Transcript 8-6 Solve Rational Equations

Quiz 8-5
1.
2.
Simplify
( 2 x  7 ) ( x  4)
 2
2
x 2
x 2
12
3

2
x  5 x  24 x  3
8-6
Solve Rational Equations
Vocabulary:
What does solve a single variable equation mean?
3x + 2 = 11
What is a factor?
What is a least common multiple?
Solving Rational Equations
Method #1: eliminate the denominators one at a time.
Method #2: Obtain common denominators for each term.
Method #3: Find the least common multiple of all the
denominators then multiply (left/right) by the LCM.
Rational Equations
x
4
5
How do you get the ‘5’ out of the denominator?
Multiply both sides by ‘5’
10
2
x
x = 20
How do you get the ‘x’ out of the denominator?
Multiply both sides by ‘x’
x=?
x 5
5* 4  *
5 1
10 x
x*2  *
x 1
2x = 10
Your turn:
21
1. Solve 7 
x
10
 12
2. Solve 18 
x
20
3. Solve 5 
( x  1)
4. Solve
30
4
6
( x  2)
Variable on both sides of the equation
3
2

x x 1
How do you get the ‘x’ out of the
left-side denominator?
x 3
2
x
* 
*
1 x x 1 1
2x
3
x 1
Multiply both sides by ‘x’
How do you get the (x+1) out of the
right-side denominator?
2 x ( x  1)
( x  1) * 3 
*
x 1
1
3x  3  2 x
Multiply both sides by (x+1)
Your turn: 5. solve
3x  3  2 x
Another example:
3
9

x  1 4x  5
The book teaches you to “cross-multiply” (yuck!)
There is no property called “cross multiply”.
There are the properties of equality.
Cross multiplication is the result of multiplying
both sides by the left denominator, then multiplying
the both sides by the right denominator.
DON’T THINK Cross multiply!! Forget it completely!!
Eliminate one Denomiator at a time.
3
9
* (4x + 5)

(4x + 5) *
x 1 4x  5
(x + 1) *
3(4 x  5)
 9 * (x + 1)
x 1
3(4 x  5)  9( x  1)
12x 15  9x  9
Or: obtain a common denominator
3
9
* (x + 1)
(4x + 5) *

(4x + 5) * x  1 4 x  5 * (x + 1)
Multiply both sides by the common denominator
(4 x  5)( x  1) 3(4 x  5)
9( x  1) (4 x  5)( x  1)

(4 x  5)( x  1) (4 x  5)( x  1)
3(4 x  5)  9( x  1)
Your Turn:
6. Solve
9
4

3x ( x  2)
7. Solve
x
1

2
x 2 x
8. Solve
x
3
3
2
4x
What about this wrinkle?
“multiply to eliminate the denominators one at a time”
4

x *   x  5 * x
x

4
x5
x
4 x  x  5x
2
Now what?
put into standard form !!!
It’s a quadratic,
solve the quadratic!
x  5x  4  0
2
Your turn:
2
x
 5x  4  0
9. solve
Rational equations with 2 solutions.
8
9
1

x5
x
Your turn:
What values of ‘x’ are not allowed?
10. Eliminate the denominator.
x( x  5) x  8   9  ( x  5)
 
   
x( x  5) x  x  5   x  ( x  5)
x( x  5)
*
1
x 2  5 x  8 x  9 x  45 x( x  5)
*

1
x( x  5)
x( x  5)
x 2  5 x  8 x  9 x  45
Your turn:
11. Solve
x  3 1 x  13


3
3x
x
Check for Extraneous solutions.
Your Turn:
12. Solve.
3x  6 x  1

2
x 4 x2
13. Check to see if either one is extraneous.
Multiply by the LCD
This is the most difficult method.
1. Determine what the LCD is.
2. Multiply both sides of the equation by the LCD.
3. Solve the resulting equation.
Multiply both sides by the LCD!!
What is the least common denominator?
2 1 4
 
3x 6 3x
Factor the
denominator
2
1
4


3x 2 * 3 3x
1. Look for what is common to each denominator. 3
2. What does the 1st denominator need to be common
with the other two denominators? Needs a ‘2’
3. What does the 2nd denominator need to be common
with the other two denominators? Needs an ‘x’
4. What does the 3rd denominator need to be common
with the other two denominators?
Needs an ‘2’
Find the least common denominator
2 1 4
 
3x 6 3x
2 * 3x
2*3 * x
2 *3x
The least common denominator is:
Multiply both sides by the LCD.
2 1 4
 
6x
3x 6 3x
6x
6x
Find the least common denominator
2 1 4
 
3x 6 3x
The LCD is: 6x
Multiply both sides by the LCD.
2 1 4
 
6x
3x 6 3x
6x
Notice how I factored 6x
so that it is easier to simplify.
(3 x * 2) * 2 6 * x *1 4 * (2 * 3 x)


3x
6
3x
4 x 8
Wow !
x=4