graphing activity, including plotting inequalities, begins with carefully drawing the axes.
Each inequality constraint gives a distinct line or boundary on this plane. These are found by converting inequalities into equations. For example, for \(x \leq 2\), we draw a vertical line at \(x = 2\), and for \(y \geq -8\), we draw a horizontal line at \(y = -8\).

• The plane is split into regions by these boundaries.
• To make graphing easier, select a standard scale on the axes for accurate plotting.

Therefore, understanding how to navigate and read a coordinate plane is essential for identifying and shading inequality regions.

Intersection of Regions

The intersection of regions refers to the common area that satisfies multiple inequality constraints at once. When graphing inequalities, we shade various parts of the plane according to each inequality's direction. The intersection is where all shaded areas overlap, and it represents the solution to the system of inequalities.
To find the intersection region:

• Graph each inequality by drawing its boundary line.
• Shade the correct side for each inequality.
• Observe the common area where all shaded portions meet.

This overlapping region represents the possible solutions that meet every requirement described by the inequalities. Using a test point method, you can verify that specific points inside this region satisfy all given inequalities. Paying attention to this intersection is crucial for solving systems of inequalities visually on a coordinate plane.