Transcript Inequalities - Ranger College

Rational Equations
Solving
Rational Equations
Rational Functions

Definition
f(x) is a rational function if and only if
p(x)
f(x) = q(x)
where p(x) and q(x) are polynomial
functions with q(x)  0

Example

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3x2 + 4x + 1
1. f(x) =
x3 – 1
Rational Equations
2
Solving Rational Equations

How do we solve equations of form:
f(x)
h(x) =
g(x)

Method 1: Clear Fractions
1. Solve:
23
x + 2 = 15
23
(x + 2) = 15 (x + 2)
x+2
23 = 15x + 30
–7 = 15x
x = –7
15
Solution Set:
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Rational Equations
{
– 7
15
}
3
Solving Rational Equations

Method 1: Clear Fractions
x+1
x+5
x+2 = x+7
x+1
x+5
(x + 2)(x + 7) = x + 7 (x + 2)(x + 7)
x+2
2. Solve:
(x + 1)(x + 7)
x2 + 8x + 7
8x + 7
x
=
=
=
=
(x + 5)(x + 2)
x2 + 7x + 10
7x + 10
3
Solution Set: { 3 }
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Rational Equations
4
Solving Rational Equations

Method 2: Cross Multiplication

Basic Principle:
c
a
= d
b

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if and only if ad = bc
x+1
x+5
1. Solve: x + 2 = x + 7
(x + 1)(x + 7) = (x + 2)(x + 5)
x2 + 8x + 7 = x2 + 7x + 10
8x + 7 = 7x + 10
x=3
Solution Set: { 3 }
Rational Equations
5
Solving Rational Equations

Method 2: Cross Multiplication

2. Solve: x – 3 = 1
x+3
7

Cross multiplying
(x – 3)(x + 3) = 7
x2 – 9 = 7
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Zero Product Property
Square Root Property
x2 – 16 = 0
x2 = 16
Rational Equations
6
Solving Rational Equations

Method 2: Cross Multiplication

2. Solve: x – 3 = 1
x+3
7
x2 – 9 = 7
Zero Product Property
x2 – 16 = 0
(x + 4)(x – 4) = 0
x + 4 = 0 OR x – 4 = 0
x = 4
x = –4
Square Root Property
x2 = 16
√ x2 = ± √16
x = ±4
Solution Set: { – 4, 4 }
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Rational Equations
7
Solving Rational Equations

Method 2: Cross Multiplication

x+1
2
3. Solve: 5x – 3 = 3
3(x + 1) = 2(5x – 3)
3x + 3 = 10x – 6
9 = 7x
x = 79
Solution Set:
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9
7
{ }
Rational Equations
8
Solving Rational Equations

Method 3: Graphical Approach

x+1
1. Solve: x – 5 = 2
y
x+1
Let y1 =
x–5
and y2 = 2
3
y1 = y2 ?
For what x
is this true ?
y1 intercepts :
–2
Horizontal : ( –1, 0 )
Vertical : ( 0, –1/5 )
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Hence: x = 11
Horizontal
Asymptote
y=1

–3
Rational Equations
3
y1
y2

(11, 2)
6
9
11
x
Vertical Asymptote
x=5
Intersection at (11, 2)
9
Think about it !
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Rational Equations
10