1. Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Equations and Inequalities
Copyright © 2017, 2013, 2009 Pearson Education, Inc. 1

2. 1.6 Other Types of Equations and Applications Rational Equations
Work Rate Problems
Equations with Radicals
Equations with Rational Exponents
Equations Quadratic in Form

3. Rational Equations
A rational equation is an equation that has a rational expression for one or more terms. To solve a rational equation, multiply each side by the least common denominator (LCD) of the terms of the equation and then solve the resulting equation. Because a rational expression is not defined when its denominator is 0, proposed solutions for which any denominator equals 0 are excluded from the solution set.

4. Solve each equation. (a) Solution
SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS
Example 1
Solve each equation.
(a)
Solution
The least common denominator is 3(x – 1), which is equal to 0 if x = 1. Therefore, 1 cannot possibly be a solution of this equation.

5. Solve each equation. (a) Solution Example 1
SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS
Example 1
Solve each equation.
(a)
Solution
Multiply by the LCD, 3(x – 1), where x ≠ 1.

6. Solve each equation. (a) Solution Example 1
SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS
Example 1
Solve each equation.
(a)
Solution
Divide out common factors.
Multiply.
Subtract 3x2; combine like terms.
Solve the linear equation.

7. Solve each equation. (a) Solution
SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS
Example 1
Solve each equation.
(a)
Solution
Proposed solution
The proposed solution meets the requirement that x ≠ 1 and does not cause any denominator to equal 0. Substitute to check for correct algebra.

8. Solve each equation. (b) Solution Example 1
SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS
Example 1
Solve each equation.
(b)
Solution
Multiply by the LCD, x – 2, where x ≠ 2.
Divide out common factors.

9. Solve each equation. (b) Solution Example 1
SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS
Example 1
Solve each equation.
(b)
Solution
Distributive property
Solve the linear equation.
Proposed solution.

10. Solve each equation. (b) Solution
SOLVING RATIONAL EQUATIONS THAT LEAD TO LINEAR EQUATIONS
Example 1
Solve each equation.
(b)
Solution
The proposed solution is 2. However, the variable is restricted to real numbers except 2. If x = 2, then not only does it cause a zero denominator, but multiplying by x – 2 in the first step is multiplying both sides by 0, which is not valid. Thus, the solution set is .

11. Solve each equation. (a) Solution Example 2
SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS
Example 2
Solve each equation.
(a)
Solution
Factor the last denominator.
Multiply by x(x – 2), x ≠ 0, 2.

12. Solve each equation. (a) Solution Example 2
SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS
Example 2
Solve each equation.
(a)
Solution
Divide out common factors.
Distributive property
Standard form
Factor.

13. Set each factor equal to 0.
SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS
Example 2
Solve each equation.
(a)
Solution
Zero-factor property
Set each factor equal to 0. or
Proposed solutions or
Because of the restriction x ≠ 0, the only valid solution is – 1. The solution set is {– 1}.

14. Solve each equation. (b) Solution Example 2
SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS
Example 2
Solve each equation.
(b)
Solution
Factor.
Multiply by (x + 1)(x – 1), x ≠ ±1.

15. Solve each equation. (b) Solution Example 2
SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS
Example 2
Solve each equation.
(b)
Solution
Divide out common factors.
Distributive property
Standard form
Divide by – 4.

16. Neither proposed solution is valid, so the solution set is .
SOLVING RATIONAL EQUATIONS THAT LEAD TO QUADRATIC EQUATIONS
Example 2
Solve each equation.
(b)
Solution
Factor. or
Zero-factor property or
Proposed solutions
Neither proposed solution is valid, so the solution set is .

17. Work Rate Problems
Problem Solving If a job can be done in t units of time, then the rate of work, r, is 1/t of the job per unit time.
The amount of work completed, A, is found by multiplying the rate of work, r, and the amount of time worked, t. This formula is similar to the distance formula, d = rt.
Amount of work completed
= rate of work × amount of time worked or

18. SOLVING A WORK RATE PROBLEM
Example 3
One printer can do a job twice as fast as another. Working together, both printers can do the job in 2 hr. How long would it take each printer, working alone, to do the job?

19. SOLVING A WORK RATE PROBLEM
Example 3
Solution
Step 1 Read the problem. We must find the time it would take each printer, working alone, to do the job.

20. SOLVING A WORK RATE PROBLEM
Example 3
Solution
Step 2 Assign a variable. Let x represent the number of hours it would take the faster printer, working alone, to do the job. The time for the slower printer to do the job alone is then 2x hours. Therefore,
and

21. Part of the Job Accomplished
SOLVING A WORK RATE PROBLEM
Example 3
Solution
Step 2 Assign a variable.
Rate
Time
Part of the Job Accomplished
Faster Printer 2
Slower Printer
A = r t

22. SOLVING A WORK RATE PROBLEM
Example 3
Solution
Step 3 Write an equation. The sum of the two parts of the job accomplished is 1, since one whole job is done.
Part of the job done by the faster printer
Part of the job done by the slower printer
One whole job + =

23. Solution Step 4 Solve. SOLVING A WORK RATE PROBLEM Example 3
Multiply each side by x, where x ≠ 0.
Distributive property
Multiply.
Add.

24. SOLVING A WORK RATE PROBLEM
Example 3
Solution
Step 5 State the answer. The faster printer would take 3 hr to do the job alone, and the slower printer would take 2(3) = 6 hr. Be sure to give both answers here.
Step 6 Check. The answer is reasonable, since the time working together (2 hr) is less than the time it would take the faster computer working alone (3 hr).

