<< Chapter < Page Chapter >> Page >

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    .

Given a system of nonlinear inequalities, sketch a graph.

  1. Find the intersection points by solving the corresponding system of nonlinear equations.
  2. Graph the nonlinear equations.
  3. Find the shaded regions of each inequality.
  4. Identify the feasible region as the intersection of the shaded regions of each inequality or the set of points common to each inequality.

Graphing a system of inequalities

Graph the given system of inequalities.

x 2 y 0 2 x 2 + y 12

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 .

x 2 y = 0 2 x 2 + y = 12 ____________        3 x 2 = 12          x 2 = 4            x = ± 2

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

x 2 y = 0 ( 2 ) 2 y = 0 4 y = 0 y = 4 ( −2 ) 2 y = 0 4 y = 0 y = 4

The two points of intersection are ( 2 , 4 ) and ( −2 , 4 ) . Notice that the equations can be rewritten as follows.

x 2 y 0 x 2 y y x 2 2 x 2 + y 12 y −2 x 2 + 12

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

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Graph the given system of inequalities.

       y x 2 1 x y 1

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

Got questions? Get instant answers now!

Access these online resources for additional instruction and practice with nonlinear equations.

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] .

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
Practice Key Terms 4

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask