Name
Date
LESSON
8-8
Class
Reteach
Solving Radical Equations and Inequalities
Solve radical equations by raising both sides of the equation to the power
n
of the index of the radical. For example, the index of a is n. Therefore,
x
⫽3
x 2 ⫽ 32
The index of x is 2. Raise
both sides to the power of 2.
x⫽9
Solve: 3x ⫺ 2 ⫽ 18
Step 1 Isolate the radical.
Divide both sides of the equation by 3 and simplify.
18
3
x ⫺ 2 ⫽ ___
________
3
3
x ⫺ 2 ⫽ 6
Step 2 Square both sides of the equation and simplify.
2
x ⫺ 2 ⫽ 62
n n
⫽ a.
Remember: a
x ⫺ 2 ⫽ 36
Step 4 Solve.
x ⫽ 38
Always check for extraneous solutions
when solving radical equations.
Step 5 Check.
3 x ⫺ 2 ⫽ 18
338 ⫺ 2 ⫽ 3 36 ⫽ 3 6 ⫽ 18
Solve each equation. Check your answer.
3 1. 42x ⫹ 11 ⫽ 12
2. 5 ⫹ x ⫺ 3 ⫽ 9
3
12
42x ⫹ 11 ⫽ ___
__________
5 ⫺ 5 ⫹ x ⫺ 3 ⫽ 9 ⫺ 5
4
4
3 2x ⫹ 11 ⫽ 3
3. 2x ⫹ 4 ⫽ 10
x 4 5
x 3 4
x 4 25
x 3 16
x 21
2x 11 27
x 19
2 21 4 2x 16; x 8
5 19 3 5
3 42 8 11 12
16 5 4
3 436 12 ✓
9✓
3 3
3
2x ⫹ 11 ⫽ 3
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c08-8_rt.indd 62
62
2 25 2 ⭈ 5
10 ✓
Holt Algebra 2
12/26/05 6:54:53 AM
Process Black
Name
LESSON
8-8
Date
Class
Reteach
Solving Radical Equations and Inequalities (continued)
Solving equations with rational exponents is similar to solving radical equations.
1
__
Solve: x ⫽ x ⫹ 20 2 .
Step 1
Think: a n ⫽ a
1
__
Raise both sides to the reciprocal power.
1 2
__
2
x ⫽ [ x ⫹ 20 2 ]
n
1 is 2.
The reciprocal of __
2
Step 2
Square both sides.
x 2 ⫽ x ⫹ 20
Step 3
Write the quadratic equation in standard form.
x 2 ⫺ x ⫺ 20 ⫽ 0
Set one side of the
Factor.
equation equal to zero.
x ⫹ 4 x ⫺ 5 ⫽ 0
Step 4
Step 5
Solve.
x ⫹ 4 ⫽ 0
or
x ⫽ ⫺4
Step 6
x ⫺ 5 ⫽ 0
x⫽5
Check for extraneous solutions.
1
__
x ⫽ x ⫹ 20 2
x ⫽ ⫺4
This is the only solution.
x⫽5
1
__
⫺4 ? ⫺4 ⫹ 20 2
1
__
⫺4 ⫽ 16 2
1
__
5 ? 5 ⫹ 20 2
1
__
5 ⫽ 25 2 ✓
Solve each equation.
