Chapter 7.4, Problem 11E

Solution

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

  $y<x^2+1$

The boundary $y=x^2+1$ is not included in the graph, and the graph of the inequality is shown as below,

  

Given Information:

The inequality is defined as,

  $y<x^2+1$

Explanation:

Consider the given inequality,

  $y<x^2+1$

Replacing the inequality symbol with the = symbol, the equation for the boundary is  $y=x^2+1$ , which is a parabola and opening upward.

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

  $\begin{array}{c}0=x^2+1\\ x^2=-1\\ =\text{ not exist}\end{array}$

And,

  $\begin{array}{c}y={\left(0\right)}^2+1\\ =0+1\\ =1\end{array}$

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

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 less 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,

  $\begin{array}{c}\left(0\right)<{\left(0\right)}^2+1\\ 0<1\end{array}$

The statement is 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 $y=x^2+1$ is not included in the graph.