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GRADUATE RECORD EXAMINATIONS®

Math Review

Chapter 2: Algebra

Copright ! 2"#" b E$%&atio'al Te(ti'g Servi&e) Allright( re(erve$) ETS* the ETS logo* GRADUATERECORD EXAMINATIONS* a'$ GRE are regi(tere$

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GRE Math Review 2 Algebra 1



The GRE®

Math Review consists of 4 chapters: Arithmetic, Algebra, Geometry, an

!ata Analysis" This is the accessible electronic format #$or% eition of the Algebra

&hapter of the Math Review" !ownloaable versions of large print #'!(% an accessible

electronic format #$or% of each of the 4 chapters of the of the Math Review, as well as a

)arge 'rint (ig*re s*pplement for each chapter are available from the GRE® website"

+ther ownloaable practice an test familiariation materials in large print an

accessible electronic formats are also available" Tactile fig*re s*pplements for the fo*r

chapters of the Math Review, along with aitional accessible practice an test

familiariation materials in other formats, are available from E T - !isability -ervices

Monay to (riay .:/0 a m to p m ew 3or time, at 156 0 758 8 158 8 . 0, or

15. 6 65/ . 85. 6 0 2 #toll free for test taers in the 9nite -tates, 9 - Territories an

&anaa%, or via email at stassets"org"

The mathematical content covere in this eition of the Math Review is the same as the

content covere in the stanar eition of the Math Review" ;owever, there are

ifferences in the presentation of some of the material" These ifferences are the res*lt of

aaptations mae for presentation of the material in accessible formats" There are also

slight ifferences between the vario*s accessible formats, also as a res*lt of specific

aaptations mae for each format"

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Figures

The Math Review incl*es fig*res" <n accessible electronic format #$or% eitions,

following each fig*re on screen is te=t escribing that fig*re" Reaers *sing vis*al

presentations of the fig*res may choose to sip parts of the te=t escribing the fig*re that

begin with >?egin sippable part of escription of @ an en with >En sippablefig*re escription"

Mathematical Equations and Expressions

The Math Review incl*es mathematical eB*ations an e=pressions" <n accessible

electronic format #$or% eitions some of the mathematical eB*ations an e=pressions

are presente as graphics" <n cases where a mathematical eB*ation or e=pression is presente as a graphic, a verbal presentation is also given an the verbal presentation

comes irectly after the graphic presentation" The verbal presentation is in green font to

assist reaers in telling the two presentation moes apart" Reaers *sing a*io alone can

safely ignore the graphical presentations, an reaers *sing vis*al presentations may

ignore the verbal presentations"

GRE Math Review 2 Algebra /



Table of Contents

+verview of the Math Review"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

+verview of this &hapter"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

2"1 +perations with Algebraic E=pressions""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""6

2"2 R*les of E=ponents"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""11

2"/ -olving )inear EB*ations"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""18

2"4 -olving C*aratic EB*ations""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""22

2" -olving )inear <neB*alities""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""2

2"6 (*nctions""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""2.

2"8 Applications"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""/0

2". &oorinate Geometry"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""41

2"7 Graphs of (*nctions"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""60

Algebra E=ercises""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""84

Answers to Algebra E=ercises"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""".2




Overview of the Math Review

The Math Review consists of 4 chapters: Arithmetic, Algebra, Geometry, an !ata

Analysis"

Each of the 4 chapters in the Math Review will familiarie yo* with the mathematical

sills an concepts that are important to *nerstan in orer to solve problems an reason

B*antitatively on the C*antitative Reasoning meas*re of the GRE®

revise General Test"

The material in the Math Review incl*es many efinitions, properties, an e=amples, as

well as a set of e=ercises #with answers% at the en of each chapter" ote, however that

this review is not intene to be all incl*sive" There may be some concepts on the test

that are not e=plicitly presente in this review" <f any topics in this review seem especially

*nfamiliar or are covere too briefly, we enco*rage yo* to cons*lt appropriate

mathematics te=ts for a more etaile treatment"

Overview of this Chapter

?asic algebra can be viewe as an e=tension of arithmetic" The main concept that

isting*ishes algebra from arithmetic is that of a variable, which is a letter that

represents a B*antity whose val*e is *nnown" The letters x an y are often *se as

variables, altho*gh any letter can be *se" Dariables enable yo* to present a wor

problem in terms of *nnown B*antities by *sing algebraic e=pressions, eB*ations,

ineB*alities, an f*nctions" This chapter reviews these algebraic tools an then progressesto several e=amples of applying them to solve real life wor problems" The chapter ens

with coorinate geometry an graphs of f*nctions as other important algebraic tools for

solving problems"

GRE Math Review 2 Algebra



2.1 Operations with Algebraic Expressions

An algebraic expression has one or more variables an can be written as a single term

or as a s*m of terms" ;ere are fo*r e=amples of algebraic e=pressions"

E=ample A: 2 x

E=ample ?: y min*s, one fo*rth

E=ample &: w c*be z , , , z sB*are, min*s z sB*are, , 6

E=ample !: the e=pression with n*merator . an enominator n p

<n the e=amples above, 2 x is a single term,

y min*s, one fo*rth has two terms,

w c*be z , , , z sB*are, min*s z sB*are, , 6 has fo*r terms, an

the e=pression with n*merator . an enominator n p has one term"

<n the e=pression w c*be z , , , z sB*are, min*s z sB*are, , 6,

the terms , z sB*are, an negative, z sB*are

are calle like terms beca*se they have the same variables, an the corresponing

variables have the same e=ponents" A term that has no variable is calle a constant term"




A n*mber that is m*ltiplie by variables is calle the coefficient of a term" (or e=ample,

in the e=pression

2, x sB*are, , 8 x, min*s ,

2 is the coefficient of the term 2, x sB*are,

8 is the coefficient of the term 8 x, an

negative is a constant term"

The same r*les that govern operations with n*mbers apply to operations with algebraic

e=pressions" +ne aitional r*le, which helps in simplifying algebraic e=pressions, is thatlie terms can be combine by simply aing their coefficients, as the following three

e=amples show"

E=ample A: 2 x x F 8 x

E=ample ?:

w c*be z , , , z sB*are, min*s z sB*are, , 6 F w c*be z , , 4, z sB*are, , 6

E=ample &:

/ x y, , 2 x, min*s x y, min*s / x F 2 x y, min*s x

A n*mber or variable that is a factor of each term in an algebraic e=pression can be

factore o*t, as the following three e=amples show"

E=ample A:

4 x 12 F 4 times, open parenthesis, x /, close parenthesis




E=ample ?:

1, y sB*are, min*s 7 y, F, / y times, open parenthesis, y min*s /, close parenthesis

E=ample &: (or val*es of x where it is efine, the algebraic e=pressionwith n*merator 8, x sB*are, , 14 x an enominator 2 x, , 4

can be simplifie as follows"

(irst factor the n*merator an the enominator to get

the algebraic e=pression with n*merator 8 x times, open parenthesis, x 2, close

parenthesis, an enominator 2 times, open parenthesis, x 2, close parenthesis"

ow, since x 2 occ*rs in both the n*merator an the enominator, it can be cancele

o*t when x 2 is not eB*al to 0, that is, when x is not eB*al to

negative 2 #since ivision by 0 is not efine%" Therefore, for all x not eB*al

to negative 2, the e=pression is eB*ivalent to 8 x over 2"

To m*ltiply two algebraic e=pressions, each term of the first e=pression is m*ltiplie by

each term of the secon e=pression, an the res*lts are ae, as the following e=ample

shows"

To m*ltiply

open parenthesis, x 2, close parenthesis, times, open parenthesis / x min*s 8, close parenthesis

first m*ltiply each term of the e=pression x 2 by each term of the e=pression / x

min*s 8 to get the e=pression

GRE Math Review 2 Algebra .



x times / x, , x times negative 8, , 2 times / x, , 2 times negative 8"

Then m*ltiply each term to get

/, x sB*are, min*s 8 x, , 6 x, min*s 14"

(inally, combine lie terms to get

/, x sB*are, min*s x, min*s 14"

-o yo* can concl*e that

open parenthesis, x 2, close parenthesis, times, open parenthesis / x min*s 8, close

parenthesis F /, x sB*are, min*s x, min*s 14"

A statement of eB*ality between two algebraic e=pressions that is tr*e for all possibleval*es of the variables involve is calle an identity" All of the statements above are

ientities" ;ere are three stanar ientities that are *sef*l"

<entity 1:

open parenthesis, a + b, close parenthesis, sB*are, F, a sB*are, , 2 a b, , b

sB*are

<entity 2:

open parenthesis, a min*s b, close parenthesis, c*be, F, a c*be, min*s /, a sB*are

b, , /a b sB*are, min*s b c*be




<entity /:

a sB*are min*s b sB*are F open parenthesis, a b, close parenthesis, times, open

parenthesis, a min*s b, close parenthesis

All of the ientities above can be *se to moify an simplify algebraic e=pressions" (or

e=ample, ientity /,

a sB*are min*s b sB*are F open parenthesis, a b, close parenthesis, times, open

parenthesis, a min*s b, close parenthesis

can be *se to simplify the algebraic e=pression

with n*merator x sB*are min*s 7 an enominator 4 x, min*s 12

as follows"

the algebraic e=pression with n*merator x sB*are min*s 7 an enominator 4 x min*s 12

