Solving Systems of Inequalities Examples

Graph (x – 3)² + (y + 5)² < 16, state the boundary and find out if (3, -1) is within the solution.

This is the equation for a circle at (3, -1) with a radius of 4. Since it is <, we will shade inside the circle. We will also make a dotted line, since the solution cannot lie directly on the outside of the circle.

If you threw a dart at (3, -1), would it land in the shaded region? Ummm, yes? No? Maybe. Actually, if you look carefully, the dart appears to have landed on the dotted line. That means it would not be part of the solution. Let's check algebraically by substituting x = 3 and y = -1.

(3 – 3)² + (-1 + 5)² < 16
0 + 16 < 16
16 ≠ 16

(3, -1) is not part of the solution to the system of inequalities.

Solve and state the boundary of:

Graphing these, the solution is the intersection of the two inequalities as pictured below.

The darker green region is the solution and that does not include the boundaries of y = 2x² – 4 and .

Write a system of inequalities that have this solution:

There are going to be 4 lines (2 horizontal and 2 vertical) as the boundaries. The white rectangle in the middle is not touched by any of the inequalities. The system of inequalities is:

y ≥ -2
y ≤ -4
x ≤ -5
x ≥ -2