Quiz 8-5Quiz 8-5

1.1.x

x

x 39

72

2.2.

SimplifySimplify

3.3.

3

3

245

122

xxx

2

)4(

2

)72(22

x

x

x

x

8-6 8-6

Solve Rational EquationsSolve Rational Equations

Vocabulary:Vocabulary:What does What does solvesolve a single variable equation mean? a single variable equation mean?

What is a What is a factor?factor?

What is a What is a least common multipleleast common multiple??

3x + 2 = 113x + 2 = 11

Solving Rational EquationsSolving Rational Equations

Method #1: eliminate the denominators one at a time.Method #1: eliminate the denominators one at a time.

Method #2: Obtain common denominators for each term. Method #2: Obtain common denominators for each term.

Method #3: Find the least common multiple of all the Method #3: Find the least common multiple of all the denominators then multiply (left/right) by the LCM. denominators then multiply (left/right) by the LCM.

Rational EquationsRational Equations

How do you get the ‘5’ out of the denominator?How do you get the ‘5’ out of the denominator?5

4x

1

5*

54*5

x

Multiply both sides by ‘5’Multiply both sides by ‘5’

x = 20x = 20

Rational EquationsRational Equations

x

102 How do you get the ‘x’ out of the denominator?How do you get the ‘x’ out of the denominator?

Multiply both sides by ‘x’Multiply both sides by ‘x’

1*

102*

x

xx 2x = 102x = 10 x = ?x = ?

Rational EquationsRational Equations

2

124

x

How do you get the ‘How do you get the ‘x-2x-2’ out of the denominator?’ out of the denominator?

Multiply both sides by ‘Multiply both sides by ‘x-2x-2’’

1

)2(*

)2(

12)2(4

x

xx

4(x – 2) = 124(x – 2) = 12

÷4 ÷4÷4 ÷4

x – 2 = 3x – 2 = 3+2 +2+2 +2

x = 5x = 5

Your turn:Your turn:

1. 1. SolveSolve12

1018

x2. 2. SolveSolve

)1(

205

x

x

217

4. 4. SolveSolve

3. 3. SolveSolve

6)2(

304

x

Variable on Variable on bothboth sides of the sides of the equationequation

1

23

xx

How do you get the ‘How do you get the ‘xx’ out of the’ out of the left-sideleft-side denominator? denominator?

1*

1

23*

1

x

xx

x

Multiply Multiply both sidesboth sides by ‘ by ‘xx’’

1

23

x

x

Variable on Variable on bothboth sides of the sides of the equationequation

1

23

xx

1

23

x

x How do you get the How do you get the (x+1) (x+1) out of theout of the right-sideright-side denominator? denominator?

Multiply Multiply both sidesboth sides by by (x+1)(x+1)1

)1(*

1

23*)1(

x

x

xx

xx 233 3 3

323 xxxx 2 2

3x

Another example:Another example:54

9

1

3

xx

The book teaches you to “cross-multiply” (yuck!)The book teaches you to “cross-multiply” (yuck!)

There is no There is no propertyproperty called “cross multiply”. called “cross multiply”.

There There areare the the properties of equalityproperties of equality..

Cross multiplication is the Cross multiplication is the resultresult of multiplying of multiplying both sides by the left denominator, then multiplyingboth sides by the left denominator, then multiplying the both sides by the right denominator.the both sides by the right denominator.

DON’T THINK DON’T THINK Cross multiplyCross multiply!! Forget it completely!!!! Forget it completely!!

Eliminate one Denomiator at Eliminate one Denomiator at a time.a time.

54

9

1

3

xx(4x + 5) *(4x + 5) * * (4x + 5)* (4x + 5)

91

)54(3

x

x(x + 1) *(x + 1) * * (x + 1)* (x + 1)

)1(9)54(3 xx

991512 xx 63 x

9153 xDistributive propertyDistributive property

-9x both sides-9x both sides

-15 both sides-15 both sides

÷3 both sides÷3 both sides

2x

OrOr: obtain a common : obtain a common denominatordenominator

54

9

1

3

xx(4x + 5) *(4x + 5) *

(4x + 5) *(4x + 5) *

)1)(54(

)1(9

)1)(54(

)54(3

xx

x

xx

x

* (x + 1) * (x + 1)

* (x + 1)* (x + 1)

)1(9)54(3 xx

Multiply Multiply bothboth sides by the common denominator sides by the common denominator

)1)(54( xx )1)(54( xx

Your Turn:Your Turn:

)2(

4

3

9

xx6. 6. SolveSolve

7. 7. Solve Solve xx

x 1

22

8. 8. Solve Solve x

x

4

33

2

What about this wrinkle?What about this wrinkle?

