MAT 171 3.7 Polynomial and Rational Inequalities A. Quadratic Inequalities Mar 15 10:35 AM B. Polynomial Inequalities Solving Polynomial Inequalities Given f (x) is a polynomial in standard form, 1. Write f in completely factored form. 2. Plot real zeroes on the x axis, noting their multiplicity. If the multiplicity is odd the function will change sign. If the multiplicity is even, there will be no change in sign. 3. Use the end behavior to determine the sign of f in the outermost intervals, then label the other intervals as f(x) < 0 or f(x) > 0 by analyzing the multiplicity of neighboring zeroes. 4. State the solution in interval notation. Mar 15 10:48 AM 1
C. Rational Inequalities D. Solving Function Inequalities Using Interval Tests E. Applications of Inequalities 2
311/8. Solve each quadratic inequality by locating the x intercept( s) ( if they 311/12. Solve each quadratic inequality by locating the x intercept( s) ( if they 3
312/22. Solve each quadratic inequality by locating the x intercept( s) ( if they 312/26. Solve each quadratic inequality by locating the x intercept( s) ( if they 4
312/28. Solve each quadratic inequality by locating the x intercept( s) ( if they 312/30. Solve each quadratic inequality by locating the x intercept( s) ( if they 5
312/34. Recall that for a square root expression to represent a real number, the radicand must be greater than or equal to zero. Applying this idea results in an inequality that can be solved using the skills from this section. Determine the domain of the following radical functions. 312/36. Recall that for a square root expression to represent a real number, the radicand must be greater than or equal to zero. Applying this idea results in an inequality that can be solved using the skills from this section. Determine the domain of the following radical functions. 6
312/42. Solve the inequality indicated using a number line and the behavior of the graph at each zero. Write all answers in interval notation. 312/48. Solve the inequality indicated using a number line and the behavior of the graph at each zero. Write all answers in interval notation. Mar 17 11:42 AM 7
312/70. Solve the inequality indicated using a number line and the behavior of the graph at each zero. Write all answers in interval notation. Mar 17 11:42 AM 312/72. Solve the inequality indicated using a number line and the behavior of the graph at each zero. Write all answers in interval notation. Mar 17 11:42 AM 8
314/95. Average speed for a round trip: Surprisingly, the average speed of a round trip is not the sum of the average speed in each direction divided by two. For a fixed distance D, consider rate r 1 in time t 1 for one direction, and rate r 2 in time t 2 for the other, giving r 1 = D/t 1 and r 2 = D/t 2. The average speed for the round trip is R = 2D / (t 1 + t 2 ). The distance from St. Louis, Missouri, to Springfield, Illinois, is approximately 80 mi. Suppose that Sione, due to the age of his vehicle, made the round trip with an average speed of 40 mph. a. Use the relationships stated to verify that r 2 = 20r 1 / (r 1 20). b. Discuss the meaning of the horizontal and vertical asymptotes in this context. c. Verify algebraically the speed returning would be greater than the speed going for 20 < r 1 < 40. In other words, solve the inequality 20r 1 / (r 1 20) > r 1 using the ideas from this section. Mar 17 11:42 AM Mar 18 9:14 AM 9
314/96. The distance from Boston, Massachusetts, to Hartford, Connecticut, is approximately 100 mi. Suppose that Stella, due to excellent driving conditions, made the round trip with an average speed of 60 mph. a. Use the relationships above to verify that r 1 = 30r 1 / (r 1 30). b. Discuss the meaning of the horizontal and vertical asymptotes in this context. c. Verify algebraically the speed returning would be greater than the speed going for 30 < r 1 < 60. In other words, solve the inequality 30r 1 / (r 1 30) > r 1 using the ideas from this section. Mar 17 11:42 AM Mar 18 10:24 AM 10
Mar 18 10:33 AM Mar 18 10:36 AM 11