1
__
1
__
4. 5x ⫹ 6 4 ⫽ 3
[ 5x ⫹ 6 4 ]
1
__
4
1 3
__
⫽3
1
__
5. 6x ⫺ 8 3 ⫽ 4
4
6x 8 3 6. x ⫽ x ⫹ 6 2
4
3
x
2
1 2
__
x 6 2 5x 6 81
6x 8 64
x2 x 6
5x 75
6x 72
x2 x 6 0
x 15
x 12
x
3 x 2 0
x3
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c08-8_rt.indd 63
63
Holt Algebra 2
12/26/05 6:54:54 AM
Process Black
*À>V̈ViÊ
3OLVING2ADICAL%QUATIONSAND)NEQUALITIES
--"
n‡n
n‡n
2EWRITEEACHEQUATIONTOISOLATETHERADICAL
^
qXX
Êr^
XÊÊÈ
3OLVEEACHEQUATION
^
X q^
^
qX
X
^
qX
qX
^
^
*À>V̈ViÊ
3OLVING2ADICAL%QUATIONSAND)NEQUALITIES
--"
ÊrÎXÊÊXÊÊn
Êr ÓXÊÊ£Ê
ÊÊÎXÊÊ£Ç
XÊÊ{Î
^
ÓÆÊXÊÊ£È
șÊÊÊ
{ÆÊXÊÊÊÊÚÚÚ
Ó
ÎÆÊXÊÊÈÎ
^^^^
^ q
X
qX XÊÊ£x
£Ê
XÊÊÊÊÚÚ
{
??
^
qX
X
XÊÊ£]ÊXÊÊÈÆÊXÊʣʈÃÊ>˜Ê
iÝÌÀ>˜iœÕÃÊ܏Ṏœ˜°
£Ê
XÊÊÊÊÚÚ
Ó
^^^^
^^^
q
X
qX
^
qX
X
XÊÊÓÆʘœÊiÝÌÀ>˜iœÕÃÊ܏Ṏœ˜Ã
^
qX
q^
X
XÊÊÈ
3OLVETHEEQUATION4HENIDENTIFYANYEXTRANEOUSSOLUTIONS
^
qX
XÊÊÓä
^
qX
q X
)DENTIFYTOWHATPOWEREACHEQUATIONMUSTBERAISEDINORDERTO
SOLVE4HENSOLVE
^^^
^^
X q
X
q
X
q^
SXD
œÊ܏Ṏœ˜Ã]ÊȘViÊLœÌ…Ê£Ê>˜`Ê
ÇÊ>ÀiÊiÝÌÀ>˜iœÕÃ
XÊÊÎÓ
??
??
SXD 3OLVEEACHEQUATIONORINEQUALITY
??
^
qX
SXD
XÊÊÓÎ
XÊÊÓÈ
^
XÊÊÓn
qX
ÓÊÊXÊÊ£
^
^
qX
qX
XÊÊÓ£
qX
ÚÚ
Ê£Ê ÊXÊÊn
Ó
3OLVE
^
!BIOLOGISTISSTUDYINGTWOSPECIESOFANIMALSINAHABITAT4HEPOPULATION
??
P
OFONEOFTHESPECIESISGROWINGACCORDINGTOPT ANDTHE
POPULATIONPOFTHEOTHERSPECIESISGROWINGACCORDINGTOPT
WHERETIMETISMEASUREDINYEARS!FTERHOWMANYYEARSWILLTHE
POPULATIONSOFTHETWOSPECIESBEEQUAL
i˜½ÃÊ܏Ṏœ˜ÊˆÃÊVœÀÀiV̰ʈ˜ÃiÞÊvœÀ}œÌÊ̅>ÌÊ̅iÊÀ>`ˆV>˜`Ê
V>˜˜œÌÊLiʘi}>̈Ûi°
ÓxÊÞi>ÀÃ
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
œÌʏ}iLÀ>ÊÓ
3OLVEEACHEQUATION
^^^^
q
X
,%33/.
n‡n
^^^^^
q
X
,iÌi>V…
3OLVING2ADICAL%QUATIONSAND)NEQUALITIES
^
X q
Sr^
XD
XÊÊ{Ç
œÌʏ}iLÀ>ÊÓ
3OLVERADICALEQUATIONSBYRAISINGBOTHSIDESOFTHEEQUATIONTOTHEPOWER
N ^
OFTHEINDEXOFTHERADICAL&OREXAMPLETHEINDEXOF
qA IS N4HEREFORE
^
qX
XÊÊΣ
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
*À>V̈ViÊ
3OLVING2ADICAL%QUATIONSAND)NEQUALITIES
n‡n
XÊÊ{{
3OLVE
!INSLEYSSOLUTION
!INSLEYAND"ENEACHSOLVETHEINEQUALITYqX
ISX"ENSSOLUTIONISX7HYARETHEIRSOLUTIONSDIFFERENT
7HICHISCORRECT
--"
XÊÊ{ä
^
^
qX
äÊÊXÊÊ£n
^
qX
^
XÊÊxÓ
3OLVEEACHINEQUALITY
^
qX
qX
XÊʙ
??
