Be sure to check all solutions for extraneous roots!
Objective and Vocabulary
The objective is to be able to solve rational equations.
Lowest Common Denominator (LCD) – The smallest denominator which has all of the original denominators as factors.
Step 1 – Find the lowest common denominator for the fractions.
5 2
3
7
x x
The lowest common multiple of 3 and x is 3x. Therefore, the LCD is 3x.
Multiply each fraction by the lowest common denominator.
5 2
3
7
x x
2
33x
5
x3x
7
x 3x
35
32
33
7x
xx x
x
Simplify
1 5 2 2 1 x
1 5 2 2 1 x
15 15 15 2 21x Subtract 15 from each side
of the equation..
2 6x
Divide each side of the equation by 2.2
2
6
23
x
x
2 6x
Simplify.
Be sure to check in original equation (above)…..
5/3 + 2/3 = 7/3 yes!
5 2
3
7
x x
1 . 1
5
4
3
8
m m
2 . a + 1
4 31
a
3 . x
x
x
x
2
3
1
22
SOLUTION
Note-it’s a good idea to put a” 1” under anyPart that is not a fraction already
M=140/3
1
5
4
3
8
m mMultiply by the lowest common denominator.
151
515
4
315
8m m
mm
m
Divide out common factors.
3 5
5
3 5 4
3
3 5 8
m m
m
m
mSimplify
Solve3m-20=120
4 31
a + 1 aMultiply by the lowest common denominator
Divide out common factors.
Simplify
a aa
a aa
a a
14
11
31 1
a a
a
a a
aa a
1 4
1
1 31
4 3 3
3
2
2
a a a a
a a a
a a a3 2
Solve
a a a
a a
a a
a a
a a
a or a
3
3 2
3 3
0 2 3
0 1 3
1 0 3 0
2
2
2
( )( )
a = 1 or -3Check each one!
Check: 1 and -3
4 31
a + 1 a
−4
1+1+
3
1=1 ,
−4
−3+1+
3
−3=1
−2 + 3 =1, , 2 −1 =1
They both check
Solution to 3
3 . x
x
x
x
2
3
1
22
x2 −4 + x2 −4x+ 3=2(x2 −5x+6)
2x2 −4x−1=2x2 −10x+126x=13
x=136
The previous examples had two or more fractions on one side of the equation. When there is a single fraction on each side of the equation, the equation can be solved as a proportion by cross-multiplying.
Extraneous roots can be found by graphing the function-see problems20-22 in the textbook on page 255. They can also be irrational (see problem 23 on the same page).