Rational Equations

• Technical Definition: An equation that contains a rational expression

• Practical Definition: An equation that has a variable in a denominator

• Example:

3

2

1

5

32

12

xxxx

A A rational equationrational equation is an equation between rational expressions.is an equation between rational expressions.

For example, For example, and and are rational equations.are rational equations.3

21

xx xx

x

x

x

2

1

1

33

2

4.4. Check the solutions. Check the solutions.

3.3. Solve the resulting polynomial equation. Solve the resulting polynomial equation.

2.2. Clear denominators by multiplying both sides of the Clear denominators by multiplying both sides of the equation by the LCM.equation by the LCM.

1.1. Find the LCM of the denominators. Find the LCM of the denominators.

To solve a rational equationTo solve a rational equation::

Solving Rational Equations

1. Find “restricted values” for the equation by setting every denominator that contains a variable equal to zero and solving

2. Find the LCD of all the fractions and multiply both sides of equation by the LCD to eliminate fractions

3. Solve the resulting equation to find apparent solutions

4. Solutions are all apparent solutions that are not restricted

After clearing denominators, After clearing denominators, a solution of the polynomial equationa solution of the polynomial equation may make a may make a denominator of the rational equation zero. denominator of the rational equation zero.

Since Since xx22 – 1 = ( – 1 = (xx – 1)( – 1)(xx + + 1), 1),

Since Since –– 11 makes both denominators zero, the rational makes both denominators zero, the rational equation has equation has no solutionsno solutions..

ExampleExample:: Solve: Solve: . .1

1

1

132

xx

x

22xx = – = – 2 2 xx = = –– 11

33xx + 1 = + 1 = xx – 1 – 1

CheckCheck..

It is critical to check all solutions.It is critical to check all solutions.

In this case, the value is In this case, the value is not a solution of the rational equationnot a solution of the rational equation..

LCM = LCM = ((xx – 1)( – 1)(xx + 1) + 1)..

1

1

1

132 xx

x)1)(1( )1)(1( xxxx

Example

1

1

2

1

1

22

mm

RV

01m 01m

012 m

011 mm

01m

OR1m 1m

SolvedAlready 1

1

2

1

11

2

mmm

LCD 112 mm

12114 mmm

11

1

2

1

11

2 LCD

mmm

2214 2 mm2214 2 mm

320 2 mm

130 mm

01 03 morm1 3 morm

6

ExampleExample: Solve:: Solve: . .158

6

3 2

xxx

x

Factor.Factor.

Polynomial Equation.Polynomial Equation.

Simplify.Simplify.

Factor.Factor.

The LCM is (The LCM is (xx – 3)( – 3)(xx – 5). – 5).xx22 – 8 – 8xx + 15 = + 15 = ((xx – 3)( – 3)(xx – 5) – 5)

xx((xx – 5) = – – 5) = – 66

xx22 – 5 – 5xx + 6 = 0 + 6 = 0

((xx – 2)( – 2)(xx – 3) = 0 – 3) = 0

xx = = 22 or or xx = = 33CheckCheck. . xx = 2 is a = 2 is a solution.solution.CheckCheck. . xx = = 3 is not a solution3 is not a solution since both sides would be since both sides would be undefinedundefined..

158

6

3 2

xxx

xOriginal Equation.Original Equation.

158

6

3 2

xxx

x)5)(3( )5)(3( xxxx

This checks in the original equation, so the solution This checks in the original equation, so the solution is 7.is 7.

Applications of Rational Expressions

• Word problems that translate to rational expressions are handled the same as all other word problems

• On the next slide we give an example of such a problem

Example

When three more than a number is divided by twice the number, the result is the same as the original number. Find all numbers that satisfy these conditions.

.

