Problem 1

Solving a Rational Equation

Solve each rational equation. Check each solution for reasonableness.

A xx ∙ 3 ∙ x

x ∙ 3 ∙ 2x2 ∙ 9

TEKS Process Standard (1)(C)

To solve a rational equation, first multiply each side by the least common denominator of the rational expressions. Doing this, however, can introduce extraneous solutions.

ESSENTIAL UNDERSTANDING

TEKS (6)(I) Solve rational equations that have real solutions.

TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Additional TEKS (1)(A), (6)(H), (6)(J)

TEKS FOCUS

•Rational equation – A rational equation contains at least one rational expression.

•Number sense – the understanding of what numbers mean and how they are related

VOCABULARY

xx ∙ 3 ∙ x

x ∙ 3 ∙ 2(x ∙ 3)(x ∙ 3)

(x ∙ 3)(x ∙ 3)J xx ∙ 3 ∙ x

x ∙ 3 R ∙ (x ∙ 3)(x ∙ 3) 2(x ∙ 3)(x ∙ 3)

x2 ∙ 3x ∙ x2 ∙ 3x ∙ 2 2x2 ∙ 2

x2 ∙ 1, so x ∙ t1

11 ∙ 3 ∙ 1

1 ∙ 3≟ 2

(1)2 ∙ 9 ∙1

∙1 ∙ 3 ∙ ∙1∙1 ∙ 3

≟ 2(∙1)2 ∙ 9

14 ∙ ∙ 1

2 ∙ ∙ 14 ✔∙ 1

2 ∙ 14 ∙ ∙ 1

4 ✔

The solutions are x ∙ 1 and x ∙ ∙1.

Factor the denominators to find the LCD.

Multiply each side by the LCD to clear denominators.

Check for reasonableness. A solution that does not satisfy the original equation is extraneous. In this case, both solutions check.

Now simplify and solve.

11-6 Solving Rational Equations

continued on next page ▶

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San Francisco

Chicago1850 mi

Problem 2

continuedProblem 1

B x ∙ 1x2 ∙ 3x ∙ 2

∙ 2xx ∙ 2

∙ x ∙ 1x ∙ 1

Use a computer algebra system (CAS) to solve this rational equation.

Step 1 On the Home Step 2 Choose Menu, Step 3 Enter the equation, screen, choose Algebra, Solve. followed by a New Document. comma and x. Then select Then press enter Add Calculator. to solve.

The calculator shows a warning because there may be extraneous solutions. The original equation restricts x so that x ≠ -2 and x ≠ -1. There is no solution.

Using Rational Equations

Flight A flight across the U.S. takes longer east to west than it does west to east. Assume that winds are constant in the eastward direction. When flying westward, the headwind decreases the airplane’s speed. When flying eastward, the tailwind increases its speed. The time for a round trip shown at the right is 73

4h. If the airplane cruises at 480 mi/h, what is the speed of the wind?

Let x = the wind speed.

Rate * Time = Distance, so Time = DistanceRate

1850

1850 480 � x1850

480 � x1850

480 � x480 � x

Going west to east

Going east to west

Distance Rate Time

continued on next page ▶

How can you use technology to solve this equation?You can use a computer algebra system to solve the equation. Calculators have limitations, and you will still need to check the reasonableness of the solutions to eliminate any extraneous solutions.

492 Lesson 11-6 Solving Rational Equations

Problem 3

How do the graphs of the two sides of the equation help you solve the equation?The x-values of the points of intersection are the solutions to the equation.

continuedProblem 2

Total time = Time west to east + Time east to west

7.75 = 1850480 + x + 1850

480 - x

(480 + x)(480 - x) 7.75 = (480 + x)(480 - x) 1850480 + x + (480 + x)(480 - x) 1850

480 - x

7.75(480 + x)(480 - x) = 1850(480 - x) + 1850(480 + x)

1,785,600 - 7.75x2 = 888,000 - 1850x + 888,000 + 1850x

-7.75x2 = -9600

x2 = -9600-7.75

x ≈ {35

Wind speed is positive, so the negative solution is not reasonable. The west-to-east wind speed is about 35 mi/h.

Check 7.75 = 1850480 + x + 1850

480 - x

7.75 ≟ 1850480 + 35 + 1850

480 - 35

7.75 ≈ 3.6 + 4.2 ✔

Multiply both sidesby the LCD,(480 + x)(480 − x).

Using a Graphing Calculator to Solve a Rational Equation

What are the solutions of the rational equation? Use a graphing calculator to solve.2

x + 2 + xx − 2 = 1

The solution is x = 0.

Check 2x + 2 + x

x - 2 = 1

20 + 2 + 0

0 - 2≟ 1

1 + 0 = 1 ✔

2.33331–.3333ERROR3.42.33331.9524

X=0

X Y1 Y2–1012345

1111111

Y1 � Y2when x � 0.

Plot1 Plot2 Plot3\Y1\Y2\Y3\Y4\Y5\Y6\Y7

= (2/(X+2))+(X/(X–2))= 1= = = ==

Enter one side of theequation as Y1. Enterthe other side as Y2.

