Transcript 10_6 solving rational equations Trout 09

10.6 Solving Rational Equations
• Goals: To solve problems involving
rational expressions
Rational Equation
An equation containing one or more rational
expressions
Steps to solve Rational Equations
1. Find the LCD
2. Multiply every term on both sides of the
equation by the LCD over 1(objective is to
cancel out the denominators)
3. Solve for the variable
a. If it is a linear equation get variables on one
side and constants on the other
b. If it is a quadratric set your equation = 0 and
factor.
Extraneous Solutions
• When both sides of the equation are mult by
a variable, the equation is transformed into
a new equation and may have an extra
solution.
• Check each solution in the original rational
equation
• Make sure that your answer does not make
the denominator 0
Solving Rational Equations
x4  x   1  1
 x4  x 
x 4 x
Multiply both sides of the equation
by the LCM of the denominators.
Least Common Multiple: Each factor
raised to the greatest exponent.
4 x  x
4  2x 2  x
LCM is x  4  x 
Solve for x:
24
1
3 5 x 24
 

4 8 12 1
2
2 23 2 2  3
18  15  2 x
33
x
2
LCM = 2  3  24
3
Solve for x:
LCM = (x + 1)(x)
1
2
 •(x + 1)(x)
(x + 1)(x)•
x 1 x
x  2x  2
1x
1x
0 x2
2  x
Solve for x:
LCM = 2x
1 1
  12 •2x
2x•
2x x
1  2   24 x
1
 x
8
Solve for x:
1
x 2 •x
x
x•
x  1  2x
2
x  2x  1  0
2
x  1
2
x 1
0
Solve for x:
2
x
4

x2 x2
x 4
x  2 x  2
2
-2 is an extraneous solution.
Solve for x:
2
x
4

x2 x2
x 4
x  2 x  2
2
-2 is an extraneous solution.
Cross products
Short cut:
x7 1

x2 4
4  x  7   1 x  2 
4x  28  x  2
1x
1x
3x  28  2
3x  30
extraneous solution? x  10
1
Cross products: 2

x 1 x  2
x 1  2x  4
5 x
Cross products:
x3 1

3x  2 5
5x 15  3x  2
2x  17
17
x
2
Cross products: x  2 x  1

x  3 x 1
x  x  2  x  4x  3
2
x
2
x
2
2
 x  2  4x  3
4x
4x
3x  2  3
5
x
3x  5
3
Assignment:
Page 453
# (2-24) even