F the algebraic e=pression with n*merator, open parenthesis, x /, close parenthesis,

times, open parenthesis, x min*s /, close parenthesis, an enominator 4 times, open

parenthesis, x min*s /, close parenthesis"

ow, since x min*s / occ*rs in both the n*merator an the enominator, it can be

cancele o*t when x min*s / is not eB*al to 0, that is, when x is not

eB*al to / #since ivision by 0 is not efine%" Therefore, for all x not eB*al to /,




the e=pression is eB*ivalent to the e=pression with n*merator x / an

enominator 4"

A statement of eB*ality between two algebraic e=pressions that is tr*e for only certainval*es of the variables involve is calle an equation" The val*es are calle the solutions

of the eB*ation"

The following are three basic types of eB*ations"

Type 1: A linear equation in one variable: for e=ample,/ x F negative 2

Type 2: A linear equation in to variables: for e=ample,

x min*s / y F 10

Type /: ! quadratic equation in one variable: for e=ample

20, y sB*are, , 6 y min*s 18 F 0

2.2 Rules of Exponents

<n the algebraic e=pression x s*perscript a, where x is raise to the power a, x is

calle a base an a is calle an exponent" ;ere are seven basic r*les of e=ponents, where

the bases x an y are nonero real n*mbers an the e=ponents a an b are integers"

R*le 1:

x to the power negative a F 1, over, x to the power a




E=ample A:

4 to the power negative / F 1, over, 4 to the power /, which is eB*al to 1 over 64

E=ample ?:

x to the power negative 10 F 1 over, x to the power 10

E=ample &:

1, over, 2 to the power negative a F 2 to the power a

R*le 2:

+pen parenthesis, x to the power a, close parenthesis, times, open parenthesis, x to the

power b, close parenthesis, F, x to the power a b

E=ample A:

+pen parenthesis, / sB*are, close parenthesis, times, open parenthesis, / to the

power 4, close parenthesis, F, / to the power 2 4, which is eB*al to / to the power

6, or 827

E=ample ?:

+pen parenthesis, y c*be, close parenthesis, times, open parenthesis, y to the power

negative 1, close parenthesis F, y sB*are

R*le /:

x to the power a, over, x to the power b, F, x to the power a min*s b, which is eB*al to

1 over, x to the power, b min*s a




E=ample A:

to the power 8, over, to the power 4, F, to the power 8 min*s 4, which is eB*al

to to the power /, or 12

E=ample ?:

t to the power /, over, t to the power ., F, t to the power negative , which is eB*al

to 1, over, t to the power

R*le 4: x to the power 0 F 1

E=ample A: 8 to the power 0 F 1

E=ample ?:

open parenthesis, negative /, close parenthesis, to the power 0, F, 1

ote that 0 to the power 0 is not efine"

R*le :

+pen parenthesis, x to the power a, close parenthesis, times, open parenthesis, y to the

power a, close parenthesis, F, open parenthesis, x y, close parenthesis, to the power a

E=ample A:

+pen parenthesis, 2 to the power /, close parenthesis, times, open parenthesis, / to

the power /, close parenthesis, F, 6 to the power /, or 216

GRE Math Review 2 Algebra 1/



E=ample ?:

+pen parenthesis, 10 z , close parenthesis, c*be, F, 10 c*be times z c*be, which is

eB*al to 1,000, z c*be

R*le 6:

+pen parenthesis, x over y, close parenthesis, to the power a, F, x to the power a, over

y to the power a

E=ample A:

+pen parenthesis, / fo*rths, close parenthesis, sB*are, F, / sB*are over 4 sB*are,

which is eB*al to 7 over 16

E=ample ?:

+pen parenthesis, r over 4t , close parenthesis, c*be, F r c*be, over, 64, t c*be

R*le 8:

+pen parenthesis, x to the power a, close parenthesis, to the power b, F, x to the

power a b

E=ample A:

+pen parenthesis, 2 to the power , close parenthesis, sB*are, F, 2 to the power 10,

which is eB*al to 1,024

E=ample ?:

+pen parenthesis, /, y to the power 6, close parenthesis, sB*are, F, open




parenthesis, / sB*are, close parenthesis, times, open parenthesis, y to the power 6,

close parenthesis, sB*are, which is eB*al to 7, y to the power 12

The r*les above are ientities that are *se to simplify e=pressions" -ometimes algebraic

e=pressions loo lie they can be simplifie in similar ways, b*t in fact they cannot" <norer to avoi mistaes commonly mae when ealing with e=ponents eep the following

si= cases in min"

&ase 1:

x to the power a times y to the power b is not eB*al to, open parenthesis, x y, close

parenthesis, to the power a b

ote that in the e=pression x to the power a times y to the power b the bases

are not the same, so R*le 2,

open parenthesis, x to the power a, close parenthesis, times, open parenthesis, x to

the power b, close parenthesis, F, x to the power a b,

oes not apply"

&ase 2:

+pen parenthesis, x to the power a, close parenthesis, to the power b is not eB*al to, x

to the power a times x to the power b

<nstea,

+pen parenthesis, x to the power a, close parenthesis, to the power b, F, x to the

power a b

an




x to the power a times x to the power b, F, x to the power a b

for e=ample,

open parenthesis, 4 sB*are, close parenthesis, c*be, F, 4 to the power 6, an 4

sB*are times 4 c*be, F, 4 to the power "

&ase /:

open parenthesis, x + y, close parenthesis, to the power a, is not eB*al to x to the

power a, , y to the power a

Recall that

open parenthesis, x + y, close parenthesis, sB*are, F, x sB*are, , 2 x y, , y

sB*are

that is, the correct e=pansion contains terms s*ch as 2 x y"

&ase 4:

+pen parenthesis, negative x, close parenthesis, sB*are, is not eB*al to the negative

of, x sB*are

<nstea,

+pen parenthesis, negative x, close parenthesis, sB*are F, x sB*are

ote caref*lly where each negative sign appears"

&ase :

The positive sB*are root of the B*antity x sB*are y sB*are, is not eB*al to x + y




&ase 6:

The e=pression with n*merator a an enominator x y, is not eB*al to a over x, , a

over y

?*t it is tr*e that

the e=pression with n*merator x y, an enominator a F, x over a, , y over a"

2.3 Solving inear E!uations

To solve an equation means to fin the val*es of the variables that mae the eB*ation

tr*e that is, the val*es that satisfy the equation" Two eB*ations that have the same

sol*tions are calle equivalent equations" (or e=ample, x 1 F 2 an 2 x 2 F 4 are

eB*ivalent eB*ations both are tr*e when x F 1, an are false otherwise" The general

metho for solving an eB*ation is to fin s*ccessively simpler eB*ivalent eB*ations so

that the simplest eB*ivalent eB*ation maes the sol*tions obvio*s"

The following two r*les are important for pro*cing eB*ivalent eB*ations"

R*le 1: $hen the same constant is ae to or s*btracte from both sies of an

eB*ation, the eB*ality is preserve an the new eB*ation is eB*ivalent to the original

eB*ation"

R*le 2: $hen both sies of an eB*ation are m*ltiplie or ivie by the same

nonero constant, the eB*ality is preserve an the new eB*ation is eB*ivalent to the

original eB*ation"




A linear equation is an eB*ation involving one or more variables in which each term in

the eB*ation is either a constant term or a variable m*ltiplie by a coefficient" one of the

variables are m*ltiplie together or raise to a power greater than 1" (or e=ample,

2 x 1 F 8 x an 10 x min*s 7 y min*s z F / are linear eB*ations, b*t

x, , y sB*are, F, 0 an xz F / are not"

"inear Equations in #ne $ariable

To solve a linear eB*ation in one variable, simplify each sie of the eB*ation by

combining lie terms" Then *se the r*les for pro*cing simpler eB*ivalent eB*ations"

E=ample 2"/"1: -olve the eB*ation

11 x, min*s 4, min*s . x, F, 2 times, open parenthesis, x 4, close parenthesis, min*s

2 x

as follows"

&ombine lie terms to get/ x min*s 4, F, 2 x . min*s 2 x

-implify the right sie to get / x min*s 4, F, .

