Problem 77 - A Graphical Approach to Precalculus with Limits

Question

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y>\frac{1}{(x-1)^{2}}\\\&y>2\end{aligned}$$

Step-by-Step Solution

Answer

Graph the region above both \( y = \frac{1}{(x-1)^2} \) and \( y = 2 \).

Step 1: Understand the Inequalities

We need to graph the region where both inequalities \( y > \frac{1}{(x-1)^2} \) and \( y > 2 \) are satisfied. This means finding the intersection of the areas above each curve in the coordinate plane.

Step 2: Graph the First Inequality

Graph the curve of \( y = \frac{1}{(x-1)^2} \). This is a vertical hyperbola opening upwards, shifted one unit to the right along the x-axis. For the inequality \( y > \frac{1}{(x-1)^2} \), shade the region above this curve.

Step 3: Graph the Second Inequality

Graph the line \( y = 2 \). For the inequality \( y > 2 \), shade the region above this horizontal line. Since \( y > 2 \) does not include the boundary, it is a dashed line in a graph.

Step 4: Identify the Overlapping Region

Look for the intersection of the shaded regions from the above steps. The solution set lies in the region where both shaded areas overlap, i.e., the area above the curve \( y = \frac{1}{(x-1)^2} \) and the line \( y = 2 \).

Step 5: Confirm and Mark the Solution Set

In addition to shading appropriately, explicitly mark the region that represents the solution set. Note that for values of \( x \) very close to 1, \( y = \frac{1}{(x-1)^2} \) rises sharply, so the intersection above \( y = 2 \) will primarily be at values of \( x \) greater than approximately 1.

Key Concepts

Coordinate Plane

The coordinate plane is a fundamental concept in graphing, comprising two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). It serves as a foundation for visually representing mathematical functions and inequalities. The point where these axes intersect is called the origin, labeled as (0,0).
Each point on the coordinate plane can be represented as a pair of numbers (x, y), where 'x' is the horizontal position and 'y' the vertical position. By plotting these coordinates, you can graph equations and inequalities to visibly showcase their solutions.
For graphing inequalities, understanding the coordinate plane is crucial. The values of x and y that satisfy inequalities like those in our problem become regions on this plane. It helps us easily identify where solutions might overlap, key in dealing with systems of inequalities.

System of Inequalities

A system of inequalities consists of two or more inequalities that need to be solved simultaneously. Each inequality delineates a region of the coordinate plane, and the solution to the system is where these regions intersect.
In our exercise, we have two inequalities:

• \( y > \frac{1}{(x-1)^2} \)
• \( y > 2 \)

Each inequality designates a unique region. The first describes all points above the curve \( y = \frac{1}{(x-1)^2} \), an upward-opening hyperbola shifted to the right by one unit. The second represents the area above the horizontal line \( y = 2 \).
The challenge is to find the common area that satisfies both conditions. Understanding how each inequality defines a part of the plane is fundamental to identifying this overlap.

Intersection of Regions

The intersection of regions refers to the common area shared by two or more graphs on the coordinate plane. When dealing with systems of inequalities, identifying this intersection is the key to finding the solution.
In our problem, we look for the overlapping region from the two inequalities:

• The area above the curve \( y = \frac{1}{(x-1)^2} \).
• The area above the line \( y = 2 \).

To graphically show this, both regions need to be shaded on the same coordinate plane, highlighting where these shadings coincide.
The combined shaded area that satisfies both inequalities forms the intersection, meaning any point in this region is a solution to the system. Notice, especially near values of \( x = 1 \), how sharply \( y = \frac{1}{(x-1)^2} \) ascends, ensuring the intersection begins only at x-values slightly larger than 1. Understanding this conceptually assists in accurately identifying where solutions truly exist.