7.3 Systems of nonlinear equations and inequalities: two variables  (Page 4/9)

Graphing a system of nonlinear inequalities

Now that we have learned to graph nonlinear inequalities, we can learn how to graph systems of nonlinear inequalities. A system of nonlinear inequalities    is a system of two or more inequalities in two or more variables containing at least one inequality that is not linear. Graphing a system of nonlinear inequalities is similar to graphing a system of linear inequalities. The difference is that our graph may result in more shaded regions that represent a solution than we find in a system of linear inequalities. The solution to a nonlinear system of inequalities is the region of the graph where the shaded regions of the graph of each inequality overlap, or where the regions intersect, called the feasible region    .

Graphing a system of inequalities

Graph the given system of inequalities.

$$\begin{array}{r}\hfill x^2-y\le 0\\ \hfill 2x^2+y\le 12\end{array}$$

These two equations are clearly parabolas. We can find the points of intersection by the elimination process: Add both equations and the variable $y$ will be eliminated. Then we solve for $x.$

$$\begin{array}{l}\underset{\_\_\_\_\_\_\_\_\_\_\_\_}{\begin{array}{c}\begin{array}{l}\hfill \\ x^2-y=0\hfill \end{array}\\ 2x^2+y=12\end{array}}\hfill \\ 3x^2=12\hfill \\ x^2=4\hfill \\ x=\pm 2\hfill \end{array}$$

Substitute the x -values into one of the equations and solve for $y.$

$$\begin{array}{r}\hfill x^2-y=0\\ \hfill {(2)}^2-y=0\\ \hfill 4-y=0\\ \hfill y=4\\ \hfill \\ \hfill {(\mathrm{-2})}^2-y=0\\ \hfill 4-y=0\\ \hfill y=4\end{array}$$

The two points of intersection are $\left(2,4\right)$ and $\left(\mathrm{-2},4\right).$ Notice that the equations can be rewritten as follows.

$$\begin{array}{l}{x}^2-y\le 0\hfill \\ x^2\le y\hfill \\ y\ge x^2\hfill \\ \hfill \\ 2x^2+y\le 12\hfill \\ y\le \mathrm{-2}{x}^2+12\hfill \end{array}$$

Graph each inequality. See [link] . The feasible region is the region between the two equations bounded by $2x^2+y\le 12$ on the top and $x^2-y\le 0$ on the bottom.

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Graph the given system of inequalities.

$$\begin{array}{l}y\ge x^2-1\hfill \\ x-y\ge -1\hfill \end{array}$$

Shade the area bounded by the two curves, above the quadratic and below the line.

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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. See [link] .
• 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. See [link] .
• 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. See [link] .
• 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. See [link] .
• Inequalities are solved the same way as equalities, but solutions to systems of inequalities must satisfy both inequalities. See [link] .