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1. 10.6 Solving Rational Equations

2. Rational Equation
An equation containing one or more rational
expressions
2
2 1 5
3
x x
x x
 
 

3. Steps to solve Rational Equations
1. Find the LCD
2. Multiply every term on both sides of the
equation by the LCD over 1
3. Cross Cancel
4. Solve for the variable
a. Linear equation: get variables on one side and
constants on the other
b. Quadratic: set your equation = 0 and factor.

4. Extraneous Solutions
• When both sides of the equation are mult by
a variable, the equation is transformed into a
new equation and may have an extra
solution.
• Check each solution in the original rational
equation
• Make sure that your answer does not make
the denominator 0

5. Solving Rational Equations
Solving Rational Equations
Multiply both sides of the equation
by the LCM of the denominators.
x
x 

4
1
1
Least Common Multiple: Each
factor raised to the greatest
exponent.
 
 x
x 4
 
LCM is 4
x x

 
x
x 
 4
x
x 

4
x
2
4  x

2

6. Solve for x:
12
8
5
4
3 x

 LCM =
x
2
15
18 

x

2
33
2
2 3
2
3
2 3 24
 
24
1
24
1

2
2 3


7. Solve for x:
x
x
2
1
1


LCM =

x
(x + 1)(x)
(x + 1)(x)• •(x + 1)(x)
2
2 
x
x

 2
1x
 1x

0 2
x
 

8. Solve for x:
12
1
2
1



x
x
LCM =

 2
1
2x
2x• •2x
x
24

x


8
1

9. Solve for x:
2
1


x
x

1
2
x
x• • x
x
2
1

x
0
1
2
2


 x
x
  0
1
2


x

10. Solve for x:
2
4
2
2


 x
x
x
2

x
4
2

x
2


x
-2 is an extraneous solution.

11. Solve for x:
2
4
2
2


 x
x
x
2

x
4
2

x
2


x
-2 is an extraneous solution.

12. Cross products
Short cut:
7 1
2 4
x
x



 
4 7
x  
extraneous solution?
 
1 2
x 
4 28 2
x x
  
1x
 1x

3 28 2
x  
3 30
x 
10
x 

13. Cross products: 2 1
1 2
x x

 
1 2 4
x x
  
5 x


14. Cross products: 3 1
3 2 5
x
x



5 15 3 2
x x
  
2 17
x 
17
2
x 

15. Cross products: 2 1
3 1
x x
x x
 

 
2
2
x x
   2
4 3
x x
 
2
x
 2
x

2 4 3
x x
   
3 2 3
x   5
3
x 
4x
 4x

3 5
x 

16. Assignment:
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