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Chapter 1: Linear Functions, Equations,
and Inequalities
1.1 Real Numbers and the Rectangular
Coordinate System
1.2 Introduction to Relations and Functions
1.3 Linear Functions
1.4 Equations of Lines and Linear Models
1.5 Linear Equations and Inequalities
1.6 Applications of Linear Functions
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1.5 Linear Equations and Inequalities
• Equations
– statements that two expressions are equal
– to solve an equation means to find all numbers
that will satisfy the equation
– the solution (or root) of an equation is said to
satisfy the equation
– solution set is the list of all solutions
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1.5 Linear Equation in One Variable
A linear equation in one variable is an equation
that can be written in the form
ax b 0, a  0.
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1.5 Linear Equations and Inequalities
• Solving Linear Equations
– analytic: paper & pencil
– graphical: often supports analytic approach
with graphs and tables
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1.5 Addition and Multiplication Properties
• Addition and Multiplication Properties of Equality
For real numbers a, b, and c,
a  b and a  c  b  c are equivalent.
If c  0, then a  b and ac  bc are equivalent.
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1.5 Solving a Linear Equation
• Example
Solve 10  3(2 x  4)  17  ( x  5)
10  6 x  12  17  x  5
 2  7 x  12
7 x  14
x2
Distributi ve property
Add x to each side
Add 2 to each side
Divide each side by 7
Check 10  3(2(2)  4)  17  (2  5)
10  0  17  7
10  10
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1.5 Solving a Linear Equation
with Fractions
x  7 2x  8

 4
• Solve
6
2
 x  7 2 x  8
6

 6(4)

