Chapter 7.4, Problem 28E

Solution

The system of inequality, whose solution is the region shaded in pink in the given figure and all the boundaries are to be included.

The obtained system of inequalities is $\left\{\begin{array}{c}-2\le x\le 2\\ y\ge 0\\ x^2+y^2\le 4\end{array}\right\}$ .

Given Information:

The figure is show as,

  

Explanation:

Consider the given system of equations,

  

It can be observed that in the given figure, one figure is circle. The general form of the circle is ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ . The circle is cantered at origin and the radius is 2. So,

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

Using the test point as origin, which is inside the circle and the appropriate inequality should be used $\le$ . So,

  $x^2+y^2\le 4$

In the given figure the value of x is lying between $-2$ to 2. So,

  $-2\le x\le 2$

For the any value of y in the above interval of x , the value of y is always positive. So,

  $y\ge 0$

Hence, the obtained system of inequalities are $\left\{\begin{array}{c}-2\le x\le 2\\ y\ge 0\\ x^2+y^2\le 4\end{array}\right\}$ .