25. Rate Problem
Note Example 3 can also be solved by using the fact that the sum of the rates of the individual printers is equal to their rate working together. Since the printers can complete the job together in 2 hr, their combined rate is 1/2 of the job per hour.

26. Rate Problem Note Multiply each side by 2x. Distributive property
Same solution found earlier

27. Power Property
If P and Q are algebraic expressions, then every solution of the equation
P = Q is also a solution of the equation Pn = Qn , for any positive integer n.

28. Caution Be very careful when using the power property
Caution Be very careful when using the power property. It does not say that the equations P = Q and Pn = Qn are equivalent; it says only that each solution of the original equation P = Q is also a solution of the new equation Pn = Qn.

29. Solving an Equation Involving Radicals
Step 1 Isolate the radical on one side of the equation.
Step 2 Raise each side of the equation to a power that is the same as the index of the radical so that the radical is eliminated.
If the equation still contains a radical, repeat Steps 1 and 2.
Step 3 Solve the resulting equation.
Step 4 Check each proposed solution in the original equation.

30. Solve Solution Example 4
SOLVING AN EQUATION CONTAINING A RADICAL (SQUARE ROOT)
Example 4
Solve
Solution
Isolate the radical.
Square each side.
Solve the quadratic equation.

31. Only 3 is a solution, so the solution set is {3}.
SOLVING AN EQUATION CONTAINING A RADICAL (SQUARE ROOT)
Example 4
Solve
Solution
Factor. or
Zero-factor property or
Proposed solutions
Only 3 is a solution, so the solution set is {3}.

32. SOLVING AN EQUATION CONTAINING TWO RADICALS
Example 5
Solve
Solution When an equation contains two radicals, begin by isolating one of the radicals on one side of the equation.
Square each side.

33. Don’t forget this term when squaring.
SOLVING AN EQUATION CONTAINING TWO RADICALS
Example 5
Solve
Solution
Be careful!
Don’t forget this term when squaring.
Isolate the remaining radical.
Square again.

34. Solve Solution or or SOLVING AN EQUATION CONTAINING TWO RADICALS
Example 5
Solve
Solution
Apply the exponents.
Distributive property
Solve the quadratic equation.
Factor or
Zero-factor property or
Proposed solutions

35. SOLVING AN EQUATION CONTAINING TWO RADICALS
Example 5
Solve
Solution or
Proposed solutions
Check each proposed solution in the original equation. Both 3 and – 1 are solutions of the original equation, so {– 1,3} is the solution set.

36. Caution Remember to isolate a radical in Step 1
Caution Remember to isolate a radical in Step 1. It would be incorrect to square each term individually as the first step in Example 5.

37. Solve Solution Example 6
SOLVING AN EQUATION CONTAINING A RADICAL (CUBE ROOT)
Example 6
Solve
Solution
Isolate a radical.
Cube each side.
Apply the exponents.
Solve the quadratic equation.

38. Both are valid solutions, and the solution set is {1/4 ,1}.
SOLVING AN EQUATION CONTAINING A RADICAL (CUBE ROOT)
Example 6
Solve
Solution
Factor. or
Zero-factor property or
Proposed solutions
Both are valid solutions, and the solution set is {1/4 ,1}.

39. Solve each equation. (a) Solution The solution set is {243}.
SOLVING EQUATIONS WITH RATIONAL EXPONENTS
Example 7
Solve each equation.
(a)
Solution
Raise each side to the power 5/3, the reciprocal of the exponent of x.
The solution set is {243}.

40. Solve each equation. (b) Solution
SOLVING EQUATIONS WITH RATIONAL EXPONENTS
Example 7
Solve each equation.
(b)
Solution
Raise each side to the power 3/2. Insert ± since this involves an even root, as indicated by the 2 in the denominator.
Proposed solutions
Both proposed solutions check in the original equation,
so the solution set is {–60,68}.

41. Equation Quadratic in Form
An equation is said to be quadratic in form if it can be written as
where a ≠ 0 and u is some algebraic expression.

42. (a) Solve Solution Since or or SOLVING EQUATIONS QUADRATIC IN FORM
Example 8
(a)
Solve
Solution
Since
Substitute.
Factor.
Zero-factor property or or

43. SOLVING EQUATIONS QUADRATIC IN FORM
Example 8
(a)
Solve
Don’t forget this step. or or
Cube each side. or or
Proposed solutions
Both proposed solutions check in the original equation, so the solution set is {– 2, 7}.

44. Remember to substitute for u.
SOLVING EQUATIONS QUADRATIC IN FORM
Example 8
(b)
Solve
Solution
Subtract 2 from each side.
Factor.
Zero-factor property or
Remember to substitute for u. or

45. (b) Solve or Solution or or SOLVING EQUATIONS QUADRATIC IN FORM
Example 8
(b)
Solve or
Solution or
Substitute again.
x-1 is the reciprocal of x. or

46. Caution When using a substitution variable in solving an equation that is quadratic in form, do not forget the step that gives the solution in terms of the original variable.

47. Solve Solution or SOLVING AN EQUATION QUADRATIC IN FORM Example 9
Let u = x2. Then u2 = x4.
Solve the quadratic equation. or
Zero-property factor

48. Solve Solution or or or SOLVING AN EQUATION QUADRATIC IN FORM
Example 9
Solve
Solution or or
Replace u with x2. or
Square root property

49. Solve Solution or or SOLVING AN EQUATION QUADRATIC IN FORM Example 9
Simplify radicals. or

50. Equation Quadratic in Form
Note Some equations that are quadratic in form are simple enough to avoid using the substitution variable technique. To solve
we could factor directly as (3x2 – 2)(4x2 – 1), set each factor equal to zero, and then solve the resulting two quadratic equations. Which method to use is a matter of personal preference.