Chapter 11, Problem 92RE

To determine

Graph each system of inequalities.

Answer to Problem 92RE

Explanation of Solution

Given information:

Graph each system of inequalities.

  $\{\begin{array}{l}{x}^2+y^2\ge 1\\ x^2+y^2\le 4\end{array}$

Calculation:

Consider the sytem of inequalities.

  $\{\begin{array}{l}{x}^2+y^2\ge 1\\ x^2+y^2\le 4\end{array}$

Let consider

  $x^2+y^2\ge 1$

To determine the part of the graph to be shaded use a test point $\left(0,0\right)$ .

Substitute $x=0,y=0$ in the first inequality of the given system,

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

Since $0\ge 1$ is false statement point $\left(0,0\right)$ does not satisfying the inequality. Hence the complete solution is all set the points which are outside the curve and does not include the point $\left(0,0\right)$

But include the points on the curve

  $x^2+y^2=1$

Now consider,

  $x^2+y^2\le 4$

Substitute $x=0,y=0$ in the first inequality of the given system,

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

Since $0\le 4$ is true statement point $\left(0,0\right)$ satisfying the inequality. Hence the complete solution is all set the points which are same side of the curve and include the point $x^2+y^2=4$ .

Now the solution corresponding to the inequalities will be the point common to the both and the graph is sketched in rectangular Cartesian coordinate system as given below.

Hence the graph of inequalities is as shown.