8'4 Equations with Rational Expressions

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8.4 Equations with Rational Expressions

• Objectives

• Determine the domain of a rational equation.
• Solve rational equations.

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• A rational equation is an equation containing one
  or more rational expressions.

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• EXAMPLE 1

Determining the Domains of Rational Equations
Find the domain of each equation.
The domain is x x ? 0 .
The domain is x x ? 1 .


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To solve a rational equation

• Find the LCD for all fractions
• multiply both sides of the equation by the LCD
  (this removes all denom.)
• solve the resulting equation, it may be linear or
  quadratic
• check each solution in original equation, some
  may not check

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• Caution on Solutions

CAUTION
When each side of an equation is multiplied by a
variable expression, the resulting solutions
may not satisfy the original equation. You
must either determine and observe the domain or
check all potential solutions in the original
equation. It is wise to do both.
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• EXAMPLE 2

Solving an Equation with Rational Expressions
The domain, which excludes 0, was found in
Example 1(a).
Multiply by the LCD, 6x.
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Distributive property
Multiply.
Subtract 30.
Divide by 1.
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• EXAMPLE 2

Solving an Equation with Rational Expressions
Check Replace x with 3 in the original equation.
Original equation
Let x 3.
?
?
True
The solution is 3 .
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• EXAMPLE 3

Solving an Equation with No Solution
Using the result from Example 1(b), we know
that the domain excludes 1 and 1, since these
values make one or more of the denominators in
the equation equal 0.
Multiply each side by the LCD, (x 1)(x 1).
Distributive property
Multiply.
Distributive property
Combine terms.
Subtract 5.
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• EXAMPLE 3

Solving an Equation with No Solution
Since 1 is not in the domain, it cannot be a
solution of the equation. Substituting 1 in the
original equation shows why.
Check
Since division by 0 is undefined, the given
equation has no solution, and the solution set is
Ø.
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• EXAMPLE 4

Solving an Equation with Rational Expressions
(a1)(a3)
(a3)(a-3)
2(a-3)(a1)
Factor each denominator to find the LCD, 2(a
3)(a 3)(a 1). The domain excludes 3, 3,
and 1.
Multiply each side by the LCD, 2(a 3)(a 3)(a
1).
2(a 3)(a 3)(a 1)
Distributive property
Distributive property
Combine terms
Note that 5 is in the domain substitute 5 in for
a in the original equation to check that the
solution set is 5 .
Subtract 5a Add 36.
Divide by 7.
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• EXAMPLE 5

Solving an Equation That Leads to a Quadratic
Equation
Multiply each side by the LCD, x(2x 1).
x(2x 1
Zero-factor property
2x 1 0 or 2x 1 0
Distributive property
Distributive property
Subtract 4x standard form.
Factor.
Factor.
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Solve each equation
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Summary on Rational Expressions and Equations

• A common mistake is to confuse an equation which
  contains with an expression which does not
  contain . Watch closely for the difference. An
  equation is solved for a number value, whereas an
  expression is simplified.

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Solving an equation Simplifying an
expression
Solve
Simplify
Change all fractions so they have the same
denominator then add or subtract the numerators
Remove fractions by multiplying by LCD
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Identify each as an expression or equation, then
simplify or solve.
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A formula containing rational expressions can be
solved for a specified variable using the same
steps.
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Solve each formula for the indicated variable