Solving Solving Rational Rational
EquationsEquationsOn to Section 2.8aOn to Section 2.8a
Solving Rational Equations
Rational Equation – an equation involving rational expressionsor fractions…can always be written in the form:
0f x
g x
where f (x) and g (x) are polynomial functions with no commonfactors. The zeros of f (x) are the solutions of the equation…
Solving Rational Equations
To solve an equation involving fractions:
• Find the LCD of all the terms in the equation.
• Clear the fractions by multiplying both sides of the equation by the LCD.
• Solve the resulting equation.
Extraneous Solutions – solutions that are not solutions of theoriginal equation; result from multiplying or dividing an equationby an expression containing variables…
Guided Practice
Solve3
4xx
The LCD is x !!!
2 3 4x x Multiply by Multiply by x x 2 4 3 0x x Subtract 4Subtract 4x x
1 3 0x x FactorFactor
1,3x Zero Factor PropertyZero Factor Property
How do we support these answers How do we support these answers numerically numerically ??????
Guided Practice
Solve1
04
xx
The LCD is x – 4!!!
44 0
4
xx x
x
2 3x 2 4 1 0x x
0.268,3.732How could we support these answers How could we support these answers graphically graphically ??????
To solve this quadratic?To solve this quadratic?
Guided Practice
Solve 2
2 1 2
1 3 4 3
x
x x x x
The LCD!!!
2
2 1 21 3 1 3
1 3 4 3
xx x x x
x x x x
1 3x x
22 5 3 0x x 2 3 1 2x x x
First, factor the last denominator…First, factor the last denominator…
Guided Practice
Solve 2
2 1 2
1 3 4 3
x
x x x x
22 5 3 0x x
2 1 3 0x x Factor!!!Factor!!!
1
2x
Confirm NumericallyConfirm Numerically
Support GraphicallySupport Graphically3x
Extraneous SolutionExtraneous Solution
Guided Practice
Solve2
3 3 60
2 2
x
x x x x
LCD:LCD: 2x x
Clear the fractions:Clear the fractions: 3 2 3 6 0x x x 2 6 3 6 0x x x
2 2 0x x 2 0x x
0, 2x
Guided Practice
Solve2
3 3 60
2 2
x
x x x x
Algebraic solutions:Algebraic solutions: 0, 2x
Both solutions are extraneous!!!Both solutions are extraneous!!!
Substituting either solution into the original equationSubstituting either solution into the original equationresults in division by zero…results in division by zero…
The original equation has no solution!!!The original equation has no solution!!!
Practice ProblemsPractice Problems
Solve
Clear fractions:
ExtraneousExtraneous
2
3 2 6
3 3
x
x x x x
23 2 6x x 2 4 3 0x x
1 3 0x x
1, 3x
Practice ProblemsPractice Problems
Solve
Clear fractions:
2
2
6 53
2
x x
x
2 26 5 3 2x x x
2 26 5 3 6x x x 22 6 11 0x x
3 311.284, 4.284
2x
Practice ProblemsPractice Problems
Solve
Clear fractions:
2
3 2 5
2 1 2
x
x x x x
3 1 2 2 5x x x
23 3 2 4 5x x x 23 1 0x x
1 130.768, 0.434
6x
Practice ProblemsPractice Problems
Solve
Clear fractions:
2
3 5 15
1 2 2
x
x x x x
3 2 5 1 15x x x
23 10 0x x
5,23
x
3 5 2 0x x
ExtraneousExtraneous
Practice ProblemsPractice Problems
Solve
Clear fractions:
2 37x
x
3 3 7x x 3 7 3 0x x
2.398, 0.441,2.838x Solve graphically!