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Section exercises

Verbal

Explain whether a system of two nonlinear equations can have exactly two solutions. What about exactly three? If not, explain why not. If so, give an example of such a system, in graph form, and explain why your choice gives two or three answers.

A nonlinear system could be representative of two circles that overlap and intersect in two locations, hence two solutions. A nonlinear system could be representative of a parabola and a circle, where the vertex of the parabola meets the circle and the branches also intersect the circle, hence three solutions.

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When graphing an inequality, explain why we only need to test one point to determine whether an entire region is the solution?

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When you graph a system of inequalities, will there always be a feasible region? If so, explain why. If not, give an example of a graph of inequalities that does not have a feasible region. Why does it not have a feasible region?

No. There does not need to be a feasible region. Consider a system that is bounded by two parallel lines. One inequality represents the region above the upper line; the other represents the region below the lower line. In this case, no points in the plane are located in both regions; hence there is no feasible region.

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If you graph a revenue and cost function, explain how to determine in what regions there is profit.

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If you perform your break-even analysis and there is more than one solution, explain how you would determine which x -values are profit and which are not.

Choose any number between each solution and plug into C ( x ) and R ( x ) . If C ( x ) < R ( x ) , then there is profit.

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Algebraic

For the following exercises, solve the system of nonlinear equations using substitution.

    x + y = 4 x 2 + y 2 = 9

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          y = x −3 x 2 + y 2 = 9

( 0 , −3 ) , ( 3 , 0 )

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          y = x x 2 + y 2 = 9

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          y = x x 2 + y 2 = 9

( 3 2 2 , 3 2 2 ) , ( 3 2 2 , 3 2 2 )

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          x = 2 x 2 y 2 = 9

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For the following exercises, solve the system of nonlinear equations using elimination.

4 x 2 −9 y 2 = 36 4 x 2 + 9 y 2 = 36

( −3 , 0 ) , ( 3 , 0 )

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x 2 + y 2 = 25 x 2 y 2 = 1

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2 x 2 + 4 y 2 = 4 2 x 2 −4 y 2 = 25 x −10

( 1 4 , 62 8 ) , ( 1 4 , 62 8 )

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y 2 x 2 = 9 3 x 2 + 2 y 2 = 8

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x 2 + y 2 + 1 16 = 2500 y = 2 x 2

( 398 4 , 199 4 ) , ( 398 4 , 199 4 )

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For the following exercises, use any method to solve the system of nonlinear equations.

−2 x 2 + y = −5      6 x y = 9

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x 2 + y = 2   x + y = 2

( 0 , 2 ) , ( 1 , 3 )

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x 2 + y 2 = 1            y = 20 x 2 −1

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x 2 + y 2 = 1            y = x 2

( 1 2 ( 5 −1 ) , 1 2 ( 1 5 ) ) , ( 1 2 ( 5 −1 ) , 1 2 ( 1 5 ) )

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2 x 3 x 2 = y             y = 1 2 x

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9 x 2 + 25 y 2 = 225 ( x −6 ) 2 + y 2 = 1

( 5 , 0 )

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x 4 x 2 = y    x 2 + y = 0

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2 x 3 x 2 = y      x 2 + y = 0

( 0 , 0 )

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For the following exercises, use any method to solve the nonlinear system.

x 2 + y 2 = 9           y = 3 x 2

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x 2 y 2 = 9            x = 3

( 3 , 0 )

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x 2 y 2 = 9            y = 3

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x 2 y 2 = 9      x y = 0

No Solutions Exist

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x 2 + y = 2 −4 x + y = −1

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x 2 + y = 2           2 y = x

No Solutions Exist

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x 2 + y 2 = 25 x 2 y 2 = 36

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x 2 + y 2 = 1          y 2 = x 2

( 2 2 , 2 2 ) , ( 2 2 , 2 2 ) , ( 2 2 , 2 2 ) , ( 2 2 , 2 2 )

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16 x 2 −9 y 2 + 144 = 0                  y 2 + x 2 = 16

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       3 x 2 y 2 = 12 ( x −1 ) 2 + y 2 = 1

( 2 , 0 )