A 4 to both sies to get / x, min*s 4, 4, F, . 4

!ivie both sies by / to get / x over / F 12 over /

-implify to get x F 4

3o* can always chec yo*r sol*tion by s*bstit*ting it into the original eB*ation"

GRE Math Review 2 Algebra 1.



ote that it is possible for a linear eB*ation to have no sol*tions" (or e=ample, the

eB*ation 2 x /, F, 2 times, open parenthesis, 8 x, close parenthesis,

has no sol*tion, since it is eB*ivalent to the eB*ation / F 14, which is false" Also, it is

possible that what loos to be a linear eB*ation t*rns o*t to be an ientity when yo* try to

solve it" (or e=ample, / x min*s 6, F, negative / times, open

parenthesis, 2 min*s x, close parenthesis is tr*e for all val*es of x, so it is an ientity"

"inear Equations in To $ariables

A linear eB*ation in two variables, x an y, can be written in the form a x b y F c,

where a, b, an c are real n*mbers an a an b are not both ero" (or e=ample,

/ x 2 y F ., is a linear eB*ation in two variables"

A sol*tion of s*ch an eB*ation is an ordered pair of n*mbers x comma y that

maes the eB*ation tr*e when the val*es of x an y are s*bstit*te into the eB*ation" (or

e=ample, both pairs 2 comma 1, an negative 2 over / comma are

sol*tions of the eB*ation / x 2 y F ., b*t 1 comma 2 is not a sol*tion" A linear

eB*ation in two variables has infinitely many sol*tions" <f another linear eB*ation in the

same variables is given, it may be possible to fin a *niB*e sol*tion of both eB*ations"

Two eB*ations with the same variables are calle a system of equations, an the

eB*ations in the system are calle simultaneous equations" To solve a system of two

eB*ations means to fin an orere pair of n*mbers that satisfies both eB*ations in the

system"




There are two basic methos for solving systems of linear eB*ations, by substitution or

by elimination" <n the s*bstit*tion metho, one eB*ation is manip*late to e=press one

variable in terms of the other" Then the e=pression is s*bstit*te in the other eB*ation"

(or e=ample, to solve the system of two eB*ations

4 x / y F 1/, an

x 2 y F 2

yo* can e=press x in the secon eB*ation in terms of y as

x F 2 min*s 2 y"

Then s*bstit*te 2 min*s 2 y for x in the first eB*ation to fin the val*e of y"

The val*e of y can be fo*n as follows"

-*bstit*te for x in the first eB*ation to get

4 times, open parenthesis, 2 min*s 2 y, close parenthesis, , / y, F, 1/

M*ltiply o*t the first term an get:

. min*s . y, , / y, F, 1/

-*btract . from both sies to get

negative . y / y F

&ombine lie terms to get negative y F

!ivie both sies by negative to get y F negative 1"




Then negative 1 can be s*bstit*te for y in either eB*ation to fin the val*e of x" $e

*se the secon eB*ation as follows:

-*bstit*te for y in the secon eB*ation to get

x , 2 times negative 1 F 2

That is, x min*s 2 F 2

A 2 to both sies to get x F 4

<n the elimination metho, the obHect is to mae the coefficients of one variable the same

in both eB*ations so that one variable can be eliminate either by aing the eB*ations

together or by s*btracting one from the other" <n the e=ample above, m*ltiplying both

sies of the secon eB*ation, x 2 y F 2, by 4 yiels

4 times, open parenthesis, x 2 y, close parenthesis, F, 4 times 2,

or 4 x . y F ."

ow yo* have two eB*ations with the same coefficient of x"

4 x / y F 1/, an

4 x . y F .

<f yo* s*btract the eB*ation 4 x . y F . from the eB*ation 4 x / y F 1/, the res*lt is

negative y F " Th*s, y F negative 1, an s*bstit*ting negative 1

for y in either of the original eB*ations yiels x F 4"




?y either metho, the sol*tion of the system is x F 4 an y F negative 1, or

the orere pair x comma y F the orere pair 4 comma negative 1"

2." Solving #ua$ratic E!uations

A quadratic equation in the variable x is an eB*ation that can be written in the form

a x sB*are bx c F 0,

where a, b, an c are real n*mbers an a is not eB*al to 0" $hen s*ch an eB*ation

has sol*tions, they can be fo*n *sing the quadratic formula:

x F the fraction with n*merator negative b pl*s or min*s the sB*are root of the B*antity b

sB*are min*s 4a c, an enominator 2a,

where the notation pl*s or min*s is shorthan for inicating two sol*tions, one that

*ses the pl*s sign an the other that *ses the min*s sign"

E=ample 2"4"1: <n the B*aratic eB*ation

2, x sB*are, min*s x, min*s 6 F 0, we have

a F 2, b F negative 1, an c F negative 6"

Therefore, the B*aratic form*la yiels




x F the fraction with n*merator, negative, open parenthesis, negative 1, close

parenthesis, pl*s or min*s the sB*are root of the B*antity, open parenthesis, negative

1, close parenthesis, sB*are, min*s 4 times 2 times negative 6, an enominator 2

times 2, which is eB*al to the fraction with n*merator 1 pl*s or min*s the sB*are root

of 47 an enominator 4, which is eB*al to the fraction with n*merator 1 pl*s or

min*s 8, an enominator 4

;ence the two sol*tions are

x F the fraction with n*merator 1 8, an enominator 4, which is eB*al to 2, an x F

the fraction with n*merator 1 min*s 8, an enominator 4, which is eB*al to negative

/ over 2"

C*aratic eB*ations have at most two real sol*tions, as in e=ample 2"4"1 above" ;owever,

some B*aratic eB*ations have only one real sol*tion" (or e=ample, the B*aratic

eB*ation x sB*are, 4 x, 4 F 0 has only one sol*tion, which is

x F negative 2" <n this case, the e=pression *ner the sB*are root symbol in the

B*aratic form*la is eB*al to 0, an so aing or s*btracting 0 yiels the same res*lt"

+ther B*aratic eB*ations have no real sol*tions for e=ample,

x sB*are, x, F 0" <n this case, the e=pression *ner the sB*are root symbol is

negative, so the entire e=pression is not a real n*mber"




-ome B*aratic eB*ations can be solve more B*icly by factoring" (or e=ample, the

B*aratic eB*ation 2, x sB*are, min*s x, min*s 6 F 0 in e=ample 2"4"1

can be factore as

open parenthesis, 2 x /, close parenthesis, times, open parenthesis, x min*s 2, close

parenthesis, F 0"

$hen a pro*ct is eB*al to 0, at least one of the factors m*st be eB*al to 0, so either

2 x / F 0 or x min*s 2 F 0"

<f 2 x / F 0, then

2 x F negative /, an x F negative / over 2"

<f x min*s 2 F 0, then x F 2"

Th*s the sol*tions are negative / over 2, an 2"

E=ample 2"4"2: The B*aratic eB*ation

, x sB*are, / x, min*s 2 F 0

can be easily factore as

open parenthesis, x, min*s 2, close parenthesis, times, open parenthesis, x 1, close

parenthesis, F 0"

Therefore, either , x, min*s 2 F 0, or x 1 F 0"

<f x, min*s 2 F 0, then x F 2 over "

<f x 1 F 0, then x F negative 1"




Th*s the sol*tions are 2 over , an negative 1"

2.% Solving inear &ne!ualities

A mathematical statement that *ses one of the following fo*r ineB*ality signs is calle an

inequality"

ote: The fo*r ineB*ality signs are given as graphics" -ince the meaning of each is given

irectly after the graphic, a >green font verbal escription of these symbols is not

incl*e"

the less than sign

the greater than sign

the less than or eB*al to sign

the greater than or eB*al to sign

<neB*alities can involve variables an are similar to eB*ations, e=cept that the two sies

are relate by one of the ineB*ality signs instea of the eB*ality sign *se in eB*ations"

(or e=ample, the ineB*ality 4 x min*s 1, followe by the less than or eB*al to

sign, followe by the n*mber 8 is a linear ineB*ality in one variable, which states that

> 4 x min*s 1 is less than or eB*al to 8" To solve an inequality means to fin the

set of all val*es of the variable that mae the ineB*ality tr*e" This set of val*es is also

nown as the solution set of an ineB*ality" Two ineB*alities that have the same sol*tion

set are calle equivalent inequalities"




The proce*re *se to solve a linear ineB*ality is similar to that *se to solve a linear

eB*ation, which is to simplify the ineB*ality by isolating the variable on one sie of the

ineB*ality, *sing the following two r*les"

R*le 1: $hen the same constant is ae to or s*btracte from both sies of anineB*ality, the irection of the ineB*ality is preserve an the new ineB*ality is

eB*ivalent to the original"

R*le 2: $hen both sies of the ineB*ality are m*ltiplie or ivie by the same

nonero constant, the irection of the ineB*ality is preserve$ if the constant is

positive b*t the irection is reverse$ if the constant is negative" <n either case, the

new ineB*ality is eB*ivalent to the original"

E=ample 2""1: The ineB*ality negative / x, , is less than or eB*al to

18 can be solve as follows"

-*btract from both sies to get negative / x is less than or eB*al to 12

!ivie both sies by negative / an reverse the irection of the ineB*ality to get

negative / x over negative / is greater than or eB*al to 12 over negative /"

That is, x is greater than or eB*al to negative 4"

Therefore, the sol*tion set of

negative / x, , , is less than or eB*al to 18




consists of all real n*mbers greater than or eB*al to negative 4"

E=ample 2""2: The ineB*ality the algebraic e=pression with n*merator

4 x 7 an enominator 11, is less than can be solve as follows"

M*ltiply both sies by 11 to get 4 x 7 is less than "

-*btract 7 from both sies to get 4 x is less than 46"

!ivie both sies by 4 to get x is less than 46 over 4"

That is, x is less than 11""

Therefore, the sol*tion set of the ineB*ality the algebraic e=pression with

n*merator 4 x 7 an enominator 11, is less than consists of all real n*mbers less

than 11""

2.' (unctions

An algebraic e=pression in one variable can be *se to efine a function of that variable"

(*nctions are *s*ally enote by letters s*ch as f , g , an h" (or e=ample, the algebraic

e=pression / x can be *se to efine a f*nction f by

f of, x F / x ,




where f of, x is calle the val*e of f at x an is obtaine by s*bstit*ting the val*e of

x in the e=pression above" (or e=ample, if x F 1 is s*bstit*te in the e=pression above, the

res*lt is f of, 1 F ."

