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Chapter 1
Equations and
Inequalities
1.2 Linear Equations and
Rational Equations
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
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Objectives:
• Solve linear equations in one variable.
• Solve linear equations containing fractions.
• Solve rational equations with variables in the
denominators.
• Recognize identities, conditional equations, and
inconsistent equations.
• Solve applied problems using mathematical models.
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Definition of a Linear Equation
A linear equation in one variable x is an equation that
can be written in the form
ax  b  0
where a and b are real numbers, and a  0
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Generating Equivalent Equations
An equation can be transformed into an equivalent
equation by one or more of the following operations:
1. Simplify an expression by removing grouping symbols
and combining like terms.
2. Add (or subtract) the same real number or variable
expression on both sides of the equation.
3. Multiply (or divide) by the same nonzero quantity on
both sides of the equation.
4. Interchange the two sides of the equation.
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Solving a Linear Equation
1. Simplify the algebraic expression on each side by
removing grouping symbols and combining like terms.
2. Collect all the variable terms on one side and all the
numbers, or constant terms, on the other side.
3. Isolate the variable and solve.
4. Check the proposed solution in the original equation.
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Example: Solving a Linear Equation
Solve and check the linear equation:
4 x  5  29
4 x  5  5  29  5
4 x  24
4 x 24

4
4
x6
Check:
4 x  5  29
4(6)  5  29?
24  5  29
29  29
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Example: Solving a Linear Equation Involving Fractions
Solve and check:
x 3 5 x 5
 
4
14
7
The LCD is 28, we will multiply both sides of the
equation by 28.
x  3
5
x  5



28 
  28    28 

 4 
 14 
 7 
7( x  3)  2(5)  4( x  5)
7 x  21  10  4 x  20
7 x  21  4 x  10
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Example: Solving a Linear Equation Involving Fractions
(continued)
11x  21  10
11x  11
x 1
Check:
x 3 5 x 5
 
4
14
7
1 3 5 1 5
 
?
4
14
7
2 5 6
 
4 14 7
1 5 12
 
2 14 14
1 7

2 14
1 1

2
2
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Solving Rational Equations
x 3 5 x 5
 
4
14
7
we were solving a linear equation with constants in
the denominators. A rational equation includes at
least one variable in the denominator. For our next
example, we will solve the rational equation
5 17 1
 
2 x 18 3x
When we solved the equation
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Example: Solving a Rational Equation
5 17 1
Solve:
 
2 x 18 3x
We check for restrictions on the variable
by setting each denominator equal to zero.
2x  0
3x  0
x0
x0
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Example: Solving a Rational Equation
(continued)
5 17 1
 
2 x 18 3x
The LCD is 18x, we will multiply both sides of the
equation by 18x.
5 
17 
1



18 x    18 x    18 x  
 2x 
 18 
 3x 
9(5)  x(17)  6
45  17 x  6
51  17x
x3
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Example: Solving a Rational Equation to Determine
When Two Equations are Equal
Find all values of x for which y1 = y2
1
1
y1 

x4 x4
1
1
22

 2
x  4 x  4 x  16
22
y2  2
x  16
We begin by finding the restricted values.
x 2  16  ( x  4)( x  4)
x  4  0  x  4
x40 x  4
The restricted values are x = – 4 and x = 4
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Example: Solving a Rational Equation to Determine
When Two Equations are Equal (continued)
1
1
22

 2
x  4 x  4 x  16
1
1
22


x  4 x  4 ( x  4)( x  4)
We multiply both sides of the equation by the LCD
1
1 
22



( x  4)( x  4) 

  ( x  4)( x  4) 

x

4
x

4
(
x

4)(
x

4)




x  4  x  4  22
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Example: Solving a Rational Equation to Determine
When Two Equations are Equal (continued)
2 x  22
x  11
11 is not a restricted value. Therefore, x = 11.
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Types of Equations: Identity, Conditional, Inconsistent
An equation that is true for at least one real number is
called a conditional equation. The examples that we
have worked so far have been conditional equations.
An equation that is true for all real numbers is called an
identity.
An equation that is not true for any real number is called
an inconsistent equation.
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Example: Categorizing an Equation
4 x  7  4( x  1)  3
4x  7  4x  4  3
4x  7  4x 1
7  1
This is a false statement. This equation is
an example of an inconsistent equation.
There is no solution.
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Example: Categorizing an Equation
7 x  9  9( x  1)  2 x
7x  9  9x  9  2x
7x  9  7x  9
99
This is a true statement. The solution set
for this equation is the set of all real numbers.
This equation is an identity.
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Example: Solving Applied Problems Using Mathematical
Models
Persons with a low sense of humor have higher levels of
depression in response to negative life events than those
with a high sense of humor. This can be modeled by the
formula
10
53 In this formula, x represents
D x
9
9
the intensity of a negative life event and D is the
average level of depression in response to that event. If
the low humor group averages a level of depression of
10 in response to a negative life event, what is the
intensity of that event?
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Example: Solving Applied Problems Using Mathematical
Models (continued)
10
53
D x
9
9
10
53
10  x 
9
9
10   53 

9(10)  9  x   9  
9  9
90  10 x  53
37  10x
x  3.7
The intensity of the event is 3.7
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