Quiz 8-5Quiz 8-5

1.1. ?339

722

xx

xx

2.2.

What is the simplified answer?What is the simplified answer?

3.3.

?33

24512

2

xxx

?2)4(

2)72(

22

xx

xx

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 EquationsMethod #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

x102

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

4 x

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

15*

54*5 xMultiply both sides by ‘5’Multiply both sides by ‘5’

Multiply both sides by ‘x’Multiply both sides by ‘x’1*102* x

xx

x = 20x = 20

2x = 102x = 10

x = ?x = ?

Your turn:Your turn:1. 1. SolveSolve

121018 x2. 2. SolveSolve

)1(205

x

x217

4. 4. SolveSolve

3. 3. SolveSolve

6)2(

304

x

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

123

xx

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

1*123*

1x

xxx

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

123

xx How do you get the (x+1) out of theHow do you get the (x+1) out of the

right-sideright-side denominator? denominator?

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

123*)1(

x

xxx

xx 233 Your turnYour turn: : 5. solve5. solve xx 233

Another example:Another example:54

913

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.

549

13

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

91)54(3

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

)1(9)54(3 xx

OrOr: obtain a common : obtain a common denominatordenominator

549

13

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

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

)1)(54()1(9

)1)(54()54(3

xx

xxx

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(

439

xx6. 6. SolveSolve

7. 7. Solve Solve xxx 122

8. 8. Solve Solve xx

433

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 x

x

)5()5(9

58

)5()5(

xx

xxxx

xxxx

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

xx9

581

Your turn:Your turn: 10. 10. Eliminate the denominator.Eliminate the denominator.

)5(459

)5(852

xxx

xxxxx

459852 xxxx

*1)5( xx

1)5(* xx

Your turn:Your turn:

xx

xx 13

31

33

11. 11. SolveSolve

Check for Extraneous Check for Extraneous solutions.solutions.

21

463

2

xx

xx

12. 12. Solve.Solve.

13. 13. 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 34

61

32

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

xx 34

3*21

32

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 34

61

32

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

The The least common denominator isleast common denominator is: : 6xMultiply Multiply both sidesboth sides by the by the LCD.LCD.

xx 34

61

32

6x 6x

Find the Find the least common least common denominatordenominator

xx 34

61

32

The The LCDLCD is: is: 6xMultiply Multiply both sidesboth sides by the by the LCD.LCD.

xx 34

61

32

6x 6x

xxx

xx

3)3*2(*4

61**6

32*)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