Problem 22 - College Algebra with Modeling and Visualization

Question

Graph the solution set to the system of inequalities. Use the graph to identify one solution. $$ \begin{array}{l} x^{2}+y \leq 4 \\ x^{2}-y \leq 3 \end{array} $$

Step-by-Step Solution

Answer

The point (0, 0) is one solution to the system of inequalities.

Step 1: Understand Each Inequality

We have two inequalities: \( x^2 + y \leq 4 \) and \( x^2 - y \leq 3 \). These are inequalities that need to be considered individually first before finding their intersection.

Step 2: Rearrange the First Inequality

The first inequality is \( x^2 + y \leq 4 \). Rearranging gives \( y \leq 4 - x^2 \). This is the region below the parabola \( y = 4 - x^2 \).

Step 3: Rearrange the Second Inequality

The second inequality is \( x^2 - y \leq 3 \). Rearranging gives \( y \geq x^2 - 3 \). This is the region above the parabola \( y = x^2 - 3 \).

Step 4: Sketch the Parabolas

Sketch the graphs of \( y = 4 - x^2 \) which opens downwards, and \( y = x^2 - 3 \), which opens upwards. Use dashed lines since we are dealing with inequalities, or solid lines if considering \( \leq \).

Step 5: Identify the Solution Region

The solution region is the overlap or intersection of the regions shaded for both inequalities. That is, where \( y \leq 4 - x^2 \) and \( y \geq x^2 - 3 \).

Step 6: Identify a Solution Point

Pick any point within the solution region. For instance, the point \( (0, 0) \) satisfies both conditions: \( 0 \leq 4 \) and \( 0 \geq -3 \), hence it is a solution.

Key Concepts

Parabolas

A parabola is an elegant, symmetrical curve that appears in various mathematical contexts. It's the graph of a quadratic equation of the form \( y = ax^2 + bx + c \). By understanding parabolas, we can determine the shape and direction of quadratic expressions in equations and inequalities.

In the context of inequalities, we deal with parabolas like those given in the exercise. The two equations involved \( y = 4 - x^2 \) and \( y = x^2 - 3 \) represent parabolas:

• \( y = 4 - x^2 \) opens downwards because the coefficient of \( x^2 \) is negative. This parabola is upside down.
• \( y = x^2 - 3 \) opens upwards as the coefficient of \( x^2 \) is positive. This curve points upwards.

Working with parabolas in inequalities implies calculating the regions where the inequality holds true relative to the parabola. This is crucial for visualizing the solution set.

Solution Set

A solution set in the realm of inequalities is a collection of all points that satisfy the given inequality. For a system of inequalities, it is the common region where all individual inequalities are true simultaneously.

In the given exercise, once each inequality \( y \leq 4 - x^2 \) and \( y \geq x^2 - 3 \) is graphed, the solution set appears as the overlapping shaded areas on the graph. This overlapping region marks all possible pairs of \( (x, y) \) that satisfy both inequalities at the same time.

To confirm a solution point, select a point within this overlapping region and validate that it satisfies both conditions. For instance, the point \( (0, 0) \) fits the criteria because:

• Substituting in the first inequality: \( 0 \leq 4 - 0^2 \), which is true.
• Substituting in the second inequality: \( 0 \geq 0^2 - 3 \), which is also true.

Thus, this point is part of the solution set.

Graphing Inequalities

Graphing inequalities involves translating mathematical expressions into visual plots to better understand the solution sets. It starts with converting each inequality into an equation to graph the respective boundary line. Once the parabolas are plotted based on their equations, shading helps illustrate the regions where the inequalities are true.

For the given system: each parabola forms a boundary. The inequalities \( y \leq 4 - x^2 \) and \( y \geq x^2 - 3 \) determine which side of each parabola represents the true region to shade.

• Graph \( y = 4 - x^2 \): This is a downward-opening parabola. Shade below it to denote \( y \leq 4 - x^2 \).
• Graph \( y = x^2 - 3 \): This represents an upward-opening parabola. Shade above it to signify \( y \geq x^2 - 3 \).

Where these shaded regions overlap is the solution to the system. This intersection helps to easily visualize which points make both inequalities true, thus solving the system graphically.