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Chapter 6 Section 6
Objectives
1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solving Equations with Rational Expressions
Distinguish between operations with rational expressions and equations with terms that are rational expressions.
Solve equations with rational expressions.
Solve a formula for a specified variable.
6.6
2
3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Distinguish between rational expressions and equations
Slide 6.6-3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Distinguish between expressions and
equations
Slide 6.6-4
Expressions EquationsNumbers Numbers
Operations Operations
Variables Variables
NO equal sign Equal sign
Simplify Solve
Uses of the LCD When adding or subtracting rational expressions, find the LCD, then add numerators
When simplifying a complex fraction, multiply numerator and denominator by the LCD
When solving an equation, multiply each side by the LCD so the denominators are eliminated.
WOW the LCD is useful!
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solution:
Identify each of the following as an expression or an equation. Then simplify the expression or solve the equation.
5
2 3 6
x x
2 4
3 9
x x
equation expression
56 6
2 3 6
x x 3
3
3
2 4
9
x x
5x 2
9
x
6 4
9 9
x x
5
Slide 6.6-5
Distinguishing between Expressions and EquationsCLASSROOM EXAMPLE 1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 2
Solve equations with rational expressions.
Slide 6.6-6
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solve equations with rational expressions
When an equation involves fractions•use the multiplication property of equality to clear the fractions•choose as multiplier the LCD of all denominators in the fractions of the equation
Please recall: The 11th Commandment
Thou shall not… divide by zero
The denominator of a rational expression cannot equal 0, since division by 0 is undefined.
Therefore, when solving an equation with rational expressions that have variables in the denominator,
The solution cannot be a number that makes the denominator equal 0.
Slide 6.6-7
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solve, and check the solution.
Solution:
2 3 6
5 3 5
m m
2 3 6
5 315 15
5
m m
9m
Check:
2 3 6
5 3 5
m m
Multiply every term of the equation by the LCD
Slide 6.6-8
Solving an Equation with Rational ExpressionsCLASSROOM EXAMPLE 2
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solving an Equation with Rational Expressions
Step 1: Multiply each side of the equation by the LCD to clear the equation of fractions. Be sure to distribute to every term on both sides.
Step 2: Solve the resulting equation.
Step 3: Check each proposed solution by substituting it into the original equation.
Reject any solutions that cause a denominator to equal 0.
Slide 6.6-9
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solution:
2 21
1 1
x
x x
Solve, and check the proposed solution.
2 21
1 11 1
x
xx x
x
1 x Reject this solution.WHY??
Slide 6.6-10
Solving an Equation with Rational ExpressionsCLASSROOM EXAMPLE 3
How do you recognize equations that could possibly have restrictions?
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solution:
2 2
2 3
2p p p p
Solve, and check the proposed solution.
4p
2 3
22 1 2
11p p p
pp p p
p p p
The solution set is {4}.
Slide 6.6-11
Solving an Equation with Rational ExpressionsCLASSROOM EXAMPLE 4
It works!
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solution:
2
8 3 3
4 1 2 1 2 1
r
r r r
Solve, and check the proposed solution.
2 1 2 1 2 18 3 3
2 1 2 1 2 1 2 12 1r r r
r
r r rr
r
0r Does it work??
Slide 6.6-12
Solving an Equation with Rational ExpressionsCLASSROOM EXAMPLE 5
The solution set is {0}.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solve, and check the proposed solution (s).
2
1 1 2
2 5 5 4x x
1 1 2
2 5 55 2 2 5
2 22 2x x x
x x xx
4x 1x
Solution:
The solution set is {−4, −1}.
or
Slide 6.6-13
Solving an Equation with Rational ExpressionsCLASSROOM EXAMPLE 6
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
2
6 1 4
5 10 5 3 10x x x x
Solve, and check the proposed solution.
Solution:
6 1 4
5 2 55 2 5 5 2
2 55
x xx x
x xx x
60x The solution set is {60}.
Slide 6.6-14
Solving an Equation with Rational ExpressionsCLASSROOM EXAMPLE 7
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 3
Solve a formula for a specified variable.
Slide 6.6-15
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solve the following formula for z.
Solution:
2 1 1 xyz xyz
x y z
Fun!
2
xyz
y x
Slide 6.6-17
Solving for a Specified Variable
2 1 1
x y z
CLASSROOM EXAMPLE 9
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
for x
z xx y
Solve each formula for the specified variable.
Solution:
xz
xx
yx y y
1
zyx
z
for s t
b sr
( )s
r rt
br
Remember to treat the variable for which you are solving as if it were the only variable, and all others as if they were contants.
br t s
Slide 6.6-16
You Try It CLASSROOM EXAMPLE 8