SXD
SXD
XÊÊÇ
Ê
^^
^^^^
q
X
qX
4HEINDEXOFq^
X IS2AISE
BOTHSIDESTOTHEPOWEROF
X
^
XÊÊÇ
SXDX
^
r X
????????
???
XÊÊx
??
??
SXD SXDX
xÊ
XÊÊÊÊÚÚ
Ó
3TEP )SOLATETHERADICAL
$IVIDEBOTHSIDESOFTHEEQUATIONBYANDSIMPLIFY
??
XÊÊÓÊ>˜`ÊXÊÊ£
3OLVEqX
XÊʙ
^
XqX
^
X
r
XÊʙÆÊXÊÊÓʈÃÊ>˜Ê
iÝÌÀ>˜iœÕÃÊ܏Ṏœ˜°
3TEP 3
QUAREBOTHSIDESOFTHEEQUATIONANDSIMPLIFY
^ D Sq X
N ^ N
A
2EMEMBERSq
D A
X 3OLVEEACHINEQUALITY
^^^
q
X
^
qX
xÊ ÊÝÊÊ£
ÊÊÚÚ
{
ÇÊÊXÊÊ£È
^
XÊÊÓ£
3OLVEEACHEQUATION#HECKYOURANSWER
^^^^
^
q X
qX
^^^
q
X
^
qX
q X
XÊÊ{
ÓÊXÊÊ£ÓÎ
3OLVE
%INSTEINSTHEORYOFRELATIVITYSTATESTHATTHEMASSOFANOBJECTINCREASES
ASTHEOBJECTSVELOCITYINCREASES4HEMASSMSVDOFANOBJECTTRAVELING
M
WITHVELOCITYVISGIVENBYMSVD?????????
WHERECISTHESPEEDOFLIGHT
^
V ???
C
ANDMISTHEMASSOFTHEOBJECTATREST)NTERMSOFCSOLVEFORTHEVELOCITY
ATWHICHTHEEFFECTIVEMASSMSVDOFTHEPARTICLEHASINCREASEDTOTWICEITS
MASSATRESTM
^
q
Êr ÎÊ
VÊÊÊÊÚÚÚ
Ê ÊÊÊC
Ó
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
AK4up.indd 81
^^^^
qX
???
??????????
^^^^
q
X
^^^^ D Sq
X
rX
X
X
X
r r X
XX
r 81
^
^
^^^^^
q TE
œÌʏ}iLÀ>ÊÓ
rX
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
^
^^
q ✓
^
q X
X
^
qX
^
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
!LWAYSCHECKFOREXTRANEOUSSOLUTIONS
WHENSOLVINGRADICALEQUATIONS
3TEP #HECK
^
rX
^
^
q
q SD
^^^
q
X
^
qX
3TEP 3OLVE
X
XÊÊx
^
^
r ✓
✓
(OLT!LGEBRA
Holt Algebra 2
5/11/06 5:11:42 PM
,%33/.
n‡n
,iÌi>V…
3OLVING2ADICAL%QUATIONSAND)NEQUALITIESCONTINUED
n‡n
%QUATIONSMAYINVOLVEMORETHANONERADICAL)NTHATCASETHESOLUTION
PROCESSISREPEATEDTOELIMINATEMULTIPLERADICALS&OREXAMPLE
^
^
qXqX
3OLVINGEQUATIONSWITHRATIONALEXPONENTSISSIMILARTOSOLVINGRADICALEQUATIONS
??
3OLVEXSXD 3TEP
??
??
