322 Chapter 7 Rational Functions

Lesson 7.5 Solving Rational EquationsWhile driving to a convention, Miss Wilson traveled 20 miles at a speed of r miles per hour and then 30 miles at a speed of r + 20 miles per hour. A rational equation can be used to model the total amount of time it took Miss Wilson to drive to the convention. If Miss Wilson’s total trip time to the convention took 1 hour, what were her driving speeds?

Solving Rational Equations

Rational equations are equations that contain variables in the denominator of a rational expression.

Example 1 Solving Rational Equations

Set up and solve a rational equation to find Miss Wilson’s driving speeds for the trip.

Solution

Use the formula d = rt, or t = dr , to model the amount of time each leg of the trip took.

First leg: t1 = 20r Second leg: t2 = 3020r +

Total trip: t = 20 3020r r+ + = 1

Multiply both sides of the equation by r(r + 20) to remove the variable in each denominator. Then solve for r.

r(r + 20) 20 3020r r+ +

= 1(r)(r + 20)

20(r + 20) + 30(r) = r2 + 20r r2 – 30r – 400 = 0 (r + 10)(r – 40) = 0 r = –10, 40

Speed cannot be negative, so Miss Wilson drove 40 miles per hour during the first leg of the trip.

r + 20 = 40 + 20 = 60

Miss Wilson drove 60 miles per hour during the second leg of the trip.

ObjectivesSolve rational equations.

7.5 Solving Rational Equations 323

Ongoing AssessmentThe total resistance of a parallel circuit is 8 ohms. There are two resistors in the parallel circuit. The value of the first resistor is half that of the second resistor. Use the equation 18

1 12= r r+ to solve for the resistance r, in ohms, of

the first resistor and for the resistance 2r of the second resistor.

Extraneous Solutions

Extraneous solutions can be introduced into an equation when multiplying both sides of the equation by the same algebraic expression. An extraneous solution is a solution to an equation derived from the original equation, but not a solution to the original equation. To determine whether a solution is extraneous, substitute it into the original equation and see if a true number sentence is formed.

Example 2 Solving Rational Equations with Extraneous Solutions

Solve the equation 86 12

442x x− −

= . Check for extraneous solutions.

Solution

Multiply both sides of the equation by (x2 – 4)(6x – 12) to eliminate the fractions.

(x2 – 4)(6x – 12) 86 12x − = 4

42x − (x2 – 4)(6x – 12) 8(x2 – 4) = 4(6x – 12)

8x2 – 32 = 24x – 48 8x2 – 24x + 16 = 0

Solve the quadratic equation by factoring.

8(x2 – 3x + 2) = 0 8(x – 2)(x – 1) = 0 x = 1, 2

Check the solutions. When 1 is substituted for x in the original rational equation, both sides equal − 43 . When 2 is substituted for x in the original equation, both denominators equal 0. So x = 1 is a solution, and x = 2 is an extraneous solution.

Ongoing Assessment

Solve the equation 121

2015 152x x− +

= . Check for extraneous solutions.

324 Chapter 7 Rational Functions

Ron can take inventory of medicines in a pharmacy in x hours. It takes Kenneth x + 4 hours to do the same job. Together, they can take the inventory in 4.8 hours. How long would it take each person to take the inventory individually?

Step 1 Understand the Problem

What information is given in the problem statement? What are you being asked to find?

Step 2 Develop a Plan

Problem-solving strategy: Use an equation.

Write a rational equation to model the situation. What part of the job does Ron do when they work together? What part

of the job does Kenneth do when they work together? What is the sum of these two expressions?

Step 3 Carry Out the Plan

Write and solve the rational equation to find x, the number of hours it takes Ron to take the inventory on his own. Then add 4 to find how long it takes Kenneth to take the inventory working alone.

Step 4 Check the Results

Substitute your result into the rational equation and make sure you obtain a true number sentence.

Lesson AssessmentThink and Discuss

1. What is a rational equation?

2. How could a graphing calculator be used to find the solutions of a rational equation?

3. Describe how to solve the equation x x+ 38 5

= . Then solve the equation and check the solutions.

4. Why is it important to check the roots of a rational equation by substituting them into the original equation?

5. What is an extraneous solution to a rational equation?

7.5 Solving Rational Equations 325

Practice and Problem Solving

Solve each equation for the variable. Check the solutions.

6. x x7

29

= + 7. a a5

7+2=

8. 2 19

2c c− +5= 9. 4 12

4b b=

+

10. xx

+−

43

21

= 11. 62 9

106 7y y+ +

=

12. p

p54

2 3=

− 13. g g+ −6 5=

14. 23 5

415t t− −

= 15. 62

53x x− −

=

16. 62

10 13w w+ = 17. 4 1n n

− = 3+1

18. 2 12

22 5

6 56

k kk

k− − ++

−= 19. 12

42x

x xx

+ +=

Solve each equation for the variable. Check for and identify any extraneous solutions.

20. 43 3

1212a a+ −

= 21. 113

42

− −xx x

=

22. 43 2

1292x

xx+

+ −−

= 23. 23 3

1892

ss

ss s+

+− −

=

24. Gina can complete a puzzle in 6 hours working alone. Hannah can complete the same puzzle in 10 hours working alone. How long would it take them to complete the puzzle working together? Set up and solve a rational equation.

25. Nancy has three hoses available to fill her goldfish pond. Individually, the hoses can fill the pond in 12 hours, 8 hours, and 6 hours, respectively. How long would it take to fill the pond if all three hoses are turned on simultaneously?

26. Reggie can install an air conditioner in 6 hours working alone. Frank can install the unit in 4 hours working alone. How long would it take them to install the air conditioner working together? Set up and solve a rational equation.

326 Chapter 7 Rational Functions

27. Carrie drove several hours last weekend to visit a college. She drove the first 260 miles of the trip at an average speed of r miles per hour. Then she drove through a 20-mile construction zone at an average speed of r – 25 miles per hour for the remainder of the trip.

a. Write rational expressions for the number of hours it took Carrie to drive each leg of the trip. 260 20

25r r,

−

b. Suppose the total trip took 4 hours 30 minutes to drive. Solve an equation for Carrie’s average driving speeds on the trip.

c. How many solutions are there to the rational equation? How did you decide which one to use in the solution? Explain.

28. Carla and Danielle are stuffing and addressing envelopes for a community fundraiser. Carla works twice as fast as Danielle. Together they need to address and stuff a total of 630 envelopes.

a. Write rational expressions for the number of envelopes completed by both volunteers in x hours.

b. Working together, Carla and Danielle can stuff and address the envelopes in 10 hours. Solve a rational equation for the number of hours it would take each of Carla and Danielle to stuff and address all 4,000 envelopes on their own.

Mixed Review

Classify each number in as many ways as possible.

29. 7 30. −34

31. 0 32. 152

33. As a car ages, its value decreases at an annual rate. Roberto bought a truck worth $24,650. The value of the truck decreased by 10.5% each year.

a. Write a formula that models the value of Roberto’s truck after t years.

b. Solve the truck value formula for the number of years, t, since Roberto bought the vehicle.

c. About how many years will it take for Roberto’s truck to be worth half its original value? Round to the nearest tenth.