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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. 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: --+-+++++ - -2 • + -1 - 5 + Find when it is Zero, Negative, and Positive o Zeros: -2, -1, 5 o Negative: (∞, -2) (-1, 5) o Positive: (-2,-1) (5,∞) Example 2 Solving Algebraically • • • Solve 2x³-7x²-10x+24>0 Analytically Use the rational zeros theorem to find possible rational zeros o ±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 4 • • • 2 2 -7 -10 24 8 4 -24 1 -6 0 F(x)=(x-4)(2x²+x-6) Factor 2x²+x-6 o (2x-3)(x+2) So f(x)=(x-4)(2x-3)(x+2) Zeros= 4, 3/2, -2 Example 2 cont. • Sign Chart Sign Change - -2 Sign Change + 3/2 Sign Change - 4 + • 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,∞) Example 3 Solving Graphically • • • • Solve x3-6x2 ≤ 2-8x graphically Rewrite the inequality so it is less than or equal to 0 o x3-6x2+8x-2 ≤ 0 Type in x3-6x2+8x-2 into the y1 of the graph on your calculator o 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 graph o 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 Potential Sign Change Potential Sign Change Potential Sign Change -3 -1/2 1 Example 5 cont. • Sign chart with undefined, zeros, positive, and negative (-) (-)(-) • • (-) und. (+)(-) -3 + (+) 0 -1/2 (+)(-) - (+) und. 1 (+)(+) + 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,∞) Example 6 Solve by Combining Fractions • Solve (5/(x+3))+(3/(x-1)) < 0 5 3 x+3 + 5(x-1) (x+3)(x-1) x-1 < 0 3(x+3) + (x+3)(x-1) 5(x-1) + 3(x+3) (x+3)(x-1) <0 Original Inequality <0 Use LCD to rewrite fractions Add Fractions Example 6 cont. 5x-5+3x+9 (x+3)(x-1) <0 Distributive property 8x+4 (x+3)(x-1) <0 Simplify <0 Divide both sides by 4 2x+1 (x+3)(x-1) 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: 0 Undefined • -1 (-)(+) Negative Solution: {-1} and [3, ∞) 0 3 (+)(+) Positive Example 8 Inequalities with Absolute Value • Solve x-2 x+3 • • ≤0 Because x+3 is in the denominator, f(x) is undefined if x = -3. The only zeros of f is 2. (-) Negative (-) + Negative (+) + Positive • Solution: (-∞, -3) and (-3,2] Matching Game The link for the game can be found here http://quizlet.com/18669267/scatter/ Grading Scale A = 60 seconds or less B = in between 60.1 and 90 seconds C = in between 90.1 and 120 seconds D = in between 120.1 and 150 seconds F = Anything greater than 150.1 seconds Quiz 1.) Combine the fraction and reduce your answer to lowest terms. x2+5/x A.) (x + 5)/x3 B.) (x3 + 5)/x C.) (x + 5)3/x 2.) Which one of these is a possible rational zero of the polynomial. 2x3+x2-4x-3 A.) ±4 B.) ±2 C.) ±3 D.) 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.) 8 B.) -1 C.) 5 D.) A and B E.) All the above Quiz Page 2 4.) The graph of f(x) = x4(x+3)2(x-1)3 changes sign at x = 0. A.) True B.) False 5.) Which is a solution to x2 < x A.) (1, ∞) B.) (0,1) C.) (0, ∞) 6.) Solve the inequality. x|x - 2| > 0 A.) (0,2)U(2,∞) B.) (-∞, 2)U(2,∞) C.) None of these answers 7.) Solve the polynomial inequality. x3 - x2 - 2x ≥ 0 A.) [-2,0]U[1,∞) B.) [-1,0]U[2,∞) C.) [0,1]U[2,∞) Quiz Page 3 8.) 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 C F.) B and D 10.) Solve the polynomial inequality. 2x3 - 5x2 - x + 6 > 0 A.) (-1, 3/2)U(2,∞) B.) [-1, 3/2]U[2,∞] C.) (-1, 3/2]U[2,∞) Answer Key 1.) 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_Cal culator/graphCalc.html www.quizlet.com