:Unknownsnumber The x

xx

x

2

3

:RV02 x0x

xxx

xx 2

2

32

223 xx

320 2 xx

1320 xx

01or 032 xx

1or 32 xx

1or 2

3 xx

To solve problems involving work, use the formula,To solve problems involving work, use the formula,

ExampleExample:: If it takes 5 hours to paint a room, what part of the If it takes 5 hours to paint a room, what part of the work is completed after 3 hours?work is completed after 3 hours?

Three-fifths of the work is completed after three hours. Three-fifths of the work is completed after three hours.

If one room can be painted in 5 hours then the If one room can be painted in 5 hours then the rate of workrate of work is is (rooms/hour). The (rooms/hour). The time workedtime worked is 3 hours. is 3 hours.

5

1

part of work completed = rate of work part of work completed = rate of work time workedtime worked..

Therefore,Therefore, part of work completed = rate of work part of work completed = rate of work time workedtime worked

5

33

5

1part of work completed part of work completed ..

ExampleExample:: If a painter can paint a room in 4 hours and her If a painter can paint a room in 4 hours and her assistant can paint the room in 6 hours, how many hours assistant can paint the room in 6 hours, how many hours will it take them to paint the room working together?will it take them to paint the room working together?

Let Let tt be the time it takes them to paint the room together. be the time it takes them to paint the room together.

164

tt

1223 tt

4.25

12125 tt

LCM = 12.LCM = 12.

Multiply by 12.Multiply by 12.

Simplify.Simplify.

Working together they will paint the room in Working together they will paint the room in 2.4 hours2.4 hours..

painterpainter

assistantassistant

rate of work rate of work time worked time worked part of part of work work completed completed

4

1 tt4

t

6

1 tt6

t

Tom can mow a lawn in 4 hours. Perry can mow the same lawn in Tom can mow a lawn in 4 hours. Perry can mow the same lawn in 5 hours. How long would it take both of them, working together 5 hours. How long would it take both of them, working together with two lawn mowers, to mow the lawn? with two lawn mowers, to mow the lawn?

UNDERSTAND the problemUNDERSTAND the problem

Question: How long will it take the two of Question: How long will it take the two of them to mow the lawn together? them to mow the lawn together?

Tom can do 1/4 of the job in one hourTom can do 1/4 of the job in one hour

Perry can do 1/5 of the job in one hourPerry can do 1/5 of the job in one hour

Data:Data: Tom takes 4 hours to mow the Tom takes 4 hours to mow the lawn. Perry takes 5 hours to mow the lawn. Perry takes 5 hours to mow the lawn.lawn.

Tom can mow a lawn in 4 hours. Perry can mow the same lawn in Tom can mow a lawn in 4 hours. Perry can mow the same lawn in 5 hours. How long would it take both of them, working together 5 hours. How long would it take both of them, working together with two lawn mowers, to mow the lawn? with two lawn mowers, to mow the lawn?

Develop and carryout a PLANDevelop and carryout a PLAN

Let Let t t represent the total number of hours represent the total number of hours it takes them working together. Then it takes them working together. Then they can mow 1/t of it in 1 hour.they can mow 1/t of it in 1 hour.

Translate to an equation.Translate to an equation.

1 1 14 5 t

Tom can do 1/4 of the job in one Tom can do 1/4 of the job in one hourhour

Perry can do 1/5 of the job in one Perry can do 1/5 of the job in one hourhour

Together they can do 1/t of Together they can do 1/t of the job in one hourthe job in one hour

Tom can mow a lawn in 4 hours. Perry can mow the same lawn in Tom can mow a lawn in 4 hours. Perry can mow the same lawn in 5 hours. How long would it take both of them, working together 5 hours. How long would it take both of them, working together with two lawn mowers, to mow the lawn? with two lawn mowers, to mow the lawn?

At a factory, smokestack A pollutes the air twice as fast as At a factory, smokestack A pollutes the air twice as fast as smokestack B.When the stacks operate together, they yield a smokestack B.When the stacks operate together, they yield a certain amount of pollution in 15 hours. Find the time it would certain amount of pollution in 15 hours. Find the time it would take each to yield that same amount of pollution operating take each to yield that same amount of pollution operating alone.alone.

