Problem 50 - Precalculus with Limits

Question

In Exercises 47-52, use a graphing utility to graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}{y \geq x^{4}-2 x^{2}+1} \\ {y \leq 1-x^{2}}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution set of the system is the common region that is identified by graphing both inequalities on a graphing utility.

Step 1: Graph the first inequality

First, graph the inequality \(y \geq x^{4}-2x^{2}+1\). The region satisfying this inequality will be the area where the y-values on the graph are greater than or equal to the defined function.

Step 2: Graph the second inequality

Secondly, plot the second inequality \(y \leq 1-x^{2}\) on the same graph. The region satisfying this inequality will be the area where y-values on the graph are less than or equal to the defined function.

Step 3: Identify the solution set

The solution to the system of inequalities is the area where both inequalities intersect or the common region of both plots. The points that lay in this region satisfy both inequalities.

Key Concepts

Understanding Systems of Inequalities

A system of inequalities consists of two or more inequalities that are considered together. Many problems in mathematics, such as those involving areas and optimal solutions, require us to examine these systems. Each inequality describes a region of the coordinate plane where the solution must lie.
To find the solution to the system, we need to identify the common area that satisfies all inequalities in the set. For example, when examining the system:

• \( y \geq x^{4} - 2x^{2} + 1 \)
• \( y \leq 1 - x^{2} \)

Each inequality will carve out a portion of the plane. The overlap of these portions is the sought-after solution set.

Utilizing a Graphing Utility

Graphing utilities, such as graphing calculators or software like Desmos, are invaluable tools in visualizing inequalities. These devices or platforms allow students to input equations and instantly see the graphical representation of each inequality.
Using a graphing utility helps to:

• Accurately plot complex functions and their respective inequalities.
• Quickly visualize the intersection of multiple inequalities.
• Identify potential errors in theoretical solutions or calculations.

With graphing utilities, checking and understanding the solution set becomes a much clearer and simpler process.

Exploring the Solution Set

The solution set of a system of inequalities is the region where all plotted inequalities intersect. For our example, the solution set is where the curves \( y \geq x^{4} - 2x^{2} + 1 \) and \( y \leq 1 - x^{2} \) meet.
Determining this region requires careful plotting and sometimes testing specific points:

• It's essential to compare points within the potential solution area to verify true solutions.
• The solution set can often include more than just an interior region but might also include boundary lines or curves.
• Accurately shading the 'solution area' on a graph helps visualize and confirm the correct results.

Delving into Quadratic Inequalities

Quadratic inequalities involve expressions of the form \( ax^2 + bx + c \). Graphing these inequalities typically results in regions defined by parabolas. In the provided exercise, \( y \leq 1 - x^2 \) is a classic example of a quadratic inequality.
Interestingly:

• The inequality \( y \leq 1 - x^2 \) forms a downward-opening parabola.
• The inequality \( y \geq x^{4} - 2x^{2} + 1 \) adds another layer of complexity, being a quartic function.
• Solving these involves not just graphing but also understanding how these curves interact and which sections lie above or below each other.

Solving quadratic inequalities fosters deep understanding of parabolic translations, reflections, and shifts as these properties influence the solution set.