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Chapter 3: Problem 83

Graph the functions given by \(y=3^{x}\) and \(y=4^{x}\) and use the graphs to solve each inequality. (a) \(4^{x}<3^{x}\) (b) \(4^{x}>3^{x}\)

Short Answer

Expert verified
For the inequality (a) \(4^{x}<3^{x}\), the solution set is all \(x\) values less than 0. For the inequality (b) \(4^{x}>3^{x}\), the solution set is all \(x\) values greater than 0.

Step by step solution

01

Graphing the functions

The exponential functions \(y=3^{x}\) and \(y=4^{x}\) need to be graphed for a set of \(x\) values. The most simple way to graph these functions is to pick a range of \(x\) values and calculate the corresponding \(y\) values. E.g., for \(y=3^{x}\), if \(x=1\), \(y=3\), if \(x=2\), \(y=9\), etc. Plot these points for both functions and draw the curves that pass through them.
02

Compare the two function graphs

Compare where the graph of \(y=4^{x}\) is lower than the graph of \(y=3^{x}\), and where it is greater. You'll find that for negative \(x\) values, \(4^{x}<3^{x}\), because as \(x\) decreases, \(4^x\) decreases at a faster rate than \(3^x\). On the other hand, for positive \(x\) values, \(4^{x}>3^{x}\). So these are the regions where each inequality holds true.
03

Conclude the results

Based on the observations from the graph, for inequality (a) \(4^{x}<3^{x}\), the solution set is x<0. And for inequality (b) \(4^{x}>3^{x}\), the solution set is x>0.

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