4.7 More Equations and Inequalities

1. 4.7 More Equations and Inequalities Solve rational equations Solve variation problems Solve polynomial inequalities Solve rational inequalities

2. Rational Equations If f(x) represents a rational function, then an equation that can be written in the form f(x) = k for some constant k is a rational equation.

3. Example: Solving a rational equation (1 of 3) Solve symbolically, graphically and numerically. Solution Symbolically

4. Example: Solving a rational equation (2 of 3) Graphically Graph Y1 = 4X/(X −1) and Y2 = 6. Their graphs intersect at (3, 6), so the solution is 3.

5. Example: Solving a rational equation (3 of 3) Numerically Enter Y1 = 4X/(X −1) and Y2 = 6. Select Table on the calculator to see this display. y1 = y2 when x = 3, so the solution is 3.

6. Direct Variation with the nth Power Let x and y denote two quantities and n be a positive number. Then y is directly proportional to the nth power of x, or y varies directly as the nth power of x, if there exists a nonzero number k such that

7. Example: Modeling a pendulum (1 of 2) The time T required for a pendulum to swing back and forth once is called its period. The length L of a pendulum is directly proportional to the square of T. A 2-foot pendulum has a 1.57-second period. a. Find the constant of proportionality k. Round to the nearest hundredth. b. Predict T for a pendulum having a length of 5 feet.

8. Example: Modeling a pendulum (2 of 2) a. Lis directly proportional to the square of T, so L= kT². L = 2 and T = 1.57, thus b. If L = 5, then 5 = 0.81T². It follows

9. Inverse Variation with the nth Power Let x and y denote two quantities and n be a positive real number. Then y is inversely proportional to the nth power of x, or y varies inversely as the nth power of x, if there exists a nonzero number k such that

10. Example: Modeling the intensity of light (1 of 2) At a distance of 3 meters, a 100-watt bulb produces an intensity of 0.88 watt per square meter. a. Find the constant of proportionality k. b. Determine the intensity at a distance of 2 meters.

11. Example: Modeling the intensity of light (2 of 2) Solution The intensity at 2 meters is 1.98 watts per square meter.

12. Polynomial Inequalities (1 of 2) In Section 3.4 a strategy for solving quadratic inequalities was presented. This strategy involves first finding boundary numbers (x-values) where equality holds. Once the boundary numbers are known, a graph or a table of test values can be used to determine the intervals where inequality holds. This strategy can be applied to other types of inequalities.

13. Polynomial Inequalities (2 of 2)

14. Solving Polynomial Inequalities (1 of 3) STEP 1: If necessary, write the inequality as p(x) < 0, where p(x) is a polynomial and the inequality symbol < may be replaced by >, ≤, or ≥. STEP 2: Solve p(x) = 0. The solutions are called boundary numbers.

15. Solving Polynomial Inequalities (2 of 3) STEP 3: Use the boundary numbers to separate the number line into disjoint intervals. On each interval, p(x) is either always positive or always negative.

16. Solving Polynomial Inequalities (3 of 3) STEP 4: To solve the inequality, either make a table of test values for p(x) or use a graph of y = p(x). For example, the solution set for p(x) < 0 corresponds to intervals where test values result in negative outputs or to intervals where the graph of y = p(x) is below the x-axis.

17. Example: Solving a polynomial inequality (1 of 5) Solve x³ > 2x² symbolically and graphically. Solution Symbolically Step 1: Write the inequality as x³ − 2x² > 0. Step 2:Replace the >symbol with an equals sign and solve the resulting equation. x³ − 2x² = 0

18. Example: Solving a polynomial inequality (2 of 5) Step 2: The boundary numbers are 0 and 2. Note that 0 is a zero of x³ − 2x²with multiplicity 2, which is even.

19. Example: Solving a polynomial inequality (3 of 5)

20. Example: Solving a polynomial inequality (4 of 5) Step 4: Complete a table of test values.

21. Example: Solving a polynomial inequality (5 of 5) Graphically Graph y = x³ − 2x²

22. Rational Inequalities To solve rational inequalities, we can use the same basic techniques that we used to solve polynomial inequalities, with one important modification: boundary numbers also occur at x-values where the denominator of any rational expression in the inequality equals 0.

23. Example: Modeling customers in a line (1 of 3) A ticket booth attendant can wait on 30 customers per hour. To keep the time waiting in line reasonable, the line length should not exceed 8 customers on average. Solve the inequality to determine the rates x at which customers can arrive before a second attendant is needed. Note that the x-values are limited to 0 ≤ x < 30.

24. Example: Modeling customers in a line (2 of 3) Solution Graph Y1 = X^2/(900 −30X) and Y2 = 8

25. Example: Modeling customers in a line (3 of 3) The only point of intersection on this interval is near (26.97, 8). We conclude that if the arrival rate is about 27 customers per hour or less, then the line length does not exceed 8 customers on average. If the arrival rate is more than 27 customers per hour, a second ticket booth attendant is needed.

26. Solving Rational Inequalities (1 of 2) STEP 2: Solve p(x) = 0 and q(x) = 0. The solutions are boundary numbers.

27. Solving Rational Inequalities (2 of 2) STEP 4: Use a table of test values or a graph to solve the inequality in Step 1.

28. Example: Solving a rational inequality symbolically (1 of 3)

29. Example: Solving a rational inequality symbolically (2 of 3) Step 2:Find the zeros of the numerator and the denominator.

30. Example: Solving a rational inequality symbolically (3 of 3) Step 4: Use a table to solve the inequality.