Chapter 7.4, Problem 13E

Solution

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

  $x^2+y^2<9$

The boundary $x^2+y^2=9$ is not included in the graph, and the graph of the inequality is shown as below,

  

Given Information:

The inequality is defined as,

  $x^2+y^2<9$

Explanation:

Consider the given inequality,

  $x^2+y^2<9$

Replacing the inequality symbol with the = symbol, the equation for the boundary is  $x^2+y^2=9$ , which is a circle with centre origin and the radius is 3.

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

  $\begin{array}{c}{x}^2+{\left(0\right)}^2=9\\ x^2=9\\ x=\pm 3\end{array}$

And,

  $\begin{array}{c}{\left(0\right)}^2+y^2=9\\ y^2=9\\ y=\pm 3\end{array}$

Therefore, the intercept is $\left(3,0\right),\left(-3,0\right)\text{ and }\left(0,3\right),\left(0,-3\right)$ .

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 to 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,

  $\begin{array}{c}{\left(0\right)}^2+{\left(0\right)}^2<9\\ 0<9\end{array}$

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 $x^2+y^2=9$ is not included in the graph.