Chapter 7.4, Problem 15E

Solution

To Graph: The given inequality by using the hand sketch and state the boundary region.

  $y>2^x$

The boundary $y=2^x$ is not included in the graph, and the graph of the inequality is shown as below,

  

Given Information:

The inequality is defined as,

  $y>2^x$

Explanation:

Consider the given inequality,

  $y>2^x$

Replacing the inequality symbol with the = symbol, the equation for the boundary is  $y=2^x$ , which is a exponential line with base 2.

Now, find the intercept of the line. Substitute 0 for x and then 0 for y in the line one by one.

  $\begin{array}{c}0=2^x\\ 2^x=\text{ not exists}\end{array}$

And,

  $\begin{array}{c}y=2^0\\ y=1\end{array}$

Therefore, the intercept are $\left(0,1\right)$ .

The boundary point is included if the inequality containing $\le \text{ or }\ge$ symbols and the boundary point is excluded if the symbols $<\text{ or }>$  is used.

As the greater then symbol was used, then the boundary is not included in the graph and drawing it by using dotted line. To determine the half-plane to be shaded, use a test point not on the boundary, say  $\left(0,0\right)$ . Then,

  $\begin{array}{c}0>2^0\\ 0>1\end{array}$

The statement is not true so the shade the half-plane where  $\left(0,0\right)$ is located. The graph will be:

  

Hence, the required graph is shown above and the boundary $y=2^x$ is not included in the graph.