Chapter 7.4, Problem 14E

Solution

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

  $x^2+y^2\ge 4$

The boundary $x^2+y^2=4$ is included in the graph, and the graph of the inequality is shown as below,

  

Given Information:

The inequality is defined as,

  $x^2+y^2\ge 4$

Explanation:

Consider the given inequality,

  $x^2+y^2\ge 4$

Replacing the inequality symbol with the = symbol, the equation for the boundary is  $x^2+y^2=4$ , which is a circle with origin centre and the radius is 2 .

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

  $\begin{array}{c}{x}^2+{\left(0\right)}^2=4\\ x^2=2\\ x=\pm 2\end{array}$

And,

  $\begin{array}{c}{\left(0\right)}^2+y^2=4\\ y^2=4\\ y=\pm 2\end{array}$

Therefore, the intercept are $\left(2,0\right)\left(-2,0\right)\text{ and }\left(0,2\right),\left(0,-2\right)$ .

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

  $\begin{array}{c}{\left(0\right)}^2+{\left(0\right)}^2\ge 4\\ 0\ge 4\end{array}$

The statement is false so shade the half-plane where  $\left(0,0\right)$

 is not located. The graph will be:

  

Hence, the required graph is shown above and the boundary $x^2+y^2=4$ is included in the graph.