Problem 34 - College Algebra with Modeling and Visualization

Question

Graph the solution set to the system of inequalities. $$ \begin{aligned} &2 x+3 y \leq 6\\\ &\frac{1}{2} x^{2}-y \leq 2 \end{aligned} $$

Step-by-Step Solution

Answer

The graph's solution set is where the shaded regions of both constraints overlap.

Step 1: Identify Inequalities

First, we will identify the system of inequalities that we need to graph. These are: 1. \(2x + 3y \leq 6\)2. \(\frac{1}{2}x^2 - y \leq 2\).

Step 2: Convert Inequalities to Equations for Graphing

Convert each inequality to an equation to find the boundary lines or curves:1. \(2x + 3y = 6\)2. \(\frac{1}{2}x^2 - y = 2\).

Step 3: Graph the First Inequality

Plot the line \(2x + 3y = 6\). To do this, find the intercepts:- When \(x=0\), \(3y = 6\) gives \(y = 2\).- When \(y=0\), \(2x = 6\) gives \(x = 3\). Connect these points to draw the line, and shade the region below this line as it represents \(2x + 3y \leq 6\).

Step 4: Graph the Second Inequality

Plot the curve \(\frac{1}{2}x^2 - y = 2\) by rewriting it as \(y = \frac{1}{2}x^2 - 2\). This is a parabola with vertex at \((0, -2)\) opening upwards. Shade the area below the parabola as it represents \(\frac{1}{2}x^2 - y \leq 2\).

Step 5: Find the Intersection of Shaded Regions

Determine the region where the shading of both inequalities overlaps. This common shaded area is the solution set for the system of inequalities.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting a line or curve and shading a specific region of the graph. The shaded area represents all the possible solutions for that inequality. When graphing the inequality \( 2x + 3y \leq 6 \), we first convert it to an equation \( 2x + 3y = 6 \) to find the line plot points. Next, we plot the intercepts to draw the line:

• For the y-intercept, let \( x = 0 \). Then, \( 3y = 6 \) gives \( y = 2 \).
• For the x-intercept, let \( y = 0 \). Then, \( 2x = 6 \) gives \( x = 3 \).

We draw a solid line through these points because the inequality includes \( \leq \). Then, we shade the area below the line to indicate that the inequality is less than or equal to that line's equation. This shaded region illustrates all points \( (x, y) \) that satisfy the inequality.Similarly, to graph \( \frac{1}{2}x^2 - y \leq 2 \), rewrite it as \( y = \frac{1}{2}x^2 - 2 \). This forms a parabola. The process to determine shading is the same, and overlapping shaded areas of both inequalities offer the combined solution.

Linear Equations

Linear equations form the backbone of many algebraic problems. A linear equation in two variables has the form \( ax + by = c \) and always produces a straight line when graphed. In this exercise, the equation \( 2x + 3y = 6 \) represents a line in a two-dimensional coordinate system.To graph a linear equation:

• Identify the x- and y-intercepts by setting each variable to zero in turn.
• Use these intercepts to plot two points on the graph.
• Connect the points to create the line.

For our specific equation, substituting values leads to the points \( (0, 2) \) and \( (3, 0) \), which are our intercepts. This straightforward nature means linear equations predictably map into lines with consistent slope and direction.The role of the intercepts aids in seeing where the line crosses the axes, providing clear insights into the relationship between \( x \) and \( y \).

Parabolic Equations

Parabolic equations graph as curved lines, known as parabolas. Typically, they take the form \( y = ax^2 + bx + c \). The equation \( y = \frac{1}{2}x^2 - 2 \) is a parabola opening upwards due to the positive coefficient \( \frac{1}{2} \) in front of \( x^2 \).Key characteristics to graph a parabola:

• Vertex: The turning point of the parabola. For our equation, the vertex is at \( (0, -2) \).
• Direction: Determined by the coefficient of \( x^2 \). Positive values open upwards, negatives downwards.
• Axis of Symmetry: A vertical line through the vertex, \( x = 0 \) here.

To plot, begin at the vertex and add points by choosing symmetric values of \( x \). Calculate their corresponding \( y \) values. This step helps shape the parabola on the graph.Parabolic curves are crucial in various applications, from physics to finance. Understanding how they behave helps solve complex problems, as we see in this system of inequalities.