X Q ÊSXDR ^
3
QUAREBOTHSIDES
XX
3TEP
7
RITETHEQUADRATICEQUATIONINSTANDARDFORM
XX
3ETONESIDEOFTHE
&
ACTOR
EQUATIONEQUALTOZERO
SXDSXD 3TEP
3OLVE
SXD OR
X
3TEP
Sq^
XDSqX
D ^^^
X
SXD
qX
4HERECIPROCALOF??
IS
3TEP
3TEP
4OSOLVEISOLATEONERADICALANDSQUAREBOTHSIDESASSHOWNBELOW
^
^
qXqX
N
4HINKSAND A
2AISEBOTHSIDESTOTHERECIPROCALPOWER
SXD
X
X
X
3OLVEEACHEQUATION#HECKEACHANSWERTOENSURETHATITDOESNOT
INCLUDEEXTRANEOUSSOLUTIONS
^^^
^^^
^^^^
^^^^
q
X
q
X
X
qX
qX
??
XSXD
4HISISTHEONLYSOLUTION
X
??
??
SD SD
??
??
pSDw
SDa
SX ??
??
SXD
TXE ??
XTXE X
X X
X
X
XX
TXETXE
X
X
--"
n‡n
(OLT!LGEBRA
^^^
^^^
q
X
qX
Ç
q
È
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
*ÀœLi“Ê-œÛˆ˜}
3OLVING2ADICAL%QUATIONSAND)NEQUALITIES
nʜÀÊ£
^^^
^^^
q
X
??
X ^^^
^^^
q
X
qX
Î
xʜÀÊÊÚÚ
Ê£Ê
™
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
q
X
^^^^^^
q
X
X
XX
x
£™
^^^^^
^^^^
^^^^^
^^^^
^^^^
q
q
X
q
q
X
X
^^^
q
X
??????
^^^
X
q
^^^^
^^^^
^^^
q
q
X
qX
œÊ܏Ṏœ˜
™
XSXD
??
QÊSXD R ^^^
^^^
q
X
qX
??
D £x°Óx
3OLVEEACHEQUATION
??
.OTICETHATNOWTHEREISONLYONERADICALINTHEEQUATION2EPEATTHE
PROCESSISOLATETHERADICALSQUAREANDSOLVE
^^^
X
SXD
qX
^^^
qX
^^^
qX
^^^
Sq
X
D #HECKFOREXTRANEOUSSOLUTIONS
X
…>i˜}i
-ULTIPLE2ADICALS
--"
œÌʏ}iLÀ>ÊÓ
,i>`ˆ˜}Ê-ÌÀ>Ìi}Þ
5SE6OCABULARY
--"
n‡n
^
4HEFORMULAS q FDCANBEUSEDTOESTIMATETHESPEEDSINMILES
PERHOURTHATACARISTRAVELINGWHENITGOESINTOASKIDWHEREFISTHE
COEFFICIENTOFFRICTIONANDDISTHELENGTHOFTHESKIDMARKSINFEET
9OUCANSOLVERADICALEQUATIONSINATHREESTEPPROCESS&OREXAMPLETO
SOLVETHEEQUATIONq^
X FOLLOWTHESTEPSBELOW
ˆÀiV̏Þ
(OWDOESTHESPEEDVARYASTHELENGTHOFTHESKIDMARKS
Ó
B (OWLONGWOULDTHESKIDMARKSBEIFHEHAD
BEENDRIVINGATASPEEDOFMIH
4OISOLATETHERADICALMEANSTOHAVEONLY
THERADICALONONESIDEOFTHEEQUATION
ANDALLOTHERVARIABLESANDCONSTANTSON
THEOTHERSIDEOFTHEEQUATION
œÆÊ«œÃÈLiÊ>˜ÃÜiÀ\ʅˆÃÊΈ`ʓ>ÀŽÃÊÜiÀiʜ˜ÞÊxÓÊvÌ]ʘœÌÊxnÊvÌ°
LœÕÌÊÎÎʓˆÉ…
D &INDHISACTUALSPEED
!SHLEYSKIDSTOASTOPONASTREETWITHASPEEDLIMITOFMIHTOAVOIDA
DOGWHORUNSINTOTHESTREETABOUTFTAHEