ALGEBRA PROJECT

UNIT 12

RATIONAL EXPRESSIONS and EQUATIONS

RATIONAL EXPRESSIONS and EQUATIONS

Lesson 1 Inverse Variation

Lesson 2 Rational Expressions

Lesson 3 Multiplying Rational Expressions

Lesson 4 Dividing Rational Expressions

Lesson 5 Dividing Polynomials

Lesson 6Rational Expressions with Like Denominators

Lesson 7Rational Expressions with Unlike Denominators

Lesson 8 Mixed Expressions and Complex Fractions

Lesson 9 Solving Rational Equations

INVERSE VARIATION

Example 1 Graph an Inverse Variation

Example 2 Graph an Inverse Variation

Example 3 Solve for x

Example 4 Solve for y

Example 5 Use Inverse Variation to Solve a Problem

Manufacturing The owner of Superfast Computer Company has calculated that the time t in hours that it takes to build a particular model of computer varies inversely with the number of people p working on the computer. The equation can be used to represent the people building a computer. Complete the table and draw a graph of the relation.

t

12108642p

t

12108642p

Original equation

Replace p with 2.

Divide each side by 2.

Simplify.

6

Solve the equation for the other values of p.

3 2 1.5 1.2 1

Answer:

Solve for

Answer:

Graph the ordered pairs: (2, 6), (4, 3), (6, 2), (8, 1.5), (10, 1.2), and (12, 1).

As the number of people p increases, the time t ittakes to build a computer decreases.

t

12108642p

33.64.56918t

12108642pAnswer:

Manufacturing The foreman of a package delivery company has found that the time t in hours that it takes to prepare packages for delivery varies inversely with the number of people p that are preparing them. The equation

can be used to represent the people preparing the packages. Complete the table and draw a graph of the relation.

Answer:

Inverse variation equation

The constant of variation is 4.

Graph an inverse variation in which y varies inversely as x and

Solve for k.

14

22

41

undefined0

–4–1

–2–2

–1–4

yx

Choose values for x and y whose product is 4.

Answer:

Graph an inverse variation in which y varies inversely as x and

Method 1 Use the product rule.

Product rule for inverse variations

Divide each side by 15.

Simplify.

If y varies inversely as x andfind x when

Method 2 Use a proportion.

Proportion rule for inverse variations

Cross multiply.

Divide each side by 15.

Answer: Both methods show that

If y varies inversely as x andfind x when

Answer: 8

If y varies inversely as x and find y when

Use the product rule.

Product rule for inverse variations

Divide each side by 4.

Simplify.

Answer:

If y varies inversely as x and find y when

Answer: –25

Physical Science When two objects are balanced on a lever, their distances from the fulcrum are inversely proportional to their weights. How far should a 2-kilogram weight be from the fulcrum if a 6-kilogram weight is 3.2 meters from the fulcrum?

Original equation

Divide each side by 2.

Simplify.

Answer: The 2-kilogram weight should be 9.6 meters from the fulcrum.

Answer: 1 m

Physical Science How far should a 10-kilogram weight be from the fulcrum if a 4 kilogram weight is 2.5 meters from the fulcrum?

RATIONAL EXPRESSIONS

Example 1 One Excluded Value

Example 2 Multiple Excluded Values

Example 3 Use Rational Expressions

Example 4 Expression Involving Monomials

Example 5 Expression Involving Polynomials

Example 6 Excluded Values

Exclude the values for which

Subtract 7 from each side.

Answer: b cannot equal –7.

The denominator cannot equal zero.

State the excluded value of

Answer: –3

State the excluded value of

Exclude the values for which

The denominator cannot equal zero.

Factor.

Use the Zero Product Property to solve for a.

or

Answer: a cannot equal –3 or 4.

State the excluded value of

Answer: 2, 3

State the excluded value of

The original mechanical advantage was 5.

Landscaping Refer toExample 3 on page 649.Suppose Kenyi finds arock that he cannot movewith a 6-foot bar, so he gets an 8-foot bar. But thistime, he places the fulcrumso that the effort arm is 6 feetlong, and the resistance armin 2 feet long.

Explain whether he has more or less mechanicaladvantage with his new setup.

Simplify.

Answer: Even though the bar is longer, because he moved the fulcrum he has a mechanical advantage of 3, so his mechanical advantage is less than before.

Use the expression for mechanical advantage to write an expression for the mechanical advantage in the new situation.

Answer: Since the mechanical advantage is 3, Kenyi canlift 3 • 180 or 540 pounds with the longer bar.

If Kenyi can apply a force of 180 pounds, what is the greatest weight he can lift with the longer bar?

Landscaping Sean and Travis are responsible for clearing an area for a garden. They come across a large rock that they cannot lift. Therefore, they use a 5-foot bar as a lever, and the fulcrum is 1 foot away from the rock.

a. Use the formula to find the mechanical advantage.

b. If they can apply a force of 200 pounds, what is the greatest weight they can lift?

Answer: 4

Answer: 800 lb

The GCF of the numeratorand denominator is

Divide the numerator anddenominator by

1

1

Simplify

Answer: Simplify.

Simplify

Answer:

Factor.

Divide the numerator and denominator by the GCF, x – 7.

1

1

Simplify

Answer: Simplify

Simplify

Answer:

Divide the numeratorand denominator bythe

1

1

Simplify State the excluded values of x.

Factor.

Simplify.Answer:

Exclude the values for which equals 0.

The denominator cannot equal zero.

Factor.

Zero Product Property

Evaluate.

Simplify.

Check Verify the excluded values by substituting them into the original expression.

Evaluate.

Simplify.

Answer: The expression is undefined when andTherefore,

Answer:

Simplify State the excluded values of w.

MULTIPLYINGRATIONAL EXPRESSIONS

Example 1 Expressions Involving Monomials

Example 2 Expressions Involving Polynomials

Example 3 Dimensional Analysis

Method 1 Divide by the greatest common factor after multiplying.

Multiply the numerators.

Multiply the denominators.

The GCF is 98xyz.

Simplify.

Find

Method 2 Divide the common factors before multiplying.

Multiply.Answer:

Divide by common factors and z.

1 x 1

6 z 2 y

31 1

1 1

Multiply.Answer:

Divide bycommon factors and r.

1 1 1

1 1 2 1 1

d 2 q

2 r 3

Find

Answer:

Answer:

a. Find

b. Find

Factor thenumerator.

Simplify.Answer:

The GCF is1

1

1

x 2

Find

Find

Factor.

The GCF is1

1

1

1

Multiply.

Simplify.Answer:

a. Find

b. Find

Answer:

Answer:

Space The velocity that a spacecraft must have in order to escape Earth’s gravitational pull is called the escape velocity. The escape velocity for a spacecraft leaving Earth is about 40,320 kilometers per hour. What is this speed in meters per second?

Answer: The escape velocity is 11,200 meters per second.

1120 10

1 1

Simplify.

Multiply.

Aviation The speed of sound, or Mach 1, is approximately 330 meters per second at sea level. What is the speed of sound in kilometers per hour?

Answer: 1188 kilometers per hour

DIVIDINGRATIONAL EXPRESSIONS

Example 1 Expression Involving Monomials

Example 2 Expression Involving Binomials

Example 3 Divide by a Binomial

Example 4 Expression Involving Polynomials

Example 5 Dimensional Analysis

Find

Multiply by

the reciprocal of

Answer: Simplify.

Divide by common factors 5, 6, and x.

1

4 11

15x3

Find

Answer:

Find

Multiply by

the reciprocal

of

Factor

or Simplify.Answer:

The GCFis

1

1

Find

Answer:

Find

Multiply by

the reciprocal

of

Factor

Simplify.Answer:

The GCF is

1

1

Find

Answer:

Find

Multiply by the

reciprocal,

Factor

Simplify.Answer:

The GCFis

1

1

Find

Answer:

Aviation In 1986, an experimental aircraft named Voyager was piloted by Jenna Yeager and Dick Rutan around the world non-stop, without refueling. The trip took exactly 9 days and covered a distance of 25,012 miles. What was the speed of the aircraft in miles per hour? Round to the nearest mile per hour.

Use the formula for rate, time, and distance.

Divide each side by 9 days.

Convert days to hours.

Answer: Thus, the speed of the aircraft was about 116 miles per hour.

Aviation Suppose that Jenna Yeager and Dick Rutan wanted to complete the trip in exactly 7 days. What would be their average speed in miles per hour for the 25,012-mile trip? 

Answer: about 149 miles per hour

DIVIDING POLYNOMIALS

Example 1 Divide a Binomial by a Monomial

Example 2 Divide a Polynomial by a Monomial

Example 3 Divide a Polynomial by a Binomial

Example 4 Long Division

Example 5 Polynomial with Missing Terms

Find

Write as a rational expression.

Divide each term by 2x.

2x 9

11

Simplify each term.

Simplify.Answer:

Find

Answer:

Find

Write as a rational expression.

Divide each term by 3y.

Simplify.Answer:

2y 1

13

Simplify each term.

y

3

Find

Answer:

1

1

Divide by the GCF.

Find

Factor the numerator.

Simplify.Answer:

Write as a rational expression.

Find

Answer:

Find

Step 1 Divide the first term of the dividend, x2, by the first term of the divisor, x.

x

Multiply x and x – 2.

Subtract.

Step 2 Divide the first term of the partial dividend, 9x – 15, by the first term of the divisor, x.

x + 9

Subtract and bring down 15.

Multiply 9 and x – 2.

Subtract.

Answer: The quotient of is

with a remainder of 3, which can be written as

Find

Answer: The quotient is with a remainder of 2.

Rename the x2 term by using a coefficient of 0.

Find

Multiply x2 and x – 5.

Subtract and bring down 34x.

Multiply 5x and x – 5.

Subtract and bring down 45.

Multiply –9 and x – 5.

Subtract.

Answer:

Find

Answer: The quotient is

RATIONAL EXPRESSIONS WITH LIKE DENOMINATORS

Example 1 Numbers in Denominator

Example 2 Binomials in Denominator

Example 3 Find a Perimeter

Example 4 Subtract Rational Expressions

Example 5 Inverse Denominators

Find

The common denominator is 15.

Add the numerators.

Simplify.Answer:

Divide by the common factor, 5.

4b

3

Find

Answer:

Divide by the common factor, c + 2.

1

1

Find

The common denominator is c + 2.

Factor the numerator.

Simplify.Answer:

Answer: 5

Find

Geometry Find an expression for the perimeter of rectangle WXYZ.

Perimeter formula

The commondenominator is

Distributive Property

Combine like terms.

Factor.

Answer: The perimeter can be represented by the

expression

Geometry Find an expression for the perimeter of rectangle PQRS.

Answer:

Find

The common denominator is

The additive inverse ofis

Distributive Property

Simplify.Answer:

Find

Answer:

Find

The denominator is the same asor . Rewrite the second expression so that it has the same denominator as the first.

Rewrite usingcommon denominators.

The common denominator is

Simplify.Answer:

Find

Answer:

RATIONAL EXPRESSIONS WITH UNLIKE DENOMINATORS

Example 1 LCM of Monomials

Example 2 LCM of Polynomials

Example 3 Monomial Denominators

Example 4 Polynomial Denominators

Example 5 Binomials in Denominators

Example 6 Polynomials in Denominators

Find the LCM of

Find the prime factors of each coefficient and variable expression.

Use each prime factor the greatest number of times it appears in any of the factorizations.

Answer:

Find the LCM of

Answer:

Find the LCM of

Express each polynomial in factored form.

Use each factor the greatest number of times it appears.

Answer:

Find the LCM of

Answer:

Find

Factor each denominator and find the LCD.

Since the denominator of is already 5z, onlyneeds to be renamed.

Multiply

by

Distributive Property

Add the numerators.

Answer: Simplify.

Find

Answer:

Find

Factor thedenominators.

The LCDis

Add thenumerators.

Simplify.Answer:

Find

Answer:

Find

Factor.

The LCD is

Add thenumerators.

Multiply.

Simplify.Answer:

Find

Answer:

Multiple-Choice Test Item

Find

A B

C D

Read the Test Item

The expression represents

the difference of two rational expressions with

unlike denominators.

Solve the Test ItemStep 1 Factor each denominator and find the LCD.

The LCD is

Step 2 Change each rational expression into anequivalent expression with the LCD. Then subtract.

Answer: C

Multiple-Choice Test Item

Find

A B

C D

Answer: C

MIXED EXPRESSIONS andCOMPLEX FRACTIONS

Example 1 Mixed Expression to Rational Expression

Example 2 Complex Fraction Involving Numbers

Example 3 Complex Fraction Involving Monomials

Example 4 Complex Fraction Involving Polynomials

Simplify

The LCD is

Add the numerators.

Distributive Property

Answer: Simplify.

Simplify

Answer:

Baking Suppose Katelyn bought 2

pounds of chocolate chip cookie dough.

If the average cookie requires ounces of dough,

how many cookies would she be able to make?

To find the total number of cookies, divide the amount of cookie dough by the amount of dough needed for each cookie.

Convert pounds to ounces and divide by common units.

Simplify.

Express each term as an improper fraction.

Simplify.

Answer: Katelyn can make 21 cookies.

