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Solving Inequalities By: Sam Milkey and Noah Bakunowicz

Solving Inequalities

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Solving Inequalities. By: Sam Milkey and Noah Bakunowicz. Polynomial Inequalities. A polynomial inequality takes the form f(x) > 0, f(x) ≥ 0, f(x) < 0, f(x) ≤ 0, or f(x) ≠ 0. To solve f(x) > 0 is to find the values of x that make f(x) positive. - PowerPoint PPT Presentation

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Page 1: Solving Inequalities

Solving InequalitiesBy: Sam Milkey and Noah Bakunowicz

Page 2: Solving Inequalities

Polynomial Inequalities• A polynomial inequality takes the form f(x) > 0, f(x) ≥ 0,

f(x) < 0, f(x) ≤ 0, or f(x) ≠ 0.

• To solve f(x) > 0 is to find the values of x that make f(x) positive.

• To solve f(x) < 0 is to find the values of x that make f(x) negative.

Page 3: Solving Inequalities

But that’s pretty boring.https://www.youtube.com/watch?v=_J7xwaOrnf8(skip to 1:00)

Page 4: Solving Inequalities

Example 1 Finding negative, positive, zero

• F(x)=(x+2)(x+1)(x-5)• Zeros: -2 (mult of 1), -1(mult of 1), 5 (mult of 1)• Number line:

• Find when it is Zero, Negative, and Positiveo Zeros: -2, -1, 5o Negative: (∞, -2) (-1, 5)o Positive: (-2,-1) (5,∞)

-2 -1 5

---

-+--

+++-

-+++

+

Page 5: Solving Inequalities

Example 2 Solving Algebraically

• Solve 2x³-7x²-10x+24>0 Analytically

• Use the rational zeros theorem to find possible rational zeroso ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24, ±1/2, ±3/2

• You can use a graph to figure out which zero to use first, in this case x=4 is good.

Page 6: Solving Inequalities

Example 2 cont.• Using synthetic division

• Factor 2x²+x-6o (2x-3)(x+2)

• So f(x)=(x-4)(2x-3)(x+2)

• Zeros= 4, 3/2, -2

4 2 -7 -10 24

48 -24

2 1 -6 0 F(x)=(x-4)(2x²+x-6)

Page 7: Solving Inequalities

Example 2 cont.• Sign Chart

• You can find the where it is negative or positive from its end behavior

• Since we wanted to find out when it is greater than 0, the solutions are (-2,3/2) and (4,∞)

-2 3/2 4

Sign Change

Sign Change

Sign Change

- + - +

Page 8: Solving Inequalities

Example 3 Solving Graphically

• Solve x3-6x2 ≤ 2-8x graphically

• Rewrite the inequality so it is less than or equal to 0o x3-6x2+8x-2 ≤ 0

• Type in x3-6x2+8x-2 into the y1 of the graph on your calculatoro Zeros are approximately 0.32, 1.46, and 4.21

• Since we want when it is less than 0, we want all of the numbers below the x-axis on the grapho Solution: (-∞,0.32] and [1.46, 4.21]

• Remember, use hard brackets because those points are solutions too!

Page 9: Solving Inequalities

Example 4 Solving with Unusual Answers

• The inequalities associated with a strictly positive polynomial function such as f(x) = (x2+7)(2x2+1) have strange solutions

o (x2+7)(2x2+1) > 0 is all real numbers

o (x2+7)(2x2+1) ≥ 0 is all real numbers

o (x2+7)(2x2+1) < is no solution

o (x2+7)(2x2+1) ≤ is no solution

Page 10: Solving Inequalities

Example 4 Cont.• The inequalities associated with a nonnegative polynomil function such as

f(x)=(x2-3x+3)(2x+5)2 also has strange answers

o (x2-3x+3)(2x+5) > 0 is (-∞,-5/2) and (-5/2,∞)

o (x2-3x+3)(2x+5) ≥ 0 is all real numbers

o (x2-3x+3)(2x+5) < 0 has no solution

o (x2-3x+3)(2x+5) ≤ 0 is a single number, -5/2

Page 11: Solving Inequalities

Example 5 Creating Sign Charts

• Let f(x) = (2x+1)/((x+3)(x-1)). Find when the function is (a) zero (b) undefined. Then make a sign chart to find when it is positive or negative.

(a). Real zeros of the function are the real zeros of the numerator. in this case 2 x+1 is the numerator

(b). f(x) is undefined when the denominator is 0. Since (x+3)(x-1) is the denominator, it is undefined at x = -3 or x = 1.

