10.6 Solving Rational Equations
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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
411
Least Common Multiple: Each factor raised to the greatest exponent.
xx 4
LCM is 4x x
xx 4
xx 4x24 x2
Solve for x:
1285
43 x
LCM =
x21518
x233
22 32
32 3 24 241
241
22 3
Solve for x:
xx2
11
LCM =
x
(x + 1)(x)
(x + 1)(x)•
•(x + 1)(x)
22 x
x 2
1x 1x
0 2x
Solve for x:
12121
xx
LCM =
21
2x
2x•
•2x
x24
x81
Solve for x:
21
xx
12x
x• • x
x2
1x
0122 xx 01 2 x
Solve for x:
24
2
2
xxx
2x42 x
2x
-2 is an extraneous solution.
Solve for x:
24
2
2
xxx
2x42 x
2x
-2 is an extraneous solution.
Cross productsShort cut:
7 12 4
xx
4 7x
extraneous solution?
1 2x 4 28 2x x 1x 1x
3 28 2x 3 30x
10x
Cross products: 2 11 2x x
1 2 4x x 5 x
Cross products: 3 13 2 5xx
5 15 3 2x x 2 17x
172
x
Cross products: 2 13 1
x xx x
2 2x x 2 4 3x x 2x 2x
2 4 3x x
3 2 3x 53
x
4x 4x
3 5x
Assignment:Page 453
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