1. Solving Rational Equations Solving Rational Equations
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Rational Equations

2. Solving Rational Equations
Rational Functions
Rational Functions and Models
Definition
f(x) is a rational function if and only if
f(x) =
Example
1. f(x) =
p(x)
q(x)
where p(x) and q(x) are polynomial
functions with q(x)  0
Rational Functions: Definition
The most common error students make in identifying rational functions is assuming that any quotient function is a rational function. Care must be taken to identify each of the numerator and denominator as a polynomial. No other type of quotient function is a rational function.
Note that some rational functions can be reduced to simple linear functions, provided the zeros of the denominator are excluded. This is the case in Example 2.
Note that not all quotient type functions are rational functions, as is illustrated in Example 3. The fractional power in the numerator indicates the cube of a square root of x. Since polynomials must have integer powers in all terms, the numerator is not a polynomial and hence the quotient f(x) is not a rational function.
3x2 + 4x + 1
x3 – 1
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Rational Equations
Section v5.0 2
10/11/2012
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Rational Equations

3. Solving Rational Equations
How do we solve equations of form:
Method 1: Clear Fractions =
g(x)
f(x)
h(x)
= 15 23
x + 2
1. Solve: = 23
x + 2
15 (x + 2)
(x + 2)
23 = 15x + 30
–7 = 15x = –7 15 x
Solution Set: { } 7 15 –
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Rational Equations

4. Solving Rational Equations
Method 1: Clear Fractions
2. Solve: =
x + 1
x + 2
x + 5
x + 7
x + 1
x + 2
(x + 2)(x + 7)
(x + 2)(x + 7) =
x + 5
x + 7
(x + 5)(x + 2) =
(x + 1)(x + 7)
x2 + 7x + 10 =
x2 + 8x + 7
7x + 10 =
8x + 7 3 = x
Solution Set: { 3 }
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Rational Equations

5. Solving Rational Equations
Method 2: Cross Multiplication
Basic Principle:
1. Solve: = a b c d
if and only if ad = bc =
x + 1
x + 2
x + 5
x + 7
(x + 2)(x + 5) =
(x + 1)(x + 7)
x2 + 7x + 10 =
x2 + 8x + 7
7x + 10 =
8x + 7 3 = x
Solution Set: { 3 }
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Rational Equations

6. Solving Rational Equations
Method 2: Cross Multiplication
2. Solve:
Cross multiplying =
x – 3 7 1
x + 3 7 =
(x – 3)(x + 3) 7 =
x2 – 9
Zero Product Property
Square Root Property =
x2 – 16 16 = x2
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Rational Equations

7. Solving Rational Equations
Method 2: Cross Multiplication
2. Solve: =
x – 3 7 1
x + 3 7 =
x2 – 9
Zero Product Property
Square Root Property =
x2 – 16 16 = x2 =
(x + 4)(x – 4) = x2 √ 16 =
x + 4 OR =
x – 4 = x 4 –4 = x 4 = x
Solution Set: { – 4, 4 }
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Rational Equations

8. Solving Rational Equations
Method 2: Cross Multiplication
3. Solve: =
x + 1
5x – 3 2 3
2(5x – 3) =
3(x + 1)
10x – 6 =
3x + 3 7x = 9 = 9 7 x
Solution Set: 9 7
{ }
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Rational Equations
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Rational Equations

9. Solving Rational Equations
Method 3: Graphical Approach
1. Solve:
x + 1
x – 5 = 2
Hence: x = 11 x y 3 6 9 11 –2 –3
Let y1
x + 1
x – 5 = y1
Horizontal Asymptote y = 1
and y2 = 2 y2
(11, 2)
y1 = y2 ? 
For what x is this true ?  
Vertical Asymptote x = 5
y1 intercepts :
Intersection at (11, 2)
Horizontal : ( –1, 0 )
Vertical : ( 0, –1/5 )
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Rational Equations
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Rational Equations

10. Think about it !
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Rational Equations