<t might be helpf*l to thin of a f*nction f as a machine that taes an inp*t, which is a

val*e of the variable x, an pro*ces the corresponing o*tp*t, f of, x" (or any

f*nction, each inp*t x gives e=actly one o*tp*t f of, x" ;owever, more than one

val*e of x can give the same o*tp*t f of, x" (or e=ample, if g is the f*nction efine

by g of, x F x sB*are, min*s 2 x, , /,

then g of, 0 F / an g of, 2 F /"

The domain of a f*nction is the set of all permissible inp*ts, that is, all permissible

val*es of the variable x" (or the f*nctions f an g efine above, the omain is the set of

all real n*mbers" -ometimes the omain of the f*nction is given e=plicitly an is

restricte to a specific set of val*es of x" (or e=ample, we can efine the f*nction h by

h of, x F x sB*are min*s 4, for, negative 2 less than oreB*al to x, which is less than or eB*al to 2"

$itho*t an e=plicit restriction, the omain is ass*me to be the set of all val*es of x for

which f of, x is a real n*mber"

E=ample 2"6"1: )et f be the f*nction efine by f of, x F the algebraic e=pression with n*merator 2 x, an enominator, x min*s 6"

<n this case, f is not efine at x F 6, beca*se 12 over 0 is not efine"

;ence, the omain of f consists of all real n*mbers e=cept for 6"




E=ample 2"6"2: )et g be the f*nction efine by

g of, x F x c*be, , the positive sB*are root of x 2, min*s 10"

<n this case, g of, x is not a real n*mber if x is less than negative 2"

;ence, the omain of g consists of all real n*mbers x s*ch that x is greater

than or eB*al to negative 2"

E=ample 2"6"/: )et h be the f*nction efine by h of, x F the absol*te

val*e of x, which is the istance between x an 0 on the n*mber line #see &hapter 1:

Arithmetic, -ection 1"%" The omain of h is the set of all real n*mbers" Also,

h of, x F h of, negative x for all real n*mbers x, which reflects the

property that on the n*mber line the istance between x an 0 is the same as the

istance between negative x an 0"

2.) Applications

Translating verbal escriptions into algebraic e=pressions is an essential initial step in

solving wor problems" Three e=amples of verbal escriptions an their translations are

given below"

E=ample A: <f the sB*are of the n*mber x is m*ltiplie by /, an then 10 is ae to

that pro*ct, the res*lt can be represente algebraically by /, x sB*are, ,

10"




E=ample ?: <f IohnJs present salary s is increase by 14 percent, then his new salary

can be represente algebraically by 1"14 s"

E=ample &: <f y gallons of syr*p are to be istrib*te among people so that one

partic*lar person gets 1 gallon an the rest of the syr*p is ivie eB*ally among the

remaining 4, then the n*mber of gallons of syr*p each of those 4 people will get can be represente algebraically by

the e=pression with n*merator y min*s 1, an enominator 4"

The remainer of this section gives e=amples of vario*s applications"

!pplications Involving !verage% Mixture% &ate% and 'ork (roblems

E=ample 2"8"1: Ellen has receive the following scores on / e=ams: .2, 84, an 70"

$hat score will Ellen nee to receive on the ne=t e=am so that the average #arithmetic

mean% score for the 4 e=ams will be . K

-ol*tion: )et x represent the score on EllenJs ne=t e=am" This initial step of assigning

a variable to the B*antity that is so*ght is an important beginning to solving the

problem" Then in terms of x, the average of the 4 e=ams is

the fraction with n*merator .2 84 70 x, an enominator 4,

which is s*ppose to eB*al ." ow simplify the e=pression an set it eB*al to .:

GRE Math Review 2 Algebra /0



the fraction with n*merator .2 84 70 x, an enominator 4, F, the fraction with

n*merator 246 x, an enominator 4, which is eB*al to ."

-olving the res*lting linear eB*ation for x, yo* get 246 x F /40, an x F 74"

Therefore, Ellen will nee to attain a score of 74 on the ne=t e=am"

E=ample 2"8"2: A mi=t*re of 12 o*nces of vinegar an oil is 40 percent vinegar,

where all of the meas*rements are by weight" ;ow many o*nces of oil m*st be ae

to the mi=t*re to pro*ce a new mi=t*re that is only 2 percent vinegarK

-ol*tion: )et x represent the n*mber of o*nces of oil to be ae" Then the total

n*mber of o*nces of the new mi=t*re will be 12 x an the total n*mber of o*nces of

vinegar in the new mi=t*re will be 0"40 times 12"

-ince the new mi=t*re m*st be 2 percent vinegar,

the fraction with n*merator 0"40 times 12 an enominator 12 x, F, 0"2"

Therefore,

0"40 times 12, F, open parenthesis, 12 x, close parenthesis, times 0"2"

M*ltiplying o*t gives 4". F / 0"2 x, so 1". F 0"2 x, an 8"2 F x"




Th*s, 8"2 o*nces of oil m*st be ae to pro*ce a new mi=t*re that is 2 percent

vinegar"

E=ample 2"8"/: <n a riving competition, Ieff an !ennis rove the same co*rse at

average spees of 1 miles per ho*r an 4 miles per ho*r, respectively" <f it too Ieff40 min*tes to rive the co*rse, how long i it tae !ennisK

-ol*tion: )et x be the time, in min*tes, that it too !ennis to rive the co*rse" The

istance d , in miles, is eB*al to the pro*ct of the rate r , in miles per ho*r, an the

time t , in ho*rs that is,

d F rt "

ote that since the rates are given in miles per hour , it is necessary to e=press the

times in ho*rs for e=ample, 40 min*tes eB*als 40 over 60 of an ho*r" Th*s, the

istance travele by Ieff is the pro*ct of his spee an his time,1 times, open parenthesis, 40 over 60, close parenthesis, miles,

an the istance travele by !ennis is similarly represente by

4 times, open parenthesis, x over 60, close parenthesis, miles"

-ince the istances are eB*al, it follows that "

1, times, open parenthesis, 40 over 60, close parenthesis, F, 4, times, open

parenthesis, x over 60, close parenthesis"

(rom this eB*ation it follows that

1 times 40 F 4 x




an

x F the fraction with n*merator 1 times 40 an enominator 4, which is

appro=imately /8"."

Th*s, it too !ennis appro=imately /8". min*tes to rive the co*rse"

E=ample 2"8"4: $oring alone at its constant rate, machine A taes / ho*rs to

pro*ce a batch of ientical comp*ter parts" $oring alone at its constant rate,

machine B taes 2 ho*rs to pro*ce an ientical batch of parts" ;ow long will it tae

the two machines, woring sim*ltaneo*sly at their respective constant rates, to

pro*ce an ientical batch of partsK

-ol*tion: -ince machine A taes / ho*rs to pro*ce a batch, machine A can pro*ce

one thir of the batch in 1 ho*r" -imilarly, machine B can pro*ce one half of

the batch in 1 ho*r" <f we let x represent the n*mber of ho*rs it taes both machines,

woring sim*ltaneo*sly, to pro*ce the batch, then the two machines will pro*ce

1 over x of the Hob in 1 ho*r" $hen the two machines wor together, aing their

inivi*al pro*ction rates, one thir an one half, gives their combine

pro*ction rate 1 over x" Therefore, it follows that "

one thir, , one half, F, 1 over x"

This eB*ation is eB*ivalent to2 over 6, , / over 6, F, 1 over x"

-o

over 6, F, 1 over x, an, 6 over , F, x"

GRE Math Review 2 Algebra //



Th*s, woring together, the machines will tae 6 over ho*rs, or 1 ho*r 12

min*tes, to pro*ce a batch of parts"

E=ample 2"8": At a fr*it stan, apples can be p*rchase for L0"1 each an pears for

L0"20 each" At these rates, a bag of apples an pears was p*rchase for L/".0" <f the

bag containe 21 pieces of fr*it, how many of the pieces were pearsK

-ol*tion: <f a represents the n*mber of apples p*rchase an p represents the n*mber

of pears p*rchase, the information can be translate into the following system of two

eB*ations"

Total &ost EB*ation: 0"1a 0"20 p F /".0

Total *mber of (r*it EB*ation: a p F 21

(rom the total n*mber of fr*it eB*ation, a F 21 min*s p"

-*bstit*ting 21 min*s p into the total cost eB*ation for a gives the eB*ation

0"1 times, open parenthesis, 21 min*s p, close parenthesis, , 0"20 p, F, /".0

-o,

0"1 times 21, min*s, 0"1 p, , 0"20 p, F, /".0,

which is eB*ivalent to

/"1, min*s, 0"1 p, , 0"20 p, F, /".0"

Therefore 0"0 p F 0"6, an p F 1/"




Th*s, of the 21 pieces of fr*it, 1/ were pears"

E=ample 2"8"6: To pro*ce a partic*lar raio moel, it costs a man*fact*rer L/0 per

raio, an it is ass*me that if 00 raios are pro*ce, all of them will be sol" $hat

m*st be the selling price per raio to ens*re that the profit #reven*e from the salesmin*s the total pro*ction cost% on the 00 raios is greater than L.,200 K

-ol*tion: <f y represents the selling price per raio, then the profit is

00 times, open parenthesis, y min*s /0, close parenthesis"

Therefore,

00 times, open parenthesis, y min*s /0, close parenthesis is greater than .,200"

