Problem 71 - A Graphical Approach to Precalculus with Limits

Question

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \leq\left(\frac{1}{2}\right)^{x}\\\&y \geq 4\end{aligned}$$

Step-by-Step Solution

Answer

Solution: The region where \( y \) is between the exponential curve and the horizontal line (including the line) is the solution.

Step 1: Understand the System of Inequalities

The system consists of two inequalities: 1. \( y \leq \left( \frac{1}{2} \right)^x \) implies that \( y \) is below or on the curve \( y = \left( \frac{1}{2} \right)^x \).2. \( y \geq 4 \) implies that \( y \) is above or on the line \( y = 4 \).

Step 2: Graph the Exponential Inequality

Start by graphing the equation \( y = \left( \frac{1}{2} \right)^x \), which is an exponential decay function. The graph has a horizontal asymptote at \( y = 0 \). For the inequality \( y \leq \left( \frac{1}{2} \right)^x \), shade below the curve.

Step 3: Graph the Horizontal Line Inequality

Graph the line \( y = 4 \), which is a horizontal line across all values of \( x \). For the inequality \( y \geq 4 \), shade the region above this line. Make sure the line is solid since \( y = 4 \) is included in the solution.

Step 4: Determine the Overlapping Region

Identify the overlapping region on the graph where both shaded areas intersect. This area represents the solution set where both inequalities are satisfied. It will be the region below \( y = \left( \frac{1}{2} \right)^x \) and above \( y = 4 \).

Key Concepts

Exponential Functions

An exponential function is a mathematical expression in which a variable appears in the exponent. It takes the general form \( y = a^x \), where \( a \) is a constant base and \( x \) is the exponent. In the given exercise, the function \( y = \left(\frac{1}{2}\right)^x \) is an example of exponential decay because the base \( \frac{1}{2} \) is a fraction between 0 and 1. Exponential functions have unique characteristics:

• A horizontal asymptote, which in our case is the line \( y = 0 \). The graph approaches but never touches this line.
• The function decreases as \( x \) increases, which is known as decay.

To graph \( y = \left(\frac{1}{2}\right)^x \), you'll notice that it passes through the point \( (0,1) \) since any number raised to the power of 0 is 1. As \( x \) increases, the value of \( y \) becomes smaller, illustrating the decay as it moves closer to the asymptote \( y = 0 \).

Systems of Inequalities

A system of inequalities consists of two or more inequalities meant to be solved simultaneously. In our exercise, we deal with:

• \( y \leq \left(\frac{1}{2}\right)^x \): Indicates that the region of allowable solutions lies on or beneath the curve of the exponential function.
• \( y \geq 4 \): This establishes a limit where \( y \) should not fall below 4, positioned along or above this horizontal line.

To solve a system of inequalities graphically, you need to:

• Graph each inequality individually.
• Shade the appropriate region for each inequality.
• Identify the overlapping shaded region, which represents the solution to the system.

For our specific system, after graphing, you'll identify the acceptable area as the space where the region shaded below \( y = \left(\frac{1}{2}\right)^x \) and the region shaded above the line \( y = 4 \) intersect.

Solution Sets

The concept of a solution set is vital in understanding systems of inequalities. A solution set contains all the possible solutions that satisfy the given inequalities.In our exercise:

• The solution set comprises the 'intersecting region' from each possible solution to the inequalities \( y \leq \left(\frac{1}{2}\right)^x \) and \( y \geq 4 \).
• This intersecting region lies beneath the exponential curve and above the horizontal line \( y = 4 \).

To visualize, after drawing both inequalities on the graph, the solution set is the shaded region that fits both conditions simultaneously. Always ensure that the boundary lines are considered correctly — here, the line at \( y = 4 \) serves as a solid boundary indicating inclusion, similarly for the curve, you consider all points below and including it, enhancing clarity of the solution set boundaries.