chapter 6 section 6 solving rational equations. a rational equation is one that contains one or more...
DESCRIPTION
1.Determine the LCD of all fractions 2.Multiply both sides (every term) of the equation by the LCD. Eliminates the fraction 3.Remove any parentheses and combine like terms on each sides 4.Solve using the properties learned earlier, denominator ≠ 0 5.Check your solution with original equation Solving Rational EquationsTRANSCRIPT
Chapter 6 Section 6
Solving Rational Equations
A rational equation is one that contains one or more rational (fractional) expressions.
Solving Rational Equationswith Integer in Denominator
1 3 8 2 5
3 5 2
3 82 5
x x
xx x
Rational Coefficient Rational Term variable in denominator
1. Determine the LCD of all fractions
2. Multiply both sides (every term) of the equation by the LCD. Eliminates the fraction
3. Remove any parentheses and combine like terms on each sides
4. Solve using the properties learned earlier, denominator ≠ 0
5. Check your solution with original equation
Solving Rational Equations
The strategy used to solve rational equations is to change the rational equation into the kind of equation we can solve.
Solving Rational Equationswith Integer in Denominator
1 LCD = (2)(3) = 62 3
(6) (6)(1) (6) (6) (6)(1)2 3 2 3
6
x x
x x x x
3
2x 6
2
3x 6 3 2 6 6x x x
Example:
Solving Rational Equationswith Integer in Denominator
2 1 2 LCD = 2010 2 20
2 1 2(20) (20) ( 2010 2 20
x x
x x
2 2)10x ( 20
10 1)2
( 20
2)20x
(2)( 2) (10)(1) ( 2) 2 4 10 2 2 10 2 4
12
x x x x x x
x
Example: Solve
Distributive Property
Combine like terms.Variable on one side.
Solving Rational Equations with a variable in the denominator
Remember that dividing by zero is undefined. Therefore, you must check your answer in the original equation. If it makes the denominator zero it is not a solution.
This is called an extraneous roots or extraneous solution.
It is only necessary to check the denominator.
Solving Rational Equations with a variable in the denominator
Example: Solve5 74 LCD = 3
3
5 7(3 ) 4 (3 ) (3 )(4) (33
xx
x x x xx
5)x
( 3
7)3
x
12 15 7 12 7 15 5 15
3
x x x x x
x
Solving Rational Equations with a variable in the denominator
Example: Solve
2 2 LCD = (m+1)(3)1 3
(3)( 1
mm
m
2)1
mm
( 3
2)( 1)3
m (3)( 2) ( 1) 2
3 6 2 2 3 2 2 6 8
m m
m m m m m
Distributive Property
Solving Rational Equations with a variable in the denominator
Remember that with Proportions we can cross multiply.
a c a c ad bcb d b d
3 2 use cross multiplication 2 4
(3)( 4) (2)( 2) 3 12 2 4
3 2 4 12 16
z z
z z z z
z z z
Example: Solve
Solving Rational Equations with a variable in the denominator
Example: Solve5 4 LCD
5( ) ( )(4) ( )( ) (
x xx
x x x x x xx
5)x
2 2
4
5 4 4 5 0 ( 5)( 1) 0
5 0 and 1 0 5 1
x
x x x x x x
x xx x
Solving Rational Equations with a variable in the denominator
Example: Solve2 5 14 LCD 2
2 2
( 2
x x xx x
x
2 5)
2x xx
( 2x
14)2x
2
2
( 5 ) 14
5 14 0 ( 7)( 2) 0
7 0 and 2 0 7 2
x x
x x x x
x xx x
-2 makes the denominator zero, therefore, -2 is an extraneous and is NOT a solution
Solving Rational Equations with a variable in the denominator
Example: Solve2
2
7 3 5 Factor: 9 ( 3)( 3)9 3 3
LCD ( 3)( 3)
( 3
x x x xx x x
x x
x
)( 3x 7)( 3
xx )( 3x
( 3)( 3)
x x
3)3x
( 3x
5)( 3)3
xx
7 ( 3)(3) ( 3)(5) 7 (3 9) 5 15 7 3 9 5 15
4 9 5 15 - 9 15 5 4 6
x x x x x x x x x
x x x x x
Remember
When adding and subtracting rational expressions, find the LCD and rewrite each expression with the common denominator.
When solving a rational equation, multiply by the LCD to eliminate the fractions.
Make sure that the equation is in standard form.ax2 + bx + c = 0
Always determine the values for the variables that will make the denominator equal zero. This is so that you can spot an extraneous solution.
HOMEWORK 6.6
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#15, 25, 33, 39, 41, 49, 53, 61