M*ltiplying o*t gives

00 y min*s 1,000 is greater than .,200,

which simplifies to

00 y is greater than 2/,200

an then to y is greater than 46"4" Th*s, the selling price m*st be greater

than L46"40 to ens*re that the profit is greater than L.,200"

!pplications Involving Interest

-ome applications involve comp*ting interest earne on an investment *ring a specifie

time perio" The interest can be comp*te as simple interest or compo*n interest"

) imple interest is base only on the initial eposit, which serves as the amo*nt on which

interest is comp*te, calle the principal, for the entire time perio" <f the amo*nt P is




investe at a si*ple annual interest rate of r percent , then the val*e V of the investment

at the en of t years is given by the form*la

V F P times, open parenthesis, 1, , rt over 100, close parenthesis,

where P an V are in ollars"

<n the case of compound interest, interest is ae to the principal at reg*lar time

intervals, s*ch as ann*ally, B*arterly, an monthly" Each time interest is ae to the

principal, the interest is sai to be compo*ne" After each compo*ning, interest is

earne on the new principal, which is the s*m of the preceing principal an the interest

H*st ae" <f the amo*nt P is investe at an annual interest rate of r percent ,

co*poun$e$ annuall+, then the val*e V of the investment at the en of t years is given

by the form*la

V F P times, open parenthesis, 1, , r over 100, close parenthesis, to the power t "

<f the amo*nt P is investe at an annual interest rate of r percent % co*poun$e$ n ti*es

per +ear , then the val*e V of the investment at the en of t years is given by the form*la

V F P times, open parenthesis, 1, , r over 100n, close parenthesis, to the power nt "




E=ample 2"8"8: <f L10,000 is investe at a simple ann*al interest rate of 6 percent,

what is the val*e of the investment after half a yearK

-ol*tion: Accoring to the form*la for simple interest, the val*e of the investment

after one half year is

L10,000 times, open parenthesis, 1, , 0"06 times one half, close parenthesis, F,L10,000 times 1"0/, which is eB*al to L10,/00"

E=ample 2"8".: <f an amo*nt P is to be investe at an ann*al interest rate of /"

percent, compo*ne ann*ally, what sho*l be the val*e of P so that the val*e of the

investment is L1,000 at the en of / yearsK

-ol*tion: Accoring to the form*la for /" percent ann*al interest, compo*ne

ann*ally, the val*e of the investment after / years is

P times, open parenthesis, 1 0"0/, close parenthesis, to the power /,

an we set it to be eB*al to L1,000"




P times, open parenthesis, 1 0"0/, close parenthesis, to the power /, F, L1,000"

To fin the val*e of P , we ivie both sies of the eB*ation by

open parenthesis 1 0"0/, close parenthesis, to the power /"

P F L1,000 over, open parenthesis, 1 0"0/, close parenthesis, to the power /, which

is appro=imately eB*al to L701"74"

Th*s, an amo*nt of appro=imately L701"74 sho*l be investe"

E=ample 2"8"7: A college st*ent e=pects to earn at least L1,000 in interest on an

initial investment of L20,000" <f the money is investe for one year at interest

compo*ne B*arterly, what is the least ann*al interest rate that wo*l achieve the

goalK

-ol*tion: Accoring to the form*la for r percent ann*al interest, compo*ne

B*arterly, the val*e of the investment after 1 year is

L20,000 times, open parenthesis, 1, , r over 400, close parenthesis, to the power 4"

GRE Math Review 2 Algebra /.



?y setting this val*e greater than or eB*al to L21,000 an solving for r , we get

L20,000 times, open parenthesis, 1, , r over 400, close parenthesis, to the power 4, is

greater than or eB*al to L21,000,

which simplifies to

open parenthesis, 1, , r over 400, close parenthesis, to the power 4, is greater than or

eB*al to 1"0"

Recall that taing the positive fo*rth root of each sie of an ineB*ality preserves theirection of the ineB*ality" #<t is also tr*e that taing the positive sB*are root or any

other positive root of each sie of an ineB*ality preserves the irection of the

ineB*ality%" 9sing this fact, we get that taing the positive fo*rth root of both sies of

open parenthesis, 1, , r over 400, close parenthesis, to the power

4, is greater than or eB*al to 1"0

yiels 1, , r over 400 is greater than or eB*al to the positive fo*rth

root of 1"0

which simplifies to r is greater than or eB*al to 400 times, open

parenthesis, the positive fo*rth root of 1"0, min*s 1, close parenthesis"

To comp*te the positive fo*rth root of 1"0, recall that for any n*mber

x greater than or eB*al to 0,

the positive fo*rth root of x, F, the positive sB*are root of the positive

sB*are root of x"




This allows *s to comp*te the positive fo*rth root 1"0 by taing the positive sB*are

root of 1"0 an then tae the positive sB*are root of the res*lt"

Therefore we can concl*e that400 times, open parenthesis, the positive fo*rth root of 1"0, min*s 1, close

parenthesis, F, 400 times, open parenthesis, the positive sB*are root of the positive

sB*are root of 1"0, min*s 1, close parenthesis, which is appro=imately 4"71"

-ince

r is greater than or eB*al to 400 times, open parenthesis, the positive fo*rth root of1"0, min*s 1, close parenthesis,

an

400 times, open parenthesis, the positive fo*rth root of 1"0, min*s 1, close

parenthesis

is appro=imately 4"71, the least ann*al interest rate is appro=imately 4"71 percent"

2., Coor$inate -eo*etr+

Two real n*mber lines that are perpenic*lar to each other an that intersect at their

respective ero points efine a rectangular coordinate system, often calle the

x + coordinate system or x + plane" The horiontal n*mber line is calle the x axis an

the vertical n*mber line is calle the + axis" The point where the two a=es intersect iscalle the origin, enote by O" The positive half of the x a=is is to the right of the origin,

an the positive half of the y a=is is above the origin" The two a=es ivie the plane into

fo*r regions calle quadrants" The fo*r B*arants are labele 1% 2% /%

and 4, as shown in Algebra (ig*re 1 below"




Algebra Figure 1

egin s/ippable part of $escription of Algebra (igure 1.

C*arant 1 is the region of the x y plane that is above the x a=is an to the right of the

y a=is" C*arant 2 is the region that is above the x a=is an to the left of the y a=is"

C*arant / is the region that is below the x a=is an to the left of the y a=is" C*arant

4 is the region that is below the x a=is an to the right of the y a=is"

There are eB*ally space tic mars along each of the a=es" Along the x a=is, to the right

of the origin the tic mars are labele 1, 2, /, an 4 an to the left of the origin the tic

mars are labele negative 1, negative 2, negative /, an negative




4" Along the y a=is, above the origin the tic mars are labele 1, 2, an / an below the

origin the tic mars are labele negative 1, negative 2, an negative /"

En$ s/ippable part of figure $escription.

Each point in the x y plane can be ientifie with an orere pair x comma y of

real n*mbers" The first n*mber in the orere pair is calle the x coordinate, an the

secon n*mber is calle the + coordinate. A point with coorinates x comma y is

locate the absol*te val*e of x *nits to the right of the y a=is if x is positive, or

to the left of the y a=is if x is negative" Also, the point is locate the absol*te

val*e of y *nits above the x a=is if y is positive, or below the x a=is if y is negative" <f x F 0, the point lies on the y a=is, an if y F 0 the point lies on the x a=is" The origin has

coorinates 0 comma 0" 9nless otherwise note, the *nits *se on the x a=is an

the y a=is are the same"

A point P with coorinates x comma y is enote by P , open parenthesis

x comma y, close parenthesis"

<n Algebra (ig*re 1 above, the point P with coorinates 4 comma 2 is 4 *nits to

the right of the y a=is an 2 *nits above the x a=is, the point P prime with

coorinates 4 comma negative 2 is 4 *nits to the right of the y a=is an 2 *nits below the

x a=is, the point P o*ble prime with coorinates negative 4 comma 2 is 4

*nits to the left of the y a=is an 2 *nits above the x a=is, an the point P

triple prime with coorinates negative 4 comma negative 2 is 4 *nits to the left of the

y a=is an 2 *nits below the x a=is"




ote that the three points

P prime with coorinates 4 comma negative 2, P o*ble prime with coorinates negative

4 comma 2, an P triple prime with coorinates negative 4 comma negative 2

have the same coorinates as P e=cept for the signs" These points are geometricallyrelate to P as follows"

P prime is the reflection of 0 about the x axis, or P prime an P are symmetric

about the x axis"

P o*ble prime is the reflection of 0 about the + axis, or P o*ble prime an P

are symmetric about the + axis"

P triple prime is the reflection of 0 about the origin, or P triple prime an P

are symmetric about the origin"

The istance between two points in the x y plane can be fo*n by *sing the 'ythagorean

theorem" (or e=ample, the istance between the two points

Q with coorinates negative 2 comma negative /, an R with coorinates 4 comma 1"

in Algebra (ig*re 2 below is the length of line segment QR" To fin this length, constr*ct

a right triangle with hypoten*se QR by rawing a vertical line segment ownwar from R

an a horiontal line segment rightwar from Q *ntil these two line segments intersect at

the point with coorinates 4 comma negative / forming a right angle, as shown

in Algebra (ig*re 2" Then note that the horiontal sie of the triangle has length

4 min*s negative 2 F 6 an the vertical sie of the triangle has length

1" min*s negative /, F, 4""