Answer: 27 cookies

Baking James bought pounds of cookie dough,

and he prefers to make large cookies. If each cookie

requires ounces of dough, how many cookies

can he make? 

Simplify

Rewrite as a division sentence.

Rewrite as multiplication by the reciprocal.

Divide by common factors a, b, and c2.

a 4 c

2

b

3

1

1 1

Simplify.Answer:

Answer:

Simplify

Simplify

The LCD of the fractionsin the numerator is

Simplify the numerator.

The numerator contains a mixed expression. Rewrite it as a rational expression first.

Rewrite as a division sentence.

Multiply by the reciprocal of

Simplify.Answer:

Factor.

Simplify

Answer:

SOLVINGRATIONAL EQUATIONS

Example 1 Use Cross Products

Example 2 Use the LCD

Example 3 Multiple Solutions

Example 4 Work Problem

Example 5 Rate Problem

Example 6 No Solution

Example 7 Extraneous Solution

Solve

Original equation

Cross multiply.

Distributive Property

Add –2x and 48 to each side.

Answer: Divide each side by 6.

Answer: –3

Solve

Solve

Originalequation

The LCD is

DistributiveProperty

Simplify.

Add.

Subtract 1 from each side.

Divide each side by 6.

Answer:

Answer: 8

Solve

Solve

DistributiveProperty

Original equation

The LCD is

Simplify.

or

Set equal to 0.

Factor.

Check Check solutions by substituting each value in the original equation.

Check Check solutions by substituting each value in the original equation.

Answer: The number 1 is an excluded value for x. Thus, the solution is 3.

Solve

Answer: 4, –1

TV Installation On Saturdays, Lee helps her father

install satellite TV systems. The jobs normally take

Lee’s father about hours. But when Lee helps,

the jobs only take them hours. If Lee were

installing a satellite system herself, how long would

the job take?

Explore Since it takes Lee’s fatherhours

to install one job, he can finish of the job

in one hour. The amount of work Lee can

do in one hour can be represented by

To determine how long it takes Lee to do

the job, use the formula

Lee’s work + her father’s work = 1 job.

Plan The time that both of them worked was

hours. Each rate multiplied by this time results

in the amount of work done by each person.

Solve Lee’s her father’stotal

work plus work equals work.

1

Multiply.

The LCD is 10t.

Distributive Property

Simplify.

Add –6t to each side.

Divide each side by 4.

Answer: The job would take Lee or hours by herself.

Examine This seems reasonable because the combined efforts of the two took longer than half of her father’s usual time.

Driveways Shawna earns extra money by shoveling

driveways. If she works alone, she can finish a large

driveway in hours. If Vince helps her, they can get

done in hours. If Vince were shoveling the

driveway himself, how long would the job take him?

Answer: hours

Transportation The schedule for the Washington, D.C., Metrorail is shown to the right. Suppose two Red Line trains leave their stations at opposite ends of the line at exactly 2:00 P.M. One train travels between the two stations in 48 minutes and the other train takes 54 minutes. At what time do the two trains pass each other?

Determine the rates of both trains. The total distance is 19.4 miles.

Train 1 Train 2

Next, since both trains left at the same time, the time both have traveled when they pass will be the same. And since they started at opposite ends of the route, the sum of their distances is equal to the total route, 19.4 miles.

19.4

48

19.4

48

t19.4

54

19.4

54

t

t min Train 2

t min Train 1

d = r t

tr

The sum of the distances is 19.4.

The LCD is 432.

Distributive Property

Simplify.

Add.

Divide each sideby 329.8.

Answer: The trains passed each other at about 25 minutesafter they left their stations, at 2:25 P.M.

Transportation Two cyclists are riding on a 5-mile circular bike trail. They both leave the bike trail entrance at 3:00 P.M. traveling in opposite directions. It usually takes the first cyclist one hour to complete the trail and it takes the second cyclist 50 minutes. At what time will they pass each other?

Answer: 3:27 P.M.

Solve

Original equation

The LCD is x – 1.

Distributive Property1

1

1

1

Simplify.

Subtract 2 from each side.

Answer: Since 1 is an excluded value for x, the number 1 is an extraneous solution. Thus, the equation has no solution.

Solve

Answer: no solutions

Solve

Original equation

The LCD is x – 2.

1

1

1

1

Distributive Property

Simplify.

Subtract 4 fromeach side.

Factor.

or Zero Product Property

Answer: The number 2 is an extraneous solution, since 2 is an excluded value for x. Thus, –2 is the solution of the equation.

Solve

Answer: –3

THIS IS THE ENDOF THE SESSION

BYE!