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Chapter 3: Problem 79

Graph the functions \(y=3^{x}\) 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
The solution for the inequality \(4^{x}<3^{x}\) is \(x<0\) and the solution for the inequality \(4^{x}>3^{x}\) is \(x>0\).

Step by step solution

01

Plotting the functions

First, plot the functions \(y = 3^{x}\) and \(y = 4^{x}\) on the same set of axes. Use the x-axis ranging say from -2 to 2 and the y-axis from 0 to 10, to make the graph readable. Typically, \(3^{x}\) will be the graph that starts at a higher point on the y-axis, but grows slower, while \(4^{x}\) starts lower but grows faster.
02

Solving the inequality \(4^{x} < 3^{x}\)

Looking at the graphs, it can be noticed that the graph of \(4^{x}\) is below the graph of \(3^{x}\) when \(x<0\). This range satisfies the inequality \(4^{x}<3^{x}\). Therefore, the solution to this inequality is \(x<0\).
03

Solving the inequality \(4^{x} > 3^{x}\)

Now, referring to the graphs again, note that the graph of \(4^{x}\) is above the graph of \(3^{x}\) when \(x>0\). This range satisfies the inequality \(4^{x}>3^{x}\). Thus, the solution to this inequality is \(x>0\).

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