1 / 22

Solving Inequalities

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.

irving
Télécharger la présentation

Solving Inequalities

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Solving Inequalities By: Sam Milkey and Noah Bakunowicz

  2. 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.

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

  4. 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 Positive • Zeros: -2, -1, 5 • Negative: (∞, -2) (-1, 5) • Positive: (-2,-1) (5,∞) --- +-- ++- +++ - + - + -2 -1 5

  5. Example 2 Solving Algebraically • Solve 2x³-7x²-10x+24>0 Analytically • Use the rational zeros theorem to find possible rational zeros • ±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.

  6. Example 2 cont. • Using synthetic division • Factor 2x²+x-6 • (2x-3)(x+2) • So f(x)=(x-4)(2x-3)(x+2) • Zeros= 4, 3/2, -2 2 -7 -10 4 24 F(x)=(x-4)(2x²+x-6) 8 4 -24 2 1 -6 0

  7. 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,∞) Sign Change Sign Change Sign Change - + - + 4 -2 3/2

  8. Example 3 Solving Graphically • Solve x3-6x2 ≤ 2-8x graphically • Rewrite the inequality so it is less than or equal to 0 • x3-6x2+8x-2 ≤ 0 • Type in x3-6x2+8x-2 into the y1 of the graph on your calculator • 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 • Solution: (-∞,0.32] and [1.46, 4.21] • Remember, use hard brackets because those points are solutions too!

  9. 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 • (x2+7)(2x2+1) > 0 is all real numbers • (x2+7)(2x2+1) ≥ 0 is all real numbers • (x2+7)(2x2+1) < is no solution • (x2+7)(2x2+1) ≤ is no solution

  10. 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 • (x2-3x+3)(2x+5) > 0 is (-∞,-5/2) and (-5/2,∞) • (x2-3x+3)(2x+5) ≥ 0 is all real numbers • (x2-3x+3)(2x+5) < 0 has no solution • (x2-3x+3)(2x+5) ≤ 0 is a single number, -5/2

  11. 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

  12. 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,∞) (-) (-) (+) (+) und. und. (-)(-) (+)(-) (+)(-) (+)(+) 0 - + - + -3 -1/2 1

  13. Example 6 Solve by Combining Fractions • Solve (5/(x+3))+(3/(x-1)) < 0 5 3 < 0 Original Inequality + x+3 x-1 5(x-1) 3(x+3) < 0 Use LCD to rewrite fractions + (x+3)(x-1) (x+3)(x-1) 5(x-1) + 3(x+3) < 0 Add Fractions (x+3)(x-1)

  14. Example 6 cont. 5x-5+3x+9 Distributive property < 0 (x+3)(x-1) 8x+4 Simplify < 0 (x+3)(x-1) 2x+1 Divide both sides by 4 < 0 (x+3)(x-1) Solution: (-∞, -3) and (-1/2, 1).

  15. 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 3 -1

  16. 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] ≤ 0 x+3 (-) (-) (+) - + + Negative Negative Positive

  17. 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

  18. 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

  19. 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,∞)

  20. 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,∞)

  21. Answer Key 1.) B 2.) C 3.) D 4.) False 5.) B 6.) A 7.) B 8.) C 9.) E 10.)

  22. 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

More Related