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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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Solving InequalitiesBy: 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.
• To solve f(x) < 0 is to find the values of x that make f(x) negative.
But that’s pretty boring.https://www.youtube.com/watch?v=_J7xwaOrnf8(skip to 1:00)
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
---
-+--
+++-
-+++
+
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.
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)
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
- + - +
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!
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
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
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
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.(+)(-)
(-)
(+)(-)
(+) (+)
(+)(+)
- + - +
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
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).
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
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
(-)
-
(-)
+
(+)
+
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
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
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,∞)
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,∞)
Answer Key1.) B 2.) C 3.) D 4.) False 5.) B 6.) A 7.) B 8.) C 9.) E 10.)
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