be sure to check all solutions for extraneous roots!
TRANSCRIPT
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).