54

xx

““multiply to eliminate the denominators one at a timemultiply to eliminate the denominators one at a time” ”

x *x * * x * x

xxx 54 2 Now what?Now what? It’s a quadratic,It’s a quadratic, solve the quadratic!solve the quadratic!

0452 xx

put into standard form !!!put into standard form !!!

Your turnYour turn: : 9. 9. solvesolve 0452 xx

54

xx

1

)5(9

)5(

8*

1

)5(

1

1*

1

)5(

x

xx

xx

1

)5(9

5

81

1

)5(

x

xx

x

Rational equations with 2 Rational equations with 2 solutions.solutions.

xx

9

5

81

Eliminate denominators one at at time.Eliminate denominators one at at time.

Multiply left/right byMultiply left/right by (x – 5) (x – 5)

CarefullCarefull: distributive property: distributive property

x

xx )5(9

1

8

1

)5(

1

)5(9

)5(

8*

1

)5(

1

1*

1

)5(

x

xx

xx

1

)5(9

5

81

1

)5(

x

xx

x

Rational equations with 2 Rational equations with 2 solutions.solutions.

xx

9

5

81

Eliminate denominators one at at time.Eliminate denominators one at at time.

Multiply left/right byMultiply left/right by (x – 5) (x – 5)

CarefullCarefull: distributive property: distributive property

x

xx )5(9

1

8

1

)5(

Rational equations with 2 Rational equations with 2 solutions.solutions.

xx

9

5

81

Eliminate denominators one at at time.Eliminate denominators one at at time.

)5(932 xxx

Combine like terms.Combine like terms.

x

xx

)5(98)5(

x

xx

)5(93

Multiply left/right Multiply left/right xx

1*

)5(9)3(

x

x

xxx

Rational equations with 2 Rational equations with 2 solutions.solutions.

xx

9

5

81

Eliminate denominators one at at time.Eliminate denominators one at at time.

)5(932 xxx Distributive property.Distributive property.

What kind of equation is this? What kind of equation is this? 45932 xxx

Quadratic:Quadratic: get into standard form then get into standard form then solvesolve

045122 xx

0)3)(15( xx

3 15, x

Your turn:Your turn:

5

5

53

3

x

x

x

x

x

x10. 10. SolveSolve

Put all factors inside parentheses to Put all factors inside parentheses to avoid silly distributive property errors avoid silly distributive property errors later on.later on.

VocabularyVocabulary

3

11

3

2

x

x

xx

Extranious SolutionExtranious Solution: : a solution obtained algebraicallya solution obtained algebraicallybut which doesn’t work in the original equation.but which doesn’t work in the original equation.

0342 xx0)1)(3( xx

3 ,1x

33

13

3

1

33

2

0

13

3

1

0

2

31

11

1

1

31

2

2

0

1

1

2

2

011

Check for Extraneous Check for Extraneous solutions.solutions.

2

1

4

632

x

x

x

x

11. 11. Solve.Solve.

12. 12. Check to see if either one is extraneous.Check to see if either one is extraneous.

Your Turn:Your Turn:

Multiply for the LCDMultiply for the LCD

Multiply both sides by the Multiply both sides by the LCDLCD!!!!

2. 2. What does the 1What does the 1stst denominator denominator needneed to be common to be common with the other two denominators?with the other two denominators? Needs a ‘2’ Needs a ‘2’

xx 3

4

6

1

3

2

What is the What is the least common denominator?least common denominator?

xx 3

4

3*2

1

3

2

1. Look for what is 1. Look for what is commoncommon to each denominator. to each denominator.

Factor theFactor the denominatordenominator

3. 3. What does the 2What does the 2ndnd denominator denominator needneed to be common with to be common with the other two denominators?the other two denominators?

33

Needs an ‘x’ Needs an ‘x’

4. 4. What does the 3What does the 3rdrd denominator denominator needneed to be common to be common with the other two denominators?with the other two denominators?

Needs an ‘2’ Needs an ‘2’

Find the Find the least common least common denominatordenominator

xx 3

4

6

1

3

2

3x 2*3 3x3x 2*3 3x2 *2 * 2 * 2 * * x * x

The The least common denominator isleast common denominator is: : 6x

Multiply Multiply both sidesboth sides by the by the LCD.LCD.

xx 3

4

6

1

3

26x 6x

Find the Find the least common least common denominatordenominator

xx 3

4

6

1

3

2 The The LCDLCD is: is: 6x

Multiply Multiply both sidesboth sides by the by the LCD.LCD.

xx 3

4

6

1

3

26x 6x

x

xx

x

x

3

)3*2(*4

6

1**6

3

2*)2*3(

Notice how I factored 6xNotice how I factored 6x so that it is easier to simplify.so that it is easier to simplify.

84 x Wow ! x = 4Wow ! x = 4