Problem 79 - College Algebra with Modeling and Visualization

Question

Shade the solutions set to the system. $$ \begin{aligned} \frac{x^{2}}{4}+\frac{y^{2}}{9} & \leq 1 \\ x+y & \geq 2 \end{aligned} $$

Step-by-Step Solution

Answer

Shade the intersection area of the ellipse inside and above the line.

Step 1: Identify the Equations

The system of inequalities consists of two parts:1. \( \frac{x^2}{4} + \frac{y^2}{9} \leq 1 \), which represents an ellipse centered at the origin with the horizontal semi-axis of 2 and vertical semi-axis of 3.2. \( x + y \geq 2 \), which represents a half-plane above and including the line \( x + y = 2 \).

Step 2: Graphing the Ellipse

First, plot the ellipse given by \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \). This ellipse is centered at (0,0), with vertices at (±2,0) and (0,±3). Shade the region inside the ellipse to satisfy \( \leq 1 \).

Step 3: Graphing the Inequality Line

Plot the line given by \( x + y = 2 \). This is a straight line with intercepts at (2,0) and (0,2). Shade the region above and including the line, as the inequality is \( \geq 2 \).

Step 4: Find the Intersection of Shaded Areas

The solution set is the intersection of the areas shaded in the ellipse and above the line. This means finding the overlapping region where both inequalities are true: inside the ellipse and above the line \( x + y = 2 \).

Step 5: Verify the Solution Boundary

Check boundary conditions to ensure inclusivity. Points on the ellipse edge \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) and on the line \( x + y = 2 \) are included as the inequalities use \( \leq \) and \( \geq \) respectively.

Key Concepts

Ellipse

An ellipse is a fascinating geometric shape that appears as a stretched circle. In mathematics, it can be defined by its equation in the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This particular equation describes an ellipse centered at the origin. The variables \( a \) and \( b \) designate the lengths of the semi-axes.
- For our example, the equation \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) indicates an ellipse with horizontal semi-axis \( a = 2 \) and vertical semi-axis \( b = 3 \).
- It means that the ellipse stretches 2 units in the x-direction and 3 units in the y-direction.
This respective ellipse captures all points

• within the boundary defined by the ellipse equation
• or that lie on it.

The inequality \( \frac{x^2}{4} + \frac{y^2}{9} \leq 1 \) expands this concept by including the entire area inside the ellipse.

Linear Inequality

Linear inequalities add a flexible dimension to mathematical equations by allowing expressions to be less than or greater than a particular value, instead of equal. Consider the equation \( x + y = 2 \). It forms a line on a graph.
- The associated linear inequality, \( x + y \geq 2 \), defines a half-plane.
- This places the line in the context of a boundary, where all the points that satisfy the inequality are located either on the line or in the area above it.

To determine where these points lie for our inequality, notice:

• All points on the line make the inequality true as it is inclusive, using \( \geq \).
• Points above the line have a higher sum than 2, which satisfies the inequality condition.
• The half-plane formed by this inequality extends infinitely in the positive y-direction.

By visualizing the inequality in this manner, we understand its impact on the solution of a system of inequalities.

Solution Set

The concept of a solution set arises when we deal with systems of inequalities. It's important to pinpoint the exact region in a graph that satisfies all given conditions simultaneously. For the current problem, we have two inequalities: an elliptical inequality and a linear inequality.

To find the solution set, we:

• Identify where these inequalities overlap in a graph, meaning both conditions are simultaneously satisfied.
• The elliptical inequality \( \frac{x^2}{4} + \frac{y^2}{9} \leq 1 \) outlines a region.
• The linear inequality \( x + y \geq 2 \) covers another area.
• Thus, the solution set is where these two shaded regions coincide.

Even if individual inequalities expand over wide areas,
the solution set pinpoints a precise intersection. It tells us where both equations hold true. Being the intersected graph area of these conditions, it represents complete solutions to the system.

Graphing Inequalities

Graphing is a powerful tool in visualizing and understanding inequalities. It transforms abstract math into clear images. Let's examine how this process would work for our given system.

Graphing the ellipse:

• The ellipse \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) is plotted first, marking its boundary.
• Be mindful to shade all regions inside the ellipse to indicate the \( \leq 1 \) portion, including the edge.

Graphing the linear inequality:

• Next, plot the line for \( x + y = 2 \).
• Shade all the area above the line, implying \( x + y \geq 2 \).

The beauty of graphing lies in revealing the relationship between boundaries and permissible areas.- The combination of shaded areas of the graph demonstrates where the solution set lies for our system.
Efficiently sketching these graphs allows us to visually comprehend and resolve the complex interpretations of inequalities in systems.