Solving Systems of Inequalities

Solving Systems of Nonlinear Inequalities

The solution of a system of nonlinear inequalities is represented on a graph as the overlap of the shaded regions that represent the solutions of the separate inequalities.
Systems of nonlinear inequalities are solved by the same method as systems of linear inequalities. The graphs may be more complicated, so in cases where it is difficult to determine which part of the coordinate plane to shade, use a test point.
Step-By-Step Example
Solving a System of Nonlinear Inequalities
Solve the system of inequalities.
{x2+y4y>x2+3x4\begin{cases}\begin{aligned}x^2+y & \leq 4\\y & > x^2+3x-4 \end{aligned}\end{cases}
Step 1
Solve the inequalities for yy, if needed.
x2+y4yx2+4\begin{aligned}x^2+y&\leq 4\\y&\leq -x^2+4\end{aligned}
The second inequality is already solved for yy .
Step 2
Graph the boundary curves using technology or by hand. Use a solid curve for:
yx2+4y\leq -x^2+4
Use a dashed curve for:
y>x2+3x4y>x^2+3x-4
Solution
Shade below the curve of:
yx2+4y\leq -x^2+4
Shade above the curve of:
y>x2+3x4y>x^2+3x-4
The solution is the region where the two shaded areas overlap.
To check the solution, select a point from the shaded region that is not on either curve and substitute the coordinates of the point into both inequalities to verify that each is true. The point (0,0)(0,0) is in the overlap of the shaded areas and not on either curve. First inequality:
x2+y402+0?404\begin{aligned}x^2+y&\leq 4\\0^2+0&\stackrel{?}{\leq}4\\0&\leq 4 & \checkmark \end{aligned}
Second inequality:
y>x2+3x40>?02+3(0)40>4\begin{aligned}y&>x^2+3x-4\\0&\stackrel{?}{>}0^2+3(0)-4\\0&>-4 & \checkmark \end{aligned}
Both inequalities are true, so the region that contains (0,0)(0,0) is the solution.