2 
 6
x  7  3(2 x  8)  24
x  7  6 x  24  24
7 x  17  24
7 x  7
x  1
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1.5 Graphical Solutions to f (x) = g(x)
• Three possible solutions
y
y
x
1 point
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y
x
No points
x
Infinitely
many points
(coincide)
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1.5 Intersection-of-Graphs Method
•
First Graphical Approach to Solving Linear
Equations
– f ( x)  g ( x), where f and g are linear functions
1. set y1  f ( x) and y2  g ( x) and graph
2. find points of intersection, if any, using intersect
in the CALC menu
–
e.g. 10  3(2 x  4)  17  ( x  5)
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1.5 Intersection-of-Graphs Method
Intersection-of-Graphs Method of Graphical Solution
To solve the equation f ( x)  g ( x) graphically, solve
y1  f ( x) and y2  g ( x).
The x-coordinate of any point of intersection of the two
graphs is a solution of the equation.
Copyright © 2011 Pearson Education, Inc.
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1.5 Application
•
The percent share of music sales (in dollars) that compact discs (CDs) held
from 1987 to 1998 can be modeled by
f ( x )  5.91x  13.7.
During the same time period, the percent share of music sales that cassette
tapes held can be modeled by
g ( x )  4.71x  64.7.
In these formulas, x = 0 corresponds to 1987, x = 1 to 1988, and so on. Use the
intersection-of-graphs method to estimate the year when sales of CDs equaled
sales of cassettes.
Solution:  4.71x  64.7  5.91x  13.7
100
1987  4.8  1992 . It follows that in 1992, both
CDs and cassettes shared about 42.1% of sales.
0
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12
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1.5 The x-Intercept Method
• Second Graphical Approach to Solving a Linear
Equation
f ( x)  g ( x)
f ( x)  g ( x)  0
– set y1  f ( x)  g ( x) and any x-intercept (or zero) is a
solution of the equation
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1.5
The x-Intercept Method
x-intercept Method of Graphical Solution
To solve the equation f ( x)  g ( x) graphically, solve
y  f ( x)  g ( x)  F ( x).
The x-intercept of the graph of F (or zero of the function F)
is a solution of the equation.
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1.5 The x-Intercept Method
• Root, solution, and zero refer to the same basic
concept:
– real solutions of f ( x)  0 correspond to the x-intercepts
of the graph y  f (x)
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1.5 Example Using the x-Intercept Method
• Solve the equation
6 x  4(3  2 x)  5( x  4)  10
6 x  4(3  2 x)  (5( x  4)  10)  0
Graph hits x-axis at
x = –2. Use Zero in
CALC menu.
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1.5 Identities and Contradictions
• A contradiction is an equation that has no solution.
– e.g. x  x  3
y2  x  3
y1  x
two parallel lines
The solution set is the empty or null set, denoted .
Copyright © 2011 Pearson Education, Inc.
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1.5 Identities and Contradictions
• An identity is an equation that is true for all values
in the domain.
– e.g. 2( x  3)  2 x  6
lines coincide
Solution set (, ).
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1.5 Identities and Contradictions
• Note:
– Contradictions and identities are not linear, since linear
equations must be of the form
ax  b  0, a  0
– linear equations - one solution
– contradictions - always false
– identities - always true
Copyright © 2011 Pearson Education, Inc.
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1.5 Solving Linear Inequalities
Addition and Multiplication Properties of Inequality
a. a  b and a  c  b  c are equivalent.
b. If c  0, then a  b and ac  bc are equivalent.
c. If c  0, then a  b and ac  bc are equivalent.
Copyright © 2011 Pearson Education, Inc.
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1.5 Solving Linear Inequalities
•
Example
3x  2(2 x  4)  2 x  1
3x  4 x  8  2 x  1
 x  8  2x  1
 3x  9
x  3 or [3,)
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1.5 Solve a Linear Inequality
with Fractions
2x  3 
x2
3
 6x  9  x  2
Reverse the inequality
symbol when multiplying
by a negative number.
9  7x  2
7  7x
1  x or
x 1
The solution set is (,1).
Copyright © 2011 Pearson Education, Inc.
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1.5 Graphical Approach to Solving
Linear Inequalities
Intersection-of-Graphs Method of Solution of a Linear Inequality
Suppose that f and g are linear functions. The solution set of f ( x)  g ( x) is
the set of all real numbers x such that the graph of f is above the graph of g.
The solution set of f ( x )  g ( x ) is the set of all real numbers x such that the
graph of f is below the graph of g.
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1.5 Intersection of Graphs Method
Example: 3x  2(2 x  4)  2 x  1
10
-10
10
y2  2 x  1
10
y1  3 x  2( 2 x  4)
-15
Solution set : [ 3,  )
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1.5 Intersection of Graphs Method
Agreement of Inclusion of Exclusion of Endpoints for Approximations
When an approximation is used for an endpoint in specifying an
interval, we continue to use parentheses in specifying inequalities
involving < or > and square brackets in specifying inequalities
involving < or >.
Copyright © 2011 Pearson Education, Inc.
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1.5 x-Intercept Method
x-intercept Method of Solution of a Linear Inequality
The solution set of F ( x )  0 is the set of all real numbers x such that
the graph of F is above the x-axis. The solution set of F ( x )  0 is the set
of all real numbers x such that the graph of F is below the x-axis.
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1.5
x-Intercept Method
Example:
Let y1  2(3 x  1) and y 2  4( x  2), then y1  y 2 can be
written as y1  y 2  0. So we graph
y1  y 2  2 (3 x  1)  4( x  2 )  0 and find the x - intercept
using the zero function in the CALC menu.
y1  2 ( 3 x  1)  4 ( x  2 )
Solution set : (-1, )
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1.5 Three-Part Inequalities
• Application
Consider error tolerances in manufacturing a can with
radius of 1.4 inches.
• r can vary by  0.02 inches
1.38  r  1.42
• Circumference C  2r varies between 2 (1.38)  8.67 inches
and 2 (1.42)  8.92 inches
8.67  C  8.92
r
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1.5 Solving a Three-Part Inequality
• Example
 2  5  3x  20
 7  3x  15
7
 x5
3
 7 
The solution set is   ,5  .
 3 
Graphical Solution
y 3  20
y 3  20
25
25
y2  5  3x
y2  5  3x
-20
-20
6
y
-20
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1
6
 2
y
-20
1
 2
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