Problem 21 - Precalculus with Limits

Question

In Exercises 19-28, use a graphing utility to graph the inequality. $$y<4^{-x-5}$$

Step-by-Step Solution

Answer

The solution to the inequality \(y < 4^{-x-5}\) is graphed as a curve with a shaded area below the curve. Any point in this shaded area represents a solution to the inequality.

Step 1: Conversion of the inequality

Firstly, the inequality needs to be written in a format suitable for graphing. Currently, the inequality is \(y < 4^{-x-5}\). This inequality can be graphed as an equation, by replacing the inequality sign with an equal sign, i.e., \(y = 4^{-x-5}\).

Step 2: Graphing

Next, this equation can be graphed using a graphing utility, which will show a curve for this function. It's important to remember that the inequality is less than, and so the solution to the inequality will be all the values of y that fall below the curve of this graph.

Step 3: Interpreting the graph

The final step is to interpret the graph. As the solution to the original inequality is all the values of y that fall below the curve of the graph, this will appear as a shaded area below the curve that represents the equation \(y = 4^{-x-5}\). So, any point in that shaded area will make the original inequality true.

Key Concepts

Using Graphing Utilities

When it comes to understanding and solving inequalities, graphing utilities serve as an essential tool, enabling students to visualize complex algebraic concepts. These digital tools allow you to plot equations and inequalities effortlessly onto a coordinate plane by simply inputting the function you wish to examine.

For example, if you have an inequality like the one in our exercise, you can convert it to an equation and use the graphing utility to plot it. This gives you a visual representation of the equation as a graph, helping to understand where the inequality applies. Most utilities will enable you to add shading to the graph, which is particularly useful in highlighting the region where the inequality holds true. By utilizing these tools, students can focus on analyzing the graph to find solutions, without getting bogged down in the tedium of manual plotting.

Converting Inequalities for Graphing

A key step before using a graphing utility is to properly prepare inequalities for graphing. The conversion process involves rewriting the inequality in a form that can be represented on a graph.

Take an inequality such as the one provided, where we have an expression of the form \(y < 4^{-x-5}\). To graph this, we first replace the inequality symbol with an equals sign, yielding the function \(y = 4^{-x-5}\). What this does is it allows us to graph the boundary — the 'fence' if you will — which divides the plane into two regions: one that satisfies the inequality, and one that does not. After graphing the boundary equation, we determine which side of the boundary represents the solution set by choosing a test point. If the original inequality holds for this point, then the entire area represented by this side is the solution region and we shade it accordingly on our graph.

Interpreting Inequality Graphs

Once the equation corresponding to the inequality is graphed and the appropriate area is shaded, it's time to interpret the inequality graph. This final step is crucial, as it helps you visualize the set of all possible solutions to the inequality. The graphed equation represents the boundary that we mentioned earlier, and the shading illustrates where the inequality is true.

In our example, \(y < 4^{-x-5}\), the area below the curve on the graph is shaded, indicating that every point within this region is a solution to the inequality. This shaded region is where the y-values are less than those on the curve. For students, interpreting these graphs means understanding not just the outcome of the inequality, but also the relationship between the variables involved. By recognizing the shaded regions, students can grasp where the solutions lie and apply this understanding to various contexts and other inequalities they may encounter.