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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


1.5 Linear Equations and Inequalities

• Solving Linear Equations– analytic: paper & pencil– graphic: often supports analytic approach with graphs

and tables

• Equations – statements that two expressions are equal– to solve an equation means to find all numbers that will

satisfy the equation– the solution to an equation is said to satisfy the

equation– solution set is the list of all solutions


1.5 Linear Equation in One Variable

• Linear Equation in One Variable

• Addition and Multiplication Properties of Equality–

– If

0 ,0 abax

equivalent are and cbcaba equivalent are and then ,0 bcacbac


1.5 Solve a Linear Equation

• Example

Solve

Check

)5(17)42(310 xx

7by sideeach Divide

sideeach to2 Add

sideeach to Add

property veDistributi

2 147 1272 51712610

xxx

xxx

1010

717010

)52(17)4)2(2(310


1.5 Solve a Linear Equation with Fractions

• Solve 42

826

7 xx

1

77

24177

242467

24)82(37

)4(62

826

76

x

x

x

xx

xx

xx


1.5 Graphical Solutions to f (x) = g(x)

• Three possible solutions

x x x

y y y

1 point No points Infinitely many points (coincide)


1.5 Intersection-of-Graphs Method

• First Graphical Approach to Solving Linear Equations

– where f and g are linear functions

1. set and graph

2. find points of intersection, if any, using intersect in the CALC menu

– e.g.

),()( xgxf )( and )(

21xgyxfy

)5(17)42(310 xx


1.5 Application

• The percent share of music sales (in dollars) that compact discs (CDs) held from 1987 to 1998 can be modeled by

During the same time period, the percent share of music sales that cassette tapes held can be modeled by

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:

.7.1391.5)( xxf

.7.6471.4)( xxg

7.1391.57.6471.4 xx

0 12

100

sales. of 42.1%about shared cassettes and CDs

both 1992,in that followsIt .19928.41987


1.5 The x-Intercept Method

• Second Graphical Approach to Solving a Linear Equation

– set and any x-intercept (or zero) is a solution of the equation

• Root, solution, and zero refer to the same basic concept:– real solutions of correspond to the x-intercepts

of the graph

0)()(

)()(

xgxf

xgxf

)()(1

xgxfy

0)( xf)(xfy


1.5 Example Using the x-Intercept Method

• Solve the equation10)4(5)23(46 xxx

0)10)4(5()23(46 xxx

Graph hits x-axis at x = –2. Use Zero in CALC menu.


two parallel lines

1.5 Identities and Contradictions

• Contradiction – equation that has no solution– e.g.

The solution set is the empty or null set, denoted

3xx

xy 1

32

xy

.



• Identity – equation that is true for all values in the domain– e.g.

Solution set ).,(

62)3(2 xx

lines coincide



• Note:– Contradictions and identities are not linear, since linear

equations must be of the form

– linear equations - one solution

– contradictions - always false

– identities - always true

0 ,0 abax


1.5 Solving Linear Inequalities

• Properties of Inequalitya.

b.

c.

• Example

equivalent are and cbcaba equivalent are and then ,0 If bcacbac equivalent are and then ,0 If bcacbac

)3[or 393

1281284312)42(23

, xx

xxxxxxxx


1.5 Solve a Linear Inequality with Fractions

32

32 x

x

296 xx

1

or 1

77

279

x

x

x

x

).1,( isset solution The

Reverse the inequality symbol when multiplying by a negative number.


1.5 Graphical Approach to Solving Linear Inequalities

• Two Methods 1. Intersection-of-Graphs

– where the solution is the set of all real numbers x such that f is above the graph of g.

– Similarly for f is below the graph of g.

– e.g.

)()( xgxf

),()( xgxf 12)42(23 xxx

)42(231

xxy

122

xy

)3,[ :setSolution

-10 1010

10

-15


1.5 Graphical Approach to Solving Linear Inequalities

2. x-intercept Method – is the set of all real numbers x such that the

graph of F is above the x-axis.

– Similarly for F (x) < 0, the graph of F is below the x-axis.

– e.g.

0)( xF

)2(4)13(2 xx

menu. CALC in thefunction theusing

intercept - thefind and 0)2(4)13(2

graph weSo .0 aswritten

becan then ,)2(4 and )13(2Let

21

21

2121

zero

xxxyy

yy

yyxyxy

)(-1, :setSolution

)2(4)13(21

xxy


1.5 Three-Part Inequalities

• Application– error tolerances in manufacturing a can with radius of

1.4 inches• r can vary by

• Circumference varies between and

inches 02.042.138.1 r

rC 2 inches 67.8)38.1(2 inches 92.8)42.1(2

92.867.8 C

r


1.5 Solving a Three-Part Inequality

• Example

Graphical Solution

20352 x

1537 x

)5,( isset solution The

537

37-

x

203

y 203

y

21

y

xy 352

xy 352

-20 -20

25 25

-20 -20 6 6

21

y

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