Problem 75 - A Graphical Approach to Precalculus with Limits

Question

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \geq \frac{1}{x-1}\\\&y \leq x\end{aligned}$$

Step-by-Step Solution

Answer

The solution is the region above \( y=\frac{1}{x-1} \) and below \( y=x \) for \( x > 1 \).

Step 1: Understand the Inequalities

We have two inequalities. The first is \( y \geq \frac{1}{x-1} \), which represents the region above the hyperbola defined by \( y = \frac{1}{x-1} \). The second is \( y \leq x \), representing the region below the line \( y = x \). Our goal is to find the intersecting region of these two inequalities.

Step 2: Graph the Hyperbola

Start by graphing \( y = \frac{1}{x-1} \). Notice that as \( x \) approaches 1, the value of \( y \) becomes undefined, so there's a vertical asymptote at \( x=1 \). The hyperbola has branches in the first and third quadrants. We shade the region above this curve since \( y \geq \frac{1}{x-1} \).

Step 3: Graph the Line

Now, graph \( y = x \), which is a straight line passing through the origin with a slope of 1. It extends infinitely in both positive and negative directions. Shade the region below this line because \( y \leq x \).

Step 4: Determine the Intersection

Identify the region where both shaded areas overlap. This region represents the solution set where both inequalities are satisfied. It will be in the first quadrant where \( x > 1 \) because the hyperbola's first quadrant branch starts at \( x=1 \).

Step 5: Highlight the Solution Region

Shade the intersection area more distinctly or darker on the graph. This region continues from just above the hyperbola \( y = \frac{1}{x-1} \) to just below the line \( y = x \) for \( x > 1 \).

Key Concepts

Graphing Inequalities

Graphing inequalities involves finding the appropriate regions on a graph that fulfill a given inequality. Each inequality represents a condition, and the solution set is where these conditions overlap. In our example, you have two inequalities:

• The hyperbola inequality: \( y \geq \frac{1}{x-1} \).
• The line inequality: \( y \leq x \).

To graph these, start by representing each inequality on the Cartesian plane. Each inequality divides the plane into two halves.
You'll need to determine which half satisfies the inequality. For \( y \geq \frac{1}{x-1} \), shade the area above the hyperbola. For \( y \leq x \), shade below the line. The solution to the system of inequalities is the overlapping shaded region. This visually shows where both conditions are true.
The critical step is identifying this overlap, particularly in the first quadrant beyond the vertical asymptote of the hyperbola. This ensures you're selecting the correct solution area.

Hyperbolas

A hyperbola is a type of curve that appears in math due to equations like \( y = \frac{1}{x-1} \). This equation shows a hyperbola because it involves a term with \( x \) in the denominator. Hyperbolas have two branches typically, and each branch extends into infinity within its quadrant.
In our inequality \( y \geq \frac{1}{x-1} \), we only consider the region above this hyperbola. This means shading above the branch of the hyperbola that sits in the right half of the coordinate plane where \( x > 1 \).
A key feature of hyperbolas is their asymptotes, lines that the curve approaches but never touches. Here, there's a vertical asymptote at \( x = 1 \), because the function is undefined at this point. As \( x \) approaches 1, \( y \) goes to infinity, indicating the hyperbola's steep inclination.

Linear Equations

Linear equations like \( y = x \) produce straight lines when graphed. This line has a slope of 1, meaning for every increase in \( x \) by 1, \( y \) increases by 1 as well. The graph of \( y = x \) passes through the origin (0,0), serving as a diagonal line cutting the coordinate plane into two equal halves.
When dealing with linear inequalities such as \( y \leq x \), you can visualize this by graphing the line and shading the appropriate region. You shade the area below the line for \( y \leq x \), which captures all points where the x-values are greater than or equal to the y-values. This region displays the set of solutions for the inequality.
The intersection where this shaded area overlaps with the area above the hyperbola (from the other inequality) provides the complete solution to the system of inequalities. It's within these intersecting regions that both inequalities hold true together.