2.8: Inequalities - Instruction

2.8: Polynomial InequalitiesAnd polynomials and graphing and designing boxes.

2(x 2)(x + 3)F(x)

ObjectivesBe able to find the positives, negatives, and zeros of a polynomial inequality via line method.Be able to graph polynomial inequalities.Be able to solve inequalities using the analytical method (synthetic division, rational zeros theorem)Know what an impossible inequality looks like and understand why its impossible.Be able to solve rational inequalities.Be able to combine functions with rational inequalities.Apply this information to real-life problems (box method, cylinder method, etc).

What is an inequality?Its not social justice. Less Than Or Equal To Greater Than Or Equal To< Less Than> Greater Than

Like Pacman!Symbols:Polynomial inequalities!To solve f(x) > 0, find the values of x that make f(x) positiveTo solve f(x) < 0, find the values of x that make f(x) negativeTo solve f(x) = 0, find the values of x that make f(x) zero

Which can help figure out which part of the number line is positive or negative. - - + +++ ++405F(x)= (x+2)(x+1)(x-5)Zeros at: -2, -1, and 5 (each multiplicity 1)F(x)= (x+3)(x+6)(x-9)Zeros at: -3 (multiplicity 2), -6(multiplicity 1), 9 (multiplicity 3).F(x)= (4x+8)(x-2)(x+7)Zeros at: -2, 2( multiplicity 1), -7(multiplicity 2)

Examples: Finding Zeros and Multiplicity Howeversome polynomial inequalities do not appear in a factored form and some dont have zero on one of the sides.

But they can be solved in various ways!

Solving Them AnalyticallyTo do this, use the rational zeros theorem, finding the possible factors of the constant coefficient over the leading coefficient.

How to find possible factors of a polynomial, statisticslectures.comExamples: Solving a Polynomial AnalyticallyF(x)= 2x-7x-10x+24In this example, enter the equation on your calculator, and find the first zero.The next step is synthetic division.2 -7 -10 24 8 4 -24__________________________________________ 2 1 -6 0

The first zero occurs at 4ContinuedContinue to factor after youve completed the synthetic divisionF(x)= (x-4)(2x + x 6)

= (x-4)(2x-3)(x-2)Zeros are: 4, 3/2, and -2.1st zeroAfter Using Rational ZerosYou may now plug in the possible rational using synthetic division.After finding a true zero, find out where the polynomial is positive or negative, as previously done

Synthetic Division Example by http://taylorlstormberg.wordpress.comGraphing!ExplainedYet another way to solve the polynomial inequality is graphically. To do this, make sure that if there are two polynomials on either side of the inequality get moved to one side. Then we make the inequality equal to zero. After plugging the equation into a graphing calculator, find the zeros of the polynomial.

Graphing polynomials from www.algebra-tutoring.comExamples: Solving Graphicallyx - 6x x-8x Solution:Rewrite! (-, 0.32] U [1.46, 4.21]x - 6x + 8x 2 0 F(x) = x - 6x + 8x 2 Zeros approx at: 0.32, 1.46, and 4.21

GRAPH!In Addition...Depending on the inequality, the solutions of the equation may vary.

If its x < 0, then the solutions are below the x-axisIf its x > 0, then the solutions are above the x-axisIf its x 0, then the solutions are on or below the x-axisIf its x 0, then the solutions are on or above the x-axis

However...there are some empty inequalities that have no solutionsImpossible Inequalities(x + 7)(2x + 1) < 0 (x + 7)(2x + 1) 0 Square roots cannot have negative numbersWhen solving for zero, x would equal -7, which would be impossible, unless imaginary numbers are present.Again, when solving for zero, x would be equal to either -7 or -1/2.Square roots of negative numbers would still be impossible unless imaginary numbers are present.Examples: Unusual Answers(2x + 7)(- x + 1) 0 solution is (-, ) also, (2x + 7)(- x + 1) 0 solutions is (-, ) BUT, (2x + 7)(- x + 1) 0 (2x + 7)(- x + 1) 0

Both solutions are emptyRational InequalitiesAlong with polynomial inequalities, there are rational inequalities as well! Not only can the zeros to rational inequalities be zero, positive, or negative, but zeros can also be undefined.Example Equation: y = (x+4)(x+3)/(x+-1)

4undefined3 - - + +++ ++Undefined zeros can be found by finding the zero of the denominator of the function01021Examples: Sign Charts for a Rational FunctionR(x) = (2x +1) / ( (x+3)(x-1))Find the real zeros. ( in numerator only)X = -1/2If x is equal to -3 or 1, r(x) is undefined.R(x) is positive if -3 x -1/2 or x 1.R(x) is negative if x -3 or -1/2 1 1

1-3-1/2 (-)------(-) (-) (-)-------(+) (-) (+)-------(+) (-) (+)-------(+) (+)Negative Positive Negative PositiveSolution(-3, -1/2) U (1,)and(-, -3) U (-1/2, 1)Combining Functions With Rational InequalitiesYou can also solve a rational inequality by combining functions. To do this, combine the two functions on the left-hand side by using the least common denominator. (multiply the two denominators together). Then you add the fractions, use distributive property, simplify, and divide.

Examples: Combining FractionsSolve 5 / (x+3) + 3/ (x-1) 0 Combine the two fractions:5(x-1)/(x+3)(x-1) + 3(x+3)/ (x+3)(x-1)5(x-1) + 3(x+3) / (x+3)(x-1)5x-5 + 3x +9 / (x+3)(x-1)8x +4 / (x+3)(x-1) (divide both sides by 4)2x +1 / (x+3)(x-1)

Applications Box ProblemA packaging company wishes to design boxes with a volume of at least 600 in. Squares are to be cut from the corners of a 20-in by 25-in piece of cardboard, with the flaps folded up to make an open box. What size squares should be cut?Volume V of a box is V(x) = x(y-2x)(y-2x)Y represents the side-lengths of cardboard, so Y would go into the volume equation.X represents the side length of the removed squares and the height of the box.Volume is now V(x) = x(25-2x)(20-2x)To obtain a volume of at least 600 in, we make the equation into an inequality.600 x(25-2x)(20-2x)6) To solve the application graphically, plug the equation from step four into the calculator and plug 600 into the second y= spot below the first equation.7) Make the window 0 x 10 (because the width of the cardboard is 20) and 0y10008) Find the intercepts between the two graphs. The solution is [1.66, 6.16]

Applications Can ProblemA cannery wants to package cans in a 2-liter(2000cm) cylindrical can. Find the radius and height of the cans if the cans have a surface area that is less than 1000 cm.Surface Area = 2r + 4000/rThe inequality for this equation is 2r + 4000/r < 1000To solve this equation plug the equation from step two into the graphing calculator. Then put 1000 in the second y= area below that.Find out where the lines cross via graphing. The surface area should be 4.62 < r < 9.65 to be less than 1000 cm.The volume equation is V = rh and V = 2000.Using this equation, we can find out that H = 2000/rTo find values of h we build a double inequality for 2000/(r)ReferencesPrecalculus 8th editionhttp://taylorlstormberg.wordpress.comstatisticslectures.comScarseths In-Class Noteswww.algebra-tutoring.comYoutube.comQuickmeme.comDeviantart.comHark.comSosmath.comhttp://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.htmlhttps://www.desmos.com/calculatorMrs. Barbers Math practice testsgallippi-advfunctions12.blogspot.comhttp://www.regentsprep.org/Regents/math/algtrig/ATE11/RationalInequalitiesLes.htm