Solving rational equations

Can the student solve rational equations?

A rational equation is an equation that has one
or more rational expressions.
Examples:
Rational Equations
3
21


xx xx
x
x
x
2
1
1
3
3
2






15
12
7
4

x
One way is to solve is to Cross
Multiply!
57127,
12
12144
841260
)7(12)15(4





xtherefore
x
x
x
x

xx
3
2
5


3
62
635
)2(3)(5




x
x
xx
xx
Check your work! 11,,
3
3
5
5
 becauseyes


Another way to Solving Rational
Eqns.
4. Check the solutions.
3. Solve the resulting equation.
2. Clear denominators by multiplying both
sides of the equation by the LCM.
1. Find the common denominator.
To solve a rational equation:















x
xx
x
xx
3
)2)((
2
5
)2)((
xx
3
2
5


The LCD is x(x+2)
Multiply both sides by this.
Now Simplify
3
62
635
)2(35




x
x
xx
xx


Easy example:
xx
12
2
13

LCD: 2x
Multiply each fraction
through by the LCD xx
12
2
13

2x2x2x
x
xx
x
x 24
2
26

246  x
18 x
18x


Factor denominator
Distribute and
set equal to zero.
Factor and solve
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 = 3
Check. x = 3 is not a solution since both sides would be undefined.
)5)(3(
6
3 


 xxx
x












 )5)(3(
6
3 xxx
x )5)(3()5)(3(  xxxx
Example: .
158
6
3 2



 xxx
x
)5)(3()5)(3(  xxxx

9
12
3
3
3
2
2




 mmm
equationthistosolutionnoistheremSince
m
mm
mm
m
mm
m
mm
m
mm
,3
3m
3m
1215
129362
12)3(3)3(2
9
12
)3)(3(
3
3
)3)(3(
3
2
)3)(3( 2












Recall that you can not divide by 0. So from the original
Equation you see that m = 3 and m = -3
m2 – 9 factors to (m+3)(m-3), therefore our LCD is (m+3)(m-3)


If R is the total resistance for a parallel circuit with two
resistors of resistances, r1 and r2 , then .
Find the resistance, r1, if the total resistance, R, is 25
ohms and r2 is 95 ohms.
Round your answer to the nearest ohm if necessary.
21
111
rrR

34
9286.33
47514
1
475
14
1
95
1
25
1
95
11
25
1
1
1
1
1
1
1






r
r
r
r
r
r
Word Problem

5
21
4
1 rr 


3
1
x
Examples Answer
4
18
10
24





x
x
x
x
3x
2
7
5
3
10

yy
5x
63
5
9
11
2
4


 mm 11
1
x


 http://www.purplemath.com/modules/solvrtnl2.ht
m
 http://www.algebra-online.com/solving-rational-
equations-3.htm
 https://www.khanacademy.org/math/algebra2/rati
onal-expressions-equations-and-functions/solving-
rational-equations/v/ex-1-multi-step-equation
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Solving rational equations

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