Algebra Figure 2

-ince line segment QR is the hypoten*se of the triangle, yo* can apply the 'ythagorean

theorem:

QR F the positive sB*are root of the B*antity 6 sB*are 4" sB*are, which is eB*al to

the positive sB*are root of 6"2, or 8""

#(or a isc*ssion of right triangles an the 'ythagorean theorem, see &hapter /:

Geometry, -ection /"/%"




EB*ations in two variables can be represente as graphs in the coorinate plane" <n the

x y plane, the graph of an equation in the variables x an y is the set of all points whose

orere pairs x comma y satisfy the eB*ation"

The graph of a linear eB*ation of the form y F mx b is a straight line in the x y plane,

where m is calle the slope of the line an b is calle the + intercept"

The x intercepts of a graph are the x val*es of the points at which the graph intersects the

x a=is" -imilarly, the + intercepts of a graph are the y val*es of the points at which the

graph intersects the y a=is"

The slope of a line passing thro*gh two points Q with

coorinates x s*b 1 comma y s*b 1, an R with coorinates x s*b 2 comma y s*b 2,

where x s*b 1 is not eB*al to x s*b 2, is efine as

the fraction with n*merator y s*b 2 min*s y s*b 1, an enominator x s*b 2 min*s x s*b

1"

This ratio is often calle >rise over r*n, where rise is the change in y when moving from

Q to R an run is the change in x when moving from Q to R" A horiontal line has a slope

of 0, since the rise is 0 for any two points on the line" -o the eB*ation of every horiontal

line has the form y F b, where b is the y intercept" The slope of a vertical line is not

efine, since the r*n is 0" The eB*ation of every vertical line has the form x F a, where a

is the x intercept"




Two lines are parallel if their slopes are eB*al" Two lines are perpendicular if their

slopes are negative reciprocals of each other" (or e=ample, the line with eB*ation

y F 2 x is perpenic*lar to the line with eB*ation

y F negative one half x, 7"

E=ample 2"."1: Algebra (ig*re / below shows the graph of the line thro*gh points

Q with coorinates negative 2 comma negative /, an R

with coorinates 4 comma 1" in the x y plane"

Algebra Figure 3





'oint Q with coorinates negative 2 comma negative / is 2 *nits to the

left, an / *nits below the origin" 'oint R with coorinates 4 comma 1" is

4 *nits to the right, an 1" *nits above the origin" )ine QR crosses the y a=is abo*t

halfway between negative 1 an negative 2, an crosses the x a=is at 2"


<n Algebra (ig*re / above, the slope of the line passing thro*gh the points

Q with coorinates negative 2 comma negative /, an R with coorinates 4 comma

1" is the fraction with n*merator 1" min*s negative /, an enominator 4 min*s

negative 2, which is eB*al to the fraction 4" over 6, or 0"8"

)ine QR appears to intersect the y a=is close to the point with coorinates 0

comma negative 1", so the y intercept of the line m*st be close to negative 1""

To get the e=act val*e of the y intercept, s*bstit*te the coorinates of any point on the

line into the eB*ation y F 0"8 x b, an solve it for b"

(or e=ample, if yo* pic the point Q with coorinates negative 2 comma

negative /, an s*bstit*te its coorinates into the eB*ation yo* get

negative / F 0"8 times negative 2, , b"

Then aing 0"8 times 2 to both sies of the eB*ation yiels

b F negative /, , 0"8 times 2, or b F negative 1""




Therefore, the eB*ation of line QR is

y F 0"8 x, min*s, 1""

3o* can see from the graph in Algebra (ig*re / that the x intercept of line QR is 2,

since QR passes thro*gh the point 2 comma 0" More generally, yo* can fin

the x intercept of a line by setting y F 0 in an eB*ation of the line an solving it for x"

-o yo* can fin the x intercept of line QR by setting y F 0 in the eB*ation

y F 0"8 x, min*s, 1" an solving it for x as follows"

-etting y F 0 in the eB*ation y F 0"8 x, min*s, 1" gives the eB*ation

0 F 0"8 x, min*s, 1"" Then aing 1" to both sies yiels

1" F 0"8 x" (inally, iviing both sies by 0"8 yiels x F 1" over

0"8, or 2"

Graphs of linear eB*ations can be *se to ill*strate sol*tions of systems of linear

eB*ations an ineB*alities, as can be seen in e=amples 2"."2 an 2"."/"

E=ample 2"."2: &onsier the system of two linear eB*ations in two variables:

4 x / y F 1/, an

x 2 y F 2

#ote that this system was solve by s*bstit*tion, an by elimination in Algebra

-ection 2"/"%

-olving each eB*ation for y in terms of x yiels




y F negative fo*r thirs x, , 1/ over /, an, y F negative one half x, , 1

Algebra (ig*re 4 below shows the graphs of the two eB*ations in the x y plane" The

sol*tion of the system of eB*ations is the point at which the two graphs intersect,

which is 4 comma negative 1"

Algebra Figure 4




egin s/ippable part of $escription of Algebra (igure ".

The graphs of both eB*ations are lines that slant ownwar as they go from left to

right" The graph of the eB*ation 4 x / y F 1/ crosses the y a=is at a n*mber that is

between 4 an , an crosses the x a=is at a n*mber that is between / an 4" The graph

of the eB*ation x 2 y F 2 intersects the y a=is at 1 an the x a=is at 2" The two graphs

intersect at the point 4 comma negative 1, which is in the fo*rth B*arant"

En$ s/ippable part of figure $escription

E=ample 2"."/: &onsier the following system of two linear ineB*alities"

x min*s / y is greater than or eB*al to negative 6, an, 2 x y is greater than or eB*al to

negative 1

-olving each ineB*ality for y in terms of x yiels

y is less than or eB*al to one thir x, , 2, an, y is greater than or eB*al to negative 2 x,

min*s 1

Each point x comma y that satisfies the first ineB*ality, y is less

than or eB*al to one thir x, , 2, is either on the line y F one thir x, , 2

or below the line beca*se the y coorinate is either eB*al to or less than one




thir x, , 2" Therefore, the graph of y is less than or eB*al to one thir x,

, 2 consists of the line y F one thir x, , 2 an the entire region below

it" -imilarly, the graph of y is greater than or eB*al to negative 2 x, min*s1 consists of the line y F negative 2 x, min*s 1 an the entire region

above it" Th*s, the sol*tion set of the system of ineB*alities consists of all of the

points that lie in the shae region shown in Algebra (ig*re below, which is the

intersection of the two regions escribe"

Algebra Figure 5




egin s/ippable part of $escription of Algebra (igure %.

The graph of the eB*ation y F one thir x, , 2 is a line that slants

*pwar as it goes from left to right, crossing the x a=is at negative 6 an the y a=is

at 2" The graph of the eB*ation y F negative 2 x, min*s 1 slants

ownwar as it goes from left to right, crossing the x a=is between negative 1 an 0

an the y a=is at negative 1" The two lines intersect in the secon B*arant" The

region below the graph of y F one thir x, , 2 an to the right of the

graph of y F negative 2 x, min*s 1 is shae"


-ymmetry with respect to the x a=is, the y a=is, an the origin is mentione earlier in this

section" Another important symmetry is symmetry with respect to the line with eB*ation

y F x" The line y F x passes thro*gh the origin, has a slope of 1, an maes a 4 egree

angle with each a=is" (or any point with coorinates a comma b, the point with

interchange coorinates b comma a is the reflection of a comma b abo*t

the line y F x that is, a comma b, an, b comma a are symmetric abo*t

the line y F x" <t follows that interchanging x an y in the eB*ation of any graph yiels

another graph that is the reflection of the original graph abo*t the line y F x.

E=ample 2"."4: &onsier the line whose eB*ation is y F 2 x " <nterchanging x an y

in the eB*ation yiels x F 2 y " -olving this eB*ation for y yiels

y F one half x, min*s halves"




The line y F 2 x an its reflection y F one half x, min*s halves are

graphe in Algebra (ig*re 6 below"

Algebra Figure 6

egin s/ippable part of $escription of Algebra (igure '.

The fig*re shows the lines y F 2 x an y F one half x, min*s halves

an the ashe line y F x between them" The three lines intersect at a point in the thir




B*arant" $hen the line y F 2 x is flippe over the line y F x, the res*lt is the line

y F one half x, min*s halves" The res*lt of this flipping on the

intercepts of the lines is that the y intercept of y F 2 x an the x intercept of

y F one half x, min*s halves are eB*al an the x intercept of

y F 2 x an the y intercept of the line y F one half x, min*s halves

are eB*al"


The line y F x is a line of symmetry for the graphs of y F 2 x an

y F one half x, min*s halves"

The graph of a B*aratic eB*ation of the form

y F a x sB*are, , bx, , c, where a, b, an c are constants an a is not eB*al to

0, is a parabola" The x intercepts of the parabola are the sol*tions of the eB*ation

a x sB*are, , bx, , c F 0" <f a is positive, the parabola opens *pwar

an the vertex is its lowest point" <f a is negative, the parabola opens ownwar an the

verte= is its highest point" Every parabola is symmetric with itself abo*t the vertical line

that passes thro*gh its verte=" <n partic*lar, the two x intercepts are eB*iistant from this

line of symmetry"

E=ample 2".": The eB*ation y F x sB*are, min*s 2 x, min*s / has

the graph shown in Algebra (ig*re 8 below"




Algebra Figure 7

egin s/ippable part of $escription of Algebra (igure ).

