Q. 49 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 49 and 50, graph each system of inequalities by hand.

$\left\{\begin{array}{l}{x}^2+y^2\le 16\\ x+y\ge 2\end{array}\right.$

Step-by-Step Solution

Answer

The graph of $\left\{\begin{array}{l}{x}^2+y^2\le 16\\ x+y\ge 2\end{array}\right.$is 

Step 1. Given data

The given system of inequality is     

$\left\{\begin{array}{l}{x}^2+y^2\le 16\\ x+y\ge 2\end{array}\right.$

Step 2. Properties of the graph of the inequality

Graph of $x^2+y^2=16$will be a circle of center $(0,0)$

now test the origin point $(0,0)$ for inequality $x^2+y^2\le 16$

$$\begin{gathered} x^2+y^2\le 16 \\ {0}^2+0^2\le 16 \\ 0<16 \end{gathered}$$

inequality satisfied so the origin will lie in the region of inequality

So the region inside the circle will be shaded

Step 3. Properties of the graph of the inequality

Graph of $x+y=2$will be a line

now test the origin point $(0,0)$ for inequality $x+y\ge 2$

$$\begin{gathered} x+y\ge 2 \\ 0+0\ge 2 \\ 0<2 \end{gathered}$$

inequality does not satisfy so the origin will not lie in the region of inequality

So the region above from the line will be shaded 

Step 4. graph of the system of inequality

The graph of the system of inequality is