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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.

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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.

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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

how to understand calculus?
Jenica Reply
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Practice Key Terms 4

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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