Chapter 4 More Nonlinear Functions and Equations

1. Chapter 4 More Nonlinear Functions and Equations

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

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

4. Solve symbolically, graphically and numerically. SolutionSymbolically Multiply by the LCD to clear the fractions. Example: Solving a rational equation

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

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

7. Direct Variation as 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 thereexists a nonzero number k such that y=kxn.

8. 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. Example: Modeling a pendulum • (a) Find the constant of proportionality k. • (b) Predict T for a pendulum havinga length of 5 feet.

9. Solution a) L is 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 Example: Modeling a pendulum

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

11. 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. Example: Modeling the intensity of light

12. Solution a) Substitute d = 3 and I = 0.88 in Solve for k. b) Let and d = 2. Then, The intensity at 2 meters is 1.98 watts per square meter. Example: Modeling the intensity of light

13. Polynomial Inequalities An inequality says that one expression is greater than, greater than or equal to, less than, or less than or equal to, another expression. Solving Inequalities Boundary numbers (x-values) are found where equality holds. A graph or a table of test values can be used to determine the intervals where the inequality holds.

14. Solving Polynomial Inequalities 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 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 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. Solve x3 ≥ 2x2 + 3x symbolically and graphically. SolutionSymbolically Step 1: Write the inequality as Step 2: Replace the inequality symbol with an equal sign and solve. Example: Solving a polynomial inequality

18. Step 2: Example: Solving a polynomial inequality The boundary numbers are –1, 0, and 3.

19. Symbolically Step 3: The boundary numbers separate the number line into four disjoint intervals: (–∞, –1), (–1, 0), (0,3), and (3,∞) Example: Solving a polynomial inequality

20. Step 4: Complete a table of test values. Example: Solving a polynomial inequality The solution set is [–1, 0] U [3, ∞).

21. Step 4: (Again) Here’s a graphing calculator display that evaluates the same test values. Example: Solving a polynomial inequality Boundary numbersare included in the solution set because the inequality involves ≥ rather than >. The solution set is [–1, 0] U [3, ∞).

22. Graphically Graph y1 = x3 – 2x2 – 3x. Example: Solving a polynomial inequality • Zeros or x-intercepts are located at –1, and 3. The graph of y1 is positive (or above thex-axis) for –1 < x < 0 or 3 < x < ∞. • If we include the boundary numbers, this resultagrees with the symbolic solution.

23. Rational Inequalities Inequalities involving rational expressions are called rational inequalities.

24. 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. Example: Modeling customers in a line

25. Solution Graph Y1 = X^2/(900 – 30X) and Y2 = 8 Example: Modeling customers in a line • 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.

26. Solving Rational Inequalities STEP 1: If necessary, write the inequalityin the form where p(x) and q(x) are polynomials. Note that > may be replaced by <, ≤, or ≥. STEP 2: Solve p(x)= 0 and q(x)= 0. The solutions are called boundary numbers.

27. Solving Rational Inequalities STEP 3: Use the boundary numbers to separate the number line into disjoint intervals.On eachinterval, is either always positive or always negative. STEP 4: Use a table of test values or a graph to solve the inequality in Step 1.

28. Solve Solution Step 1: Write in the form Example: Solving a rational inequality symbolically

29. Step 2: Find the zeros of the numerator and the denominator. Example: Solving a rational inequality symbolically • Step 3: The boundary numbers are – 1, 0 and 1, which separate the number line into four disjoint intervals:(–∞, –1), (–1, 0), (0, 1), (1, ∞).

30. Step 4: Use a table to solve the inequality. The interval notation is (–1, 0) U [1, ∞).(–1 and 0 make the inequality undefined.) Example: Solving a rational inequality symbolically

31. Rational Inequality Caution Caution: When solving a rational inequality, it is essential not to multiply or divide each side of the inequality by the LCD if the LCD contains a variable. This technique often leads to an incorrect solution set.