The fig*re shows the parabola, which is an *pwar facing, 9 shape c*rve, an the

vertical ashe line x F 1"


The graph inicates that the x intercepts of the parabola are negative 1 an

/" The val*es of the x intercepts can be confirme by solving the B*aratic eB*ation

x sB*are, min*s 2 x, min*s / F 0 to get

GRE Math Review 2 Algebra



x F negative 1 an x F /" The point 1 comma negative 4 is the verte= of the

parabola, an the line x F 1 is its line of symmetry" The y intercept is the y coorinate

of the point on the parabola at which x F 0, which is

y F 0 sB*are, min*s 2 times 0, min*s / F negative /"

The graph of an eB*ation of the form

open parenthesis, x min*s a, close parenthesis, sB*are, , open parenthesis, y min*s

b, close parenthesis, sB*are, F, r sB*are

is a circle with its center at the point a comma b an with rai*s r "

E=ample 2"."6: Algebra (ig*re . below shows the graph of two circles in the x y

plane" The larger of the two circles is centere at the origin an has rai*s 10, so its

eB*ation is x sB*are, , y sB*are, F, 100" The smaller of the two

circles has center 6 comma negative an rai*s /, so its eB*ation is

open parenthesis, x min*s 6, close parenthesis, sB*are, , open parenthesis, y ,

close parenthesis, sB*are F, 7"




Algebra Figure 8

egin s/ippable part of $escription of Algebra (igure ,.

The center of the smaller circle is in the fo*rth B*arant an lies insie the large

circle" The two circles intersect at two points, both of which are in the fo*rth

B*arant"





2. -raphs of (unctions

The coorinate plane can be *se for graphing f*nctions" To graph a f*nction in the

x y plane, yo* represent each inp*t x an its corresponing o*tp*t f of, x as a point

x comma y, where y F f of, x" <n other wors, yo* *se the x a=is for

the inp*t an the y a=is for the o*tp*t"

?elow are several e=amples of graphs of elementary f*nctions"

E=ample 2"7"1: &onsier the linear f*nction efine by

f of, x F negative one half x, , 1"

<ts graph in the x y plane is the line with the linear eB*ation

y F negative one half x, , 1"

E=ample 2"7"2: &onsier the B*aratic f*nction efine by

g of, x F x sB*are"

The graph of g is the parabola with the B*aratic eB*ation y F x sB*are"

The graph of both the linear eB*ation y F negative one half x, , 1 an

the B*aratic eB*ation y F x sB*are are shown in Algebra (ig*re 7 below"




Algebra Figure 9

egin s/ippable part of $escription of Algebra (igure .

The graph of the linear eB*ation y F negative one half x, , 1 is a line

that slants ownwar as it goes from left to right, intersecting the y a=is at 1, an the x

a=is at 2" The graph of the B*aratic eB*ation y F x sB*are is an *pwar

facing parabola, whose verte= is at the origin"


ote that the graphs f an g in Algebra (ig*re 7 above intersect at two points" These

are the points at which g of, x F f of, x" $e can fin these points

algebraically as follows"




-et g of, x F f of, x an get x sB*are, F, negative one

half x, , 1, which is eB*ivalent to

x sB*are, , one half x, min*s, 1, F, 0 or 2, x sB*are, , x, min*s 2, F, 0"

Then solve the eB*ation 2, x sB*are, , x, min*s 2, F, 0 for x *sing

the B*aratic form*la getting x F the fraction with n*merator

negative 1 pl*s or min*s the sB*are root of the B*antity 1 16, an enominator 4,

which represents the x coorinates of the two sol*tions

x F the fraction with n*merator negative 1 pl*s the positive sB*are root of 18, an

enominator 4, which is appro=imately 0"8., an x F the fraction with n*merator

negative 1 min*s the positive sB*are root of 18, an enominator 4, which is

appro=imately negative 1"2."

$ith these inp*t val*es, the corresponing y coorinates can be fo*n *sing either f

or g :

g of, the fraction with n*merator negative 1 the positive sB*are root of 18 an

enominator 4, F, open parenthesis, the fraction with n*merator negative 1 the

positive sB*are root of 18 an enominator 4, close parenthesis, sB*are, which is

appro=imately 0"61, an

g of, the fraction with n*merator negative 1 min*s the positive sB*are root of 18 an

enominator 4, F, open parenthesis, the fraction with n*merator negative 1 min*s the

positive sB*are root of 18 an enominator 4, close parenthesis, sB*are, which is

appro=imately 1"64"




Th*s, the two intersection points can be appro=imate by

the point 0"8. comma 0"61 an the point negative 1"2. comma 1"64"

E=ample 2"7"/: &onsier the absol*te val*e f*nction efine by h of, x F

the absol*te val*e of x" ?y *sing the efinition of absol*te val*e #see &hapter 1:

Arithmetic, -ection 1"%, h can be e=presse as a pieceise defined f*nction:

h of, x F x, for x greater than or eB*al to 0, an h of, x F negative x, for x less than 0

The graph of this f*nction is D shape an consists of two linear pieces,

y F x an y F negative x, Hoine at the origin, as shown in Algebra

(ig*re 10 below"




Algebra Figure 10




E=ample 2"7"4: This e=ample is base on Algebra (ig*re 11 below, which is the graph

of 2 parabolas"

Algebra Figure 11

+ne of the parabolas is the *pwar facing parabola y F x sB*are an the

other loos lie the parabola y F x sB*are, b*t instea of facing *pwar, it

faces to the right" The verte= of both parabolas is at the origin"


The half of the *pwar facing parabola to the right of the y a=is is a soli c*rve, an

the half to the left of the y a=is is a ashe c*rve" The half of the right facing parabola

above the x a=is is a soli c*rve an the half below the x a=is is a ashe c*rve"





These f*nctions are relate to the absol*te val*e f*nction absol*te val*e of x an

the B*aratic f*nction x sB*are, respectively, in simple ways"

The graph of f of, x F the absol*te val*e of x, , 2 is the graph of

y F the absol*te val*e of x shifte *pwar by 2 *nits, as shown in Algebra

(ig*re 12 below"

Algebra Figure 12


Algebra (ig*re 12 shows the graph of y F the absol*te val*e of x, , 2 as

a soli D shape c*rve an the graph of y F the absol*te val*e of x as a

ashe D shape c*rve"





-imilarly, the graph of the f*nction k of, x F the absol*te val*e of x,

min*s, 2 is the graph of y F the absol*te val*e of x shifte ownwar by

*nits"

The graph of g of, x F open parenthesis, x 1, close parenthesis,

sB*are is the graph of y F x sB*are shifte to the left by 1 *nit, as shown in

Algebra (ig*re 1/ below"

Algebra Figure 13


Algebra (ig*re 1/ shows the graph of g of, x F open parenthesis,

x 1, close parenthesis, sB*are as a soli parabola an the graph of y F x

sB*are as a ashe parabola"





-imilarly, the graph of the f*nction j of, x F open parenthesis, x min*s 4,

close parenthesis, sB*are is the graph of y F x sB*are shifte to the right by 4

*nits" To o*ble chec the irection of the shift, yo* can plot some corresponing val*es

of the original f*nction an the shifte f*nction"

<n general, for any f*nction h of, x an any positive n*mber c, the following are

tr*e"

The graph of h of, x, , c is the graph of h of x shifted upard by c *nits"

The graph of h of, x, min*s, c is the graph of h of x shifted donard byc *nits"

The graph of h of, the B*antity, x c is the graph of h of, x shifted to the

left by c *nits"

The graph of h of, the B*antity, x min*s c is the graph of h of, x shifted to

the right by c *nits"

E=ample 2"7"6: &onsier the f*nctions efine by

f of, x F 2 times the absol*te val*e of the B*antity x min*s 1, an g of, x F negative x

sB*are, over 4"

These f*nctions are relate to the absol*te val*e f*nction absol*te val*e of x an

the B*aratic f*nction x sB*are, respectively, in more complicate ways than in

the preceing e=ample"




The graph of f of, x F 2 times the absol*te val*e of the B*antity x

min*s 1, is the graph of y F the absol*te val*e of x shifte to the right by 1

*nit an then stretche, or ilate, vertically away from the x a=is by a factor of 2, as

shown in Algebra (ig*re 14 below"

Algebra Figure 14

egin s/ippable part of $escription of Algebra (igure 1".

Algebra (ig*re 14 shows the graph of f of, x F 2 times the absol*te

val*e of the B*antity x min*s 1 as a soli D shape c*rve an the graph of y F

the absol*te val*e of x as a ashe D shape c*rve" The bottom of the D in the graph

of f of, x F 2 times the absol*te val*e of the B*antity x min*s 1 is 1




*nit to the right of the bottom of the D in the graph of y F the absol*te val*e

of x, an the D shape of the graph of f of, x F 2 times the absol*te

val*e of the B*antity x min*s 1 is narrower than the D shape of the graph of

y F the absol*te val*e of x"


-imilarly, the graph of the f*nction h of, x F one half times the

absol*te val*e of the B*antity x min*s 1 is the graph of y = the absol*te val*e

of x shifte to the right by 1 *nit an then shr*n, or contracte, vertically towar the

x a=is by a factor of one half"

The graph of g of, x F negative x sB*are over 4 is the graph of

y F x sB*are contracte vertically towar the x a=is by a factor of one fo*rth anthen reflecte in the x a=is, as shown in Algebra (ig*re 1 below"




Algebra Figure 15

egin s/ippable part of $escription of Algebra (igure 1%.

