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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 LCD2. 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 EquationsSolving Rational Equations
Multiply both sides of the equation by the LCM of the denominators.
xx
4
11
Least Common Multiple: Each factor raised to the greatest exponent.
xx 4
LCM is 4x x
xx 4
xx 4
x24 x2
Solve for x:
128
5
4
3 x LCM =
x21518
x2
33
22 32
32 3 24 24
124
1
22 3
Solve for x:
xx
2
1
1
LCM =
x
(x + 1)(x)
(x + 1)(x)•
•(x + 1)(x)
22 x
x 2
1x 1x
0 2x
Solve for x:
121
2
1
xx
LCM =
21
2x
2x•
•2x
x24
x8
1
Solve for x:
21
x
x
12x
x• • x
x2
1x
0122 xx
01 2 x
Solve for x:
2
4
2
2
xx
x
2x42 x
2x
-2 is an extraneous solution.
Solve for x:
2
4
2
2
xx
x
2x42 x
2x
-2 is an extraneous solution.
Cross productsShort cut:
7 1
2 4
x
x
4 7x
extraneous solution?
1 2x 4 28 2x x 1x 1x
3 28 2x 3 30x
10x
Cross products: 2 1
1 2x x
1 2 4x x 5 x
Cross products: 3 1
3 2 5
x
x
5 15 3 2x x 2 17x
17
2x
Cross products: 2 1
3 1
x x
x x
2 2x x 2 4 3x x 2x 2x
2 4 3x x
3 2 3x 5
3x
4x 4x
3 5x
Assignment:Page 453
# (2-24) even