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       3 x 2 y 2 = 12 ( x −1 ) 2 + y 2 = 4

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3 x 2 y 2 = 12     x 2 + y 2 = 16

( 7 , −3 ) , ( 7 , 3 ) , ( 7 , −3 ) , ( 7 , 3 )

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x 2 y 2 6 x 4 y 11 = 0                     x 2 + y 2 = 5

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x 2 + y 2 −6 y = 7            x 2 + y = 1

( 1 2 ( 73 −5 ) , 1 2 ( 7 73 ) ) , ( 1 2 ( 73 −5 ) , 1 2 ( 7 73 ) )

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x 2 + y 2 = 6          x y = 1

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Graphical

For the following exercises, graph the inequality.

For the following exercises, graph the system of inequalities. Label all points of intersection.

x 2 + y < −5 y > 5 x + 10

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x 2 + y 2 < 25 3 x 2 y 2 > 12

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x 2 y 2 > −4 x 2 + y 2 < 12

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x 2 + 3 y 2 > 16 3 x 2 y 2 < 1

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Extensions

For the following exercises, graph the inequality.

y e x y ln ( x ) + 5

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y log ( x ) y e x

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For the following exercises, find the solutions to the nonlinear equations with two variables.

4 x 2 + 1 y 2 = 24 5 x 2 2 y 2 + 4 = 0

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6 x 2 1 y 2 = 8 1 x 2 6 y 2 = 1 8

( −2 70 383 , −2 35 29 ) , ( −2 70 383 , 2 35 29 ) , ( 2 70 383 , −2 35 29 ) , ( 2 70 383 , 2 35 29 )

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x 2 x y + y 2 −2 = 0                 x + 3 y = 4

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x 2 x y −2 y 2 −6 = 0                 x 2 + y 2 = 1

No Solution Exists

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x 2 + 4 x y −2 y 2 −6 = 0                             x = y + 2

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Technology

For the following exercises, solve the system of inequalities. Use a calculator to graph the system to confirm the answer.

x y < 1 y > x

x = 0 , y > 0 and 0 < x < 1 , x < y < 1 x

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Real-world applications

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.

Two numbers add up to 300. One number is twice the square of the other number. What are the numbers?

12, 288

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The squares of two numbers add to 360. The second number is half the value of the first number squared. What are the numbers?

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A laptop company has discovered their cost and revenue functions for each day: C ( x ) = 3 x 2 −10 x + 200 and R ( x ) = −2 x 2 + 100 x + 50. If they want to make a profit, what is the range of laptops per day that they should produce? Round to the nearest number which would generate profit.

2–20 computers

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A cell phone company has the following cost and revenue functions: C ( x ) = 8 x 2 −600 x + 21,500 and R ( x ) = −3 x 2 + 480 x . What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

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Questions & Answers

Cos45/sec30+cosec30=
dinesh Reply
Cos 45 = 1/ √ 2 sec 30 = 2/√3 cosec 30 = 2. =1/√2 / 2/√3+2 =1/√2/2+2√3/√3 =1/√2*√3/2+2√3 =√3/√2(2+2√3) =√3/2√2+2√6 --------- (1) =√3 (2√6-2√2)/((2√6)+2√2))(2√6-2√2) =2√3(√6-√2)/(2√6)²-(2√2)² =2√3(√6-√2)/24-8 =2√3(√6-√2)/16 =√18-√16/8 =3√2-√6/8 ----------(2)
exercise 1.2 solution b....isnt it lacking
Miiro Reply
I dnt get dis work well
john Reply
what is one-to-one function
Iwori Reply
what is the procedure in solving quadratic equetion at least 6?
Qhadz Reply
Almighty formula or by factorization...or by graphical analysis
Damian
I need to learn this trigonometry from A level.. can anyone help here?
wisdom Reply
yes am hia
Miiro
tanh2x =2tanhx/1+tanh^2x
Gautam Reply
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation
favour Reply
f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb
favour
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
i am in
Cliff
1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
helo
Akash
hlo
Akash
Hello
Hudheifa
which of these functions is not uniformly continuous on 0,1
Basant Reply
Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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