Algebra (ig*re 1 shows the graph of g of, x F negative x sB*are over

4 as a soli parabola an the graph of y F x sB*are as a ashe parabola" The

graph of g of, x F negative x sB*are over 4 has verte= at the origin,

opens ownwar, an is wier than the parabola y F x sB*are"





<n general, for any f*nction h of, x an any positive n*mber c, the following are

tr*e"

The graph of c times h of, x is the graph of h of, x stretched vertically by a

factor of c if c is greater than 1"

The graph of c times h of, x is the graph of h of, x shrunk vertically by a

factor of c if 0 is less than c, which is less than 1"




Algebra Exercises

1" (in an algebraic e=pression to represent each of the following"

a" The sB*are of y is s*btracte from , an the res*lt is m*ltiplie by /8"

b" Three times x is sB*are, an the res*lt is ivie by 8"

c" The pro*ct of open parenthesis, x 4, close parenthesis an y is ae to 1."

2" -implify each of the following algebraic e=pressions"

a" /, x sB*are, min*s 6, , x 11, min*s x sB*are, , x

b" / times, open parenthesis, x, min*s 1, close parenthesis, min*s x,

, 4

c" the e=pression with n*merator x sB*are min*s 16 an

enominator x min*s 4, where x is not eB*al to 4

" open parenthesis, 2 x, , , close parenthesis, times, open

parenthesis, / x, min*s 1, close parenthesis

/"

a" $hat is the val*e of f of, x F /, x sB*are,

min*s 8 x, , 2/, when x is eB*al to negative 2 K




b" $hat is the val*e of h of, x F x c*be, min*s

2, x sB*are, , x, min*s 2, when x F 2 K

c" $hat is the val*e of k of, x F thirs x, min*s 8, when

x F 0 K

4" <f the f*nction g is efine for all nonero n*mbers y by g of, y F y over

the absol*te val*e of y, fin the val*e of each of the following"

a" g of, 2

b" g of, negative 2

c" g of, 2, min*s, g of, negative 2

" 9se the r*les of e=ponents to simplify the following"

a" n to the power , times, n to the power negative /

b" s to the power 8, times, t to the power 8

c" r to the power 12 over r to the power 4

" open parenthesis 2a over b, close parenthesis, to the power




e" open parenthesis, w to the power , close parenthesis, to the power negative /

f" to the power 0, times, d to the power /

g"

the e=pression with n*merator x to the power 10 times y to the power negative 1, an

enominator x to the power negative times y to the power

h"

open parenthesis, / x over y, close parenthesis, sB*are, ivie by, open parenthesis, 1

over y, close parenthesis, to the power

6" -olve each of the following eB*ations for x"

a" x, min*s 8 F 2.

b" 12 min*s x F x /0

c" times, open parenthesis, x 2, close parenthesis, F 1, min*s, / x

" open parenthesis, x 6, close parenthesis, times, open

parenthesis, 2 x, min*s 1, close parenthesis, F 0

e" x sB*are, , x, min*s 14

f" x sB*are, min*s x, min*s 1 F 0




8" -olve each of the following systems of eB*ations for x an y"

a" x y F 24, an, x min*s y F 1.

b" / x min*s y F negative , an, x 2 y F /

c" 1 x min*s 1. min*s 2 y F negative / x, , y, an, 10 x, , 8 y, , 20 F 4 x, , 2

." -olve each of the following ineB*alities for x"

a" negative / x is greater than 8 x

b" 2 x 16 is greater than or eB*al to 10 min*s x

c" 16 x is greater than . x, min*s 12

7" (or a given two igit positive integer, the tens igit is more than the *nits igit" The

s*m of the igits is 11" (in the integer"

10" <f the ratio of 2 x to y is / to 4, what is the ratio of x to y K




b" $rite an algebraic e=pression for the profit per chair"

18" Algebra (ig*re 16 below shows right triangle PQR in the x y plane" (in the

following"

a" &oorinates of point Q

b" )engths of line segment PQ, line segment QR, an line segment PR

c" 'erimeter of triangle PQR

" Area of triangle PQR

e" -lope, y intercept, an eB*ation of the line passing thro*gh points P an R

Algebra Figure 16

egin s/ippable part of $escription of Algebra (igure 1'.




The fig*re shows right triangle PQR in the x y plane" Derte= R lies on the x a=is an has

coorinates comma 0" Derte= P lies in the secon B*arant an has coorinates

negative 2 comma 6" Derte= Q lies on the x a=is irectly below verte= P " Angle

PQR is a right angle"

En$ s/ippable part of figure $escription"

1." <n the x y plane, fin the following"

a" -lope an y intercept of the line with eB*ation 2 y x F 6

b" EB*ation of the line passing thro*gh the point / comma 2 with y intercept 1

c" The y intercept of a line with slope / that passes thro*gh the point negative 2

comma 1

" The x intercepts of the graphs in parts a, b, an c

17" (or the parabola y = x sB*are, min*s 4 x, min*s 12 in the

x y plane, fin the following"

a" The x intercepts

b" The y intercept

c" &oorinates of the verte=

20" (or the circle open parenthesis, x min*s 1, close

parenthesis, sB*are, , open parenthesis, y 1, close parenthesis, sB*are, F, 20

in the x y plane, fin the following"




Answers to Algebra Exercises

1"

a"

/8 times, open parenthesis, min*s, y sB*are, close parenthesis, or 1., min*s, /8, y

sB*are

b"

the fraction, open parenthesis, / x, close parenthesis, sB*are, over 8, or the fraction, 7, x

sB*are over 8

c"

1. , open parenthesis, x 4, close parenthesis, times y, or, 1. x y 4 y

2"

a" 2, x sB*are, 6 x,

b" 14 x 1

c" x 4

" 6, x sB*are, 1/ x, min*s

/"

a" 47

GRE Math Review 2 Algebra .0



b" 0

c" negative 8

4"

a" 1

b" negative 1

c" 2

"

a" n sB*are

b" open parenthesis, st , close parenthesis, to the power 8

c" r to the power .

" the fraction /2, a to the power , over, b to the power

e" 1 over, w to the power 1

f" d c*be

g" the fraction x to the power 1, over, y to the power 6

h" 7, x sB*are, y c*be




6"

a" 8

b" negative /

c" negative 7 over .

" the two sol*tions are negative 6, an one half

e" the two sol*tions are negative 8 an 2

f" the two sol*tions are

the fraction with n*merator 1 the positive sB*are root of , an enominator 2, an

the fraction with n*merator 1 min*s the positive sB*are root of , an enominator 2

8"

a" x F 21 y F /

b" x F negative 1, y F 2

c" x F one half y F negative /




."

a" x is less than negative 8 over 4

b" x is greater than or eB*al to negative / over 1/

c" x is less than 4

7" ./

10" 1 to .

11" L220

12" L/

1/" L.00 at 10N an L2,200 at .N

14" 4. miles per ho*r an 6 miles per ho*r

1"

a" L4,/20

b" L10.

GRE Math Review 2 Algebra ./



16"

a" P F c y, min*s x

b"

'rofit per chair: P over c, F, the fraction with n*merator c y min*s x, an enominator c,

which is eB*al to y min*s the fraction x over c

18"

a" The coorinates of point Q are negative 2 comma 0

b" The length of PQ is 6, the length of QR is 8, an the length of PR is the positive

sB*are root of ."

c" 1/ the positive sB*are root of .

" 21

e" -lope: negative 6 over 8 y intercept: /0 over 8

eB*ation of line:

y F negative 6 sevenths x, , /0 over 8, or 8 y 6 x F /0

1."

a" -lope: negative one half y intercept: /




b" y F x over /, , 1

c" 8

" 6, negative /, an negative 8 over /

17"

a" x F negative 2, an x F 6

b" y F negative 12

c" coorinates 2 comma negative 16

20"

a" coorinates 1 comma negative 1

b" the positive sB*are root of 20

c" 20 pi

21"

a" !omain: the set of all real n*mbers" The graph is a horiontal line with y intercept

negative 4 an no x intercept"

b" !omain: the set of all real n*mbers" The graph is a line with slope negative

700, y intercept 100, an x intercept 1 ninth"




c" !omain: the set of all real n*mbers" The graph is a parabola opening ownwar with

verte= at negative 20 comma , line of symmetry x F negative 20,

y intercept negative /7 an x intercepts negative 20 pl*s or min*s the

positive sB*are root of "

" !omain: the set of n*mbers greater than or eB*al to negative 2" The graph is half

a parabola opening to the right with verte= at negative 2 comma 0, x intercept

negative 2, an y intercept the positive sB*are root of 2"

e" !omain: the set of all real n*mbers" The graph is two half lines Hoine at the origin:

one half line is the negative x a=is an the other is a line starting at the origin with slope

2" Every nonpositive n*mber is an x intercept, an the y intercept is 0" The f*nction is

eB*al to the following piecewise efine f*nction

f of, x F 2 x, for x greater than or eB*al to 0, an f of, x F 0 for x less than 0

math review 2 algebra

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