1/x is the fraction of the pollution produced by A in 1 hour.1/x is the fraction of the pollution produced by A in 1 hour.

1/2x is the fraction of the pollution produced by B in 1 hour.1/2x is the fraction of the pollution produced by B in 1 hour.

1/15 is the fraction of the total pollution produced by A and B in 1 hour.1/15 is the fraction of the total pollution produced by A and B in 1 hour.

1 1 1+ =

x 2x 15

distance = rate distance = rate time time and and time =time = ..

To solve problems involving motion, use the formulas,To solve problems involving motion, use the formulas,

rate

distance

ExamplesExamples: : 1.1. If a car travels at 60 miles per hour for 3 hours, If a car travels at 60 miles per hour for 3 hours, what distance has it traveled?what distance has it traveled?

2.2. How long does it take an airplane to travel 1200 How long does it take an airplane to travel 1200 miles flying at a speed of 250 miles per hour?miles flying at a speed of 250 miles per hour?

It takes 4.8 hours for the plane make its trip.It takes 4.8 hours for the plane make its trip.

time = time = = = = 4.8.= 4.8.rate

distance

250

1200

Since Since distancedistance = 1200 (mi) and = 1200 (mi) and raterate = 250 (mi/h), = 250 (mi/h),

Since Since raterate = 60 (mi/h) and = 60 (mi/h) and timetime = 3 = 3 h, then h, then

The car travels 180 miles.The car travels 180 miles.distance = rate distance = rate timetime = 60 = 60 3 = 180.3 = 180.

ExampleExample: A traveling salesman drives from home to a client’s : A traveling salesman drives from home to a client’s store 1store 1550 miles away. On the return trip he drives 10 miles per hour 0 miles away. On the return trip he drives 10 miles per hour slowerslower and and addsadds one-half hour in driving time. one-half hour in driving time.

Let Let rr be the rate of travel (speed) in miles per hour. be the rate of travel (speed) in miles per hour.

300300rr – – 300(300(rr – 10)– 10) = = rr((rr –– 10) 10)

Trip to client Trip to client

Trip Trip homehome

distance distance raterate timetime

150150 rrr

150

150150 rr – 10 – 1010

150

r

2

1150

10

150

rrLCM = 2LCM = 2rr ((rr – 10). – 10).

Example continuedExample continued

At what speed was the salesperson driving on the way to the client’s At what speed was the salesperson driving on the way to the client’s store?store?

Multiply by LCM.Multiply by LCM.

0 = 0 = rr22 – 10 – 10rr – – 30003000

The salesman drove from home to the client’s store atThe salesman drove from home to the client’s store at 660 miles per hour0 miles per hour..

The return trip The return trip tooktook one-half hour one-half hour longer longer..

At At 660 mph the time taken to drive the 10 mph the time taken to drive the 1550 miles 0 miles from the salesman’s home to the clients store is from the salesman’s home to the clients store is = = 2.5 h2.5 h..

60

150

At 50 mph (ten miles per hour At 50 mph (ten miles per hour slowerslower) the ) the time taken to make the return trip of 1time taken to make the return trip of 1550 miles is 0 miles is = = 33 h h..

50

150

rr = = 660 or –0 or – 5050

Example continuedExample continued

CheckCheck::

(–50) is (–50) is irrelevant.)irrelevant.)

300300rr – – 330000rr + 3000 + 3000 = = rr22 –– 10 10rr

0 = (0 = (rr – 60)( – 60)(rr + 50) + 50)

An airplane flies 1062 km with the wind. InAn airplane flies 1062 km with the wind. In the same the same amount of time it can fly 738 km against the wind. The amount of time it can fly 738 km against the wind. The speed of the plane in still air is 200 km/h. Find the speed speed of the plane in still air is 200 km/h. Find the speed of the wind.of the wind.

r = 36 km/hr = 36 km/h

INEQUALITIES

Rational