• Sign Chart

-3 -1/2 1

Potential Sign Change

Potential Sign Change

Potential Sign Change

Page 12: Solving Inequalities

Example 5 cont.• Sign chart with undefined, zeros, positive, and negative

• f(x) is negative if x < 3 or -1/2 < x < 1, so the solutions are (-∞, -3) and (-1/2, 1)

• f(x) is positive if -3 < x < -1/2 or x > 1, so the solutions are (-3, -1/2) and (1,∞)

-3 -1/2 1

(-)

(-)(-) und. 0 und.(+)(-)

(-)

(+)(-)

(+) (+)

(+)(+)

- + - +

Page 13: Solving Inequalities

Example 6 Solve by Combining Fractions

• Solve (5/(x+3))+(3/(x-1)) < 0

5

x+3+

3

x-1< 0 Original Inequality

(x+3)(x-1)

5(x-1)+

(x+3)(x-1)

3(x+3)< 0 Use LCD to rewrite fractions

(x+3)(x-1)

5(x-1) + 3(x+3)< 0 Add Fractions

Page 14: Solving Inequalities

Example 6 cont.

5x-5+3x+9

(x+3)(x-1)< 0 Distributive property

(x+3)(x-1)

8x+4< 0 Simplify

(x+3)(x-1)

2x+1< 0 Divide both sides by 4

Solution: (-∞, -3) and (-1/2, 1).

Page 15: Solving Inequalities

Example 7 Inequalities Involving Radicals

• Solve (x-3)√(x+1) ≥ 0.

• Because of the factor √(x+1), f(x) is undefined if x < -1.• The zeros of f are 3 and -1.• Sign Chart:

• Solution: {-1} and [3, ∞)

0 0

Undefined Negative Positive

(-)(+) (+)(+)

-1 3

Page 16: Solving Inequalities

Example 8 Inequalities with Absolute Value

• Solve x-2

• Because x+3 is in the denominator, f(x) is undefined if x = -3.

• The only zeros of f is 2.

• Solution: (-∞, -3) and (-3,2]

x+3≤ 0

Negative Negative Positive

(-)

-

(-)

+

(+)

+

Page 17: Solving Inequalities

Matching GameThe link for the game can be found here

http://quizlet.com/18669267/scatter/

Grading Scale

A = 60 seconds or lessB = in between 60.1 and 90 secondsC = in between 90.1 and 120 secondsD = in between 120.1 and 150 secondsF = Anything greater than 150.1 seconds

Page 18: Solving Inequalities

Quiz1.) Combine the fraction and reduce your answer to lowest terms.

x2+5/xA.) (x + 5)/x3

B.) (x3 + 5)/xC.) (x + 5)3/x

2.) Which one of these is a possible rational zero of the polynomial.2x3+x2-4x-3

A.) ±4B.) ±2C.) ±3D.) All the above

3.) Determine the x values that cause the polynomial function to be a zero.f(x) = (2x2+5)(x-8)2(x+1)3

A.) 8B.) -1C.) 5

D.) A and BE.) All the above

Page 19: Solving Inequalities

Quiz Page 24.) The graph of f(x) = x4(x+3)2(x-1)3 changes sign at x = 0.

A.) TrueB.) False

5.) Which is a solution to x2 < xA.) (1, ∞)B.) (0,1)C.) (0, ∞)

6.) Solve the inequality. x|x - 2| > 0A.) (0,2)U(2,∞)B.) (-∞, 2)U(2,∞)C.) None of these answers

7.) Solve the polynomial inequality. x3 - x2 - 2x ≥ 0A.) [-2,0]U[1,∞)B.) [-1,0]U[2,∞)C.) [0,1]U[2,∞)

Page 20: Solving Inequalities

Quiz Page 38.) Complete the factoring if needed and solve the polynomial inequality.

(x + 1)(x2 - 3x + 2) < 0

A.) [-1,0]U[2,∞)B.) (-∞,0)U(2,3)

C.) (-∞,-1)U(1,2)

9.) Dunder Mifflin Paper Company wishes to design paper boxes with a volume of not more than 100 in3. Squares are to be cut from the corners of a 12-in. by 15-in. piece of cardboard, with the flaps folded up to make an open box. What size squares should be cut from the cardboard.

A.) 0 in. ≤ x ≤ 0.69 in.B.) 0 in. ≥ x ≥ 0.69 in.C.) 4.20 ≤ x ≤ 6 in.

D.) 4.20 ≥ x ≥ 6 in.E.) A and CF.) B and D

10.) Solve the polynomial inequality. 2x3 - 5x2 - x + 6 > 0A.) (-1, 3/2)U(2,∞)B.) [-1, 3/2]U[2,∞]C.) (-1, 3/2]U[2,∞)

Page 21: Solving Inequalities

Answer Key1.) B 2.) C 3.) D 4.) False 5.) B 6.) A 7.) B 8.) C 9.) E 10.)

Page 22: Solving Inequalities

Work Cited• Precalculus Graphical, Numerical, Algebraic; Eighth Edition

• https://www.youtube.com/watch?v=_J7xwaOrnf8 (malakai333)

• www.graphsketch.com

• http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

• www.quizlet.com