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Quiz 8-5Quiz 8-5
1.1.
2.2.
SimplifySimplify
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
x
102
How do you get the ‘5’ out of the denominator?How do you get the ‘5’ out of the denominator?5
4x
How do you get the ‘x’ out of the denominator?How do you get the ‘x’ out of the denominator?
1
5*5
4*5x
Multiply 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. 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 ‘x’ out of theHow do you get the ‘x’ out of the left-sideleft-side denominator? denominator?
1*1
23*1
x
xx
x
Multiply Multiply both sidesboth sides by ‘x’ by ‘x’
1
23
x
x 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(*1
23*)1(
x
x
xx
xx 233 Your turnYour turn: : 5. solve5. solve xx 233
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
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
)5(
)5(9
5
8
)5(
)5(
x
x
xxx
x
xx
xx
Rational equations with 2 Rational equations with 2 solutions.solutions.
xx
9
5
81
Your turn:Your turn: 10. 10. Eliminate the denominator.Eliminate the denominator.
)5(
459
)5(
852
xx
x
xx
xxx
459852 xxxx
*1
)5( xx1
)5(*
xx
What values of ‘x’ are What values of ‘x’ are not allowed?not allowed?
Your turn:Your turn:
x
x
x
x 13
3
1
3
3
11. 11. SolveSolve
Check for Extraneous Check for Extraneous solutions.solutions.
2
1
4
632
x
x
x
x
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 by the LCDMultiply by the LCDThis is the most difficult method.This is the most difficult method.1.1.Determine what the LCD is.Determine what the LCD is.2.2.Multiply both sides of the equation by the LCD.Multiply both sides of the equation by the LCD.3.3.Solve the resulting equation.Solve the resulting equation.
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