There appears to beonly one intersectionpoint, at x � 0.

If you substitute 35 for x, will the equation check exactly?No; since 35 is an approximation it is likely that the values will be nearly equal, but probably not equal.

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PRACTICE and APPLICATION EXERCISES

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M E W O RK

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Select Tools to Solve Problems (1)(C) Use either pencil and paper or technology to solve each equation. Check each solution for reasonableness.

1. 1x + x

2 = x + 42x 2. 11

3x- 1

3= -4

x2 3. x + 6x = -5 4. 5

2x - 23 = 1

x + 56

5. Apply Mathematics (1)(A) The speed s of an airplane is given by s = dt , where d

represents the distance and t is the time.

a. A plane flies 700 miles from New York to Chicago at a speed of 360 mi/h. Find the time for the trip.

b. On the return trip from Chicago to New York, a tail wind helps the plane move faster. The total flying time for the round trip is 3.5 h. Find the speed x of the tail wind.

Select Tools to Solve Problems (1)(C) Use either pencil and paper or technology to solve each equation. Check each solution for reasonableness.

6. 3x = 5 7. 2

x - 1 = 4 8. 5x - 2x - 4 = -3

9. 2x = x

2 10. 2x + 3 = x - 3

2 11. 2x - 1 + 3

x + 1 = 4

Solve each equation for the given variable.

12. m = 2EV 2; E 13. c

E - 1mc = 0; E 14. /

T 2 =g

4p2 ; T 15. 1c - c

a2 + b2 = 0; c

16. Use a Problem-Solving Model (1)(B) You and a classmate have volunteered to contact every member of your class by phone to inform them of an upcoming event. You can complete the calls in six days if you work alone. Your classmate can complete them in four days. How long will it take to complete the calls working together?

17. Apply Mathematics (1)(A) A tapered cylinder is made by decreasing the radius of a rod continuously as you move from one end to the other. The rate at which it tapers is the taper per foot. You can

use the formula T = 24(R - r)L to calculate the taper per foot.

The lengths R, r, and L are measured in inches.

a. Solve this equation for L.

b. What is L for T = 0.75, 0.85, and 0.95, if R = 4 in.; r = 3 in.?

Solve each equation. Check each solution for reasonableness.

18. 15x + 9x - 7

x + 2 = 9 19. 2x + 2

- 1x

= -4x(x + 2)

20. 1b + 1

+ 1b - 1

= 2b2 - 1

21. 1x - 5

= xx 2 - 25

22. kk + 1 + k

k - 2 = 2 23. 5x 2 - 7x + 12

- 23 - x

= 5x - 4

24. 102y + 8

-7y + 8

y 2 - 16= -8

2y - 8 25. 2

x + 3- 3

4 - x= 2x - 2

x 2 - x - 12

R

L

r

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494 Lesson 11-6 Solving Rational Equations

TEXAS Test Practice

32. What is the solution of x + 1x = -2?

A. 1, -1 B. 0 only C. - 12

only D. -1 only

33. Which of the following is equivalent to 6124213

?

F. 212 G. 312 H. 512 J. 612

34. An investment of $750 will be worth $1500 after 12 years of continuous compounding at a fixed interest rate. What is that interest rate?

A. 2.00% B. 5.78% C. 6.93% D. 200%

35. A librarian orders 48 fiction and nonfiction books for the school library. A fiction book costs $15 and a nonfiction book costs $20. The total cost of the order was $900. How many nonfiction books did the librarian order? Show your work.

26. Evaluate Reasonableness (1)(B) Describe and correct the error made in solving the equation.

27. Create Representations to Communicate Mathematical Ideas (1)(E) Suppose you drive an average of 15,000 miles per year, and your car gets 24 miles per gallon. Suppose gasoline costs $3.60 a gallon.

a. How much money do you spend each year on gasoline?

b. You plan to trade in your car for one that gets x more miles per gallon. Write an expression to represent the new yearly cost of gasoline.

c. Write an expression to represent your total savings on gasoline per year.

d. Suppose you can save $600 a year with the new car. How many miles per gallon does the new car get?

28. Connect Mathematical Ideas (1)(F) Write and solve a problem that can be modeled by a rational equation.

29. Apply Mathematics (1)(A) An automatic pitching machine can pitch all its baseballs in 1 14 hours. One attendant can retrieve all the baseballs pitched by one machine in 3 12 hours. At least how many attendants working at the same rate should be hired so that the baseballs from 10 machines are all retrieved in less than 8 hours?

30. Create Representations to Communicate Mathematical Ideas (1)(E) Write a rational equation that has the following.

a. one solution b. two solutions c. no real solution

31. Apply Mathematics (1)(A) The average hourly wage H (x) of workers in an industry

is modeled by the function H (x) = 16.24x0.062x + 39.42, where x represents the number

of years since 1970. In what year does the model predict that wages will be $25/h?

− 2( + 2) = ( + 1)( − 2) − 2 − 4 =

0 = −2 =

x x x xx x

2 + 2x2 x

2 − − 2x x

=−x 2−2x + 2

+ 1xx

There is no square root of a negative number, so the equation has no solution.

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