# How do you solve the system x^2+y^2<36 and 4x^2+9y^2>36 by graphing?

Mar 29, 2017

#### Explanation:

Let us graph the circle ${x}^{2} + {y}^{2} = 36$, which is a circle with center $\left(0 , 0\right)$ and radius $6$ and ${x}^{2} + {y}^{2} < 36$ is the set of all points lying inside the circle but not on the circle, as we have inequality. It would have included points on the circle had the inequality been ${x}^{2} + {y}^{2} \le 36$. The graph of ${x}^{2} + {y}^{2} < 36$, appears as follows:
graph{x^2+y^2<36 [-14, 14, -7, 7]}

Similarly, $4 {x}^{2} + 9 {y}^{2} > 36$ is the graph is the set of all points lying outside the ellipse but not on the circle. Its graph appears as follows:
graph{4x^2+9y^2>36 [-14, 14, -7, 7]}

And solution of the system of inequalities ${x}^{2} + {y}^{2} = 36$ and $4 {x}^{2} + 9 {y}^{2} > 36$ is the set of intersection of points in the two.

Thus it includes points outside the ellipse $4 {x}^{2} + 9 {y}^{2} = 36$ but inside the circle ${x}^{2} + {y}^{2} = 36$ but does not include points on the circle and the ellipse. The region appears as shown below.