# Summary: Systems of Nonlinear Equations and Inequalities

## Key Concepts

- There are three possible types of solutions to a system of equations representing a line and a parabola: (1) no solution, the line does not intersect the parabola; (2) one solution, the line is tangent to the parabola; and (3) two solutions, the line intersects the parabola in two points.
- There are three possible types of solutions to a system of equations representing a circle and a line: (1) no solution, the line does not intersect the circle; (2) one solution, the line is tangent to the parabola; (3) two solutions, the line intersects the circle in two points.
- There are five possible types of solutions to the system of nonlinear equations representing an ellipse and a circle: (1) no solution, the circle and the ellipse do not intersect; (2) one solution, the circle and the ellipse are tangent to each other; (3) two solutions, the circle and the ellipse intersect in two points; (4) three solutions, the circle and ellipse intersect in three places; (5) four solutions, the circle and the ellipse intersect in four points.
- An inequality is graphed in much the same way as an equation, except for > or <, we draw a dashed line and shade the region containing the solution set.
- Inequalities are solved the same way as equalities, but solutions to systems of inequalities must satisfy both inequalities.

## Glossary

**feasible region**the solution to a system of nonlinear inequalities that is the region of the graph where the shaded regions of each inequality intersect

**nonlinear inequality**an inequality containing a nonlinear expression

**system of nonlinear equations**a system of equations containing at least one equation that is of degree larger than one

**system of nonlinear inequalities**a system of two or more inequalities in two or more variables containing at least one inequality that is not linear

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