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Algebraic Fractions and Rational Equations. In this discussion, we will look at examples of simplifying Algebraic Fractions using the 4 rules of fractions. +. -. ×. ÷. Addition. Simplify. +. Addition. Simplify. +. Subtraction. Simplify. -. Subtraction. Simplify. 4. -. - PowerPoint PPT Presentation
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Algebraic Fractions and Algebraic Fractions and Rational EquationsRational Equations
In this discussion, we will look at examples of simplifying Algebraic Fractions using the 4 rules of fractions.
AdditionAddition
bd
bcad
d
c
b
a
xx
2
31Simplify
)2(
)(3)2(1
xx
xx
)2(
32
xx
xx
)2(
22
xx
x
AdditionAddition
bd
bcad
d
c
b
a
y
x
x 2
3Simplify
)2(
))(()2(3
yx
xxy
xy
xy
2
6 2
SubtractionSubtraction
bd
bcad
d
c
b
a
p
t
p 2
2Simplify
€
(2)(2p) − (p)(t)
(p)(2p)
22
4
p
ptp
22
)4(
p
tp
p
t
2
)4(
SubtractionSubtraction
bd
bcad
d
c
b
a
4
33
2
12
xxSimplify
)4)(2(
)33(2)12(4
xx
8
6648
xx
8
22 x
8
)1(2
x4
4
1x
MultiplicationMultiplication
bd
ac
d
c
b
a
5
5
3
x
xxSimplify
)5(3
)5)((
x
xx
)5(3
52
x
xx
MultiplicationMultiplication
bd
ac
d
c
b
a
62
5
15
3
x
xSimplify
)62)(15(
)5)(3(
x
x
)3)(2)(15(
)5)(3(
x
xFactoring (2x-6)Factoring (2x-6)
3
)2)(3(
1
6
1
DivisionDivision
bc
ad
d
c
b
a
Simplify
14
4
7
2 yx
y
x
4
14
7
2
)4)(7(
)14)(2(
y
x
2
2
y
x
EquationsEquationsOnce you know how to simplify algebraic fractions, youCan solve equations containing them
For example
743
2xx
35
2
2
1
xx
Click an equation to see it solved
743
2xx
Solve
7)4)(3(
))(3()2)(4(
xx
712
38
xx
712
5x
845 x5
84 x
Solve 35
2
2
1
xx
3)5)(2(
)2)(2()1)(5(
xx
310
4255
xx 310
97
x
3097 x
217 x
3x
The equation in the previous example is called a rational equation.
and are rational equations.
3
21
xx xx
x
x
x
2
1
1
33
2
4. Check the solutions.
3. Solve the resulting polynomial equation.
2. Clear denominators by multiplying both sides of the equation by the LCM.
1. Find the LCM of the denominators.
To solve a rational equation:
Examples: 1. Solve: .3
1
3
1
x
x
xMultiply by LCM = (x – 3).
Solve for x.
1 = x + 1
x = 0
Check.
Simplify.
3
1
3
1
(0)
(0)
(0)
3
1
3
1
True.
2. Solve: .1
21
xx
x – 1 = 2x
Find the LCM.LCM = x(x – 1).
Multiply by LCM.
Simplify.
x = –1 Solve.
1
2
1
xx)1( )1( xxxx
After clearing denominators, a solution of the polynomial equation may make a denominator of the rational equation zero.
Since x2 – 1 = (x – 1)(x + 1),
Since – 1 makes both denominators zero, the rational equation has no solutions.
Example: Solve: .1
1
1
132
xx
x
2x = – 2 x = – 1
3x + 1 = x – 1
Check.
It is critical to check all solutions.
In this case, the value is not a solution of the rational equation.
LCM = (x – 1)(x + 1).
1
1
1
132 xx
x)1)(1( )1)(1( xxxx
16
Example: Solve: .158
6
3 2
xxx
x
Factor.
Polynomial Equation.
Simplify.
Factor.
The LCM is (x – 3)(x – 5).x2 – 8x + 15 = (x – 3)(x – 5)
x(x – 5) = – 6
x2 – 5x + 6 = 0
(x – 2)(x – 3) = 0
x = 2 or x = 3Check. x = 2 is a solution.
Check. x = 3 is not a solution since both sides would be undefined.
158
6
3 2
xxx
xOriginal Equation.
158
6
3 2
xxx
x)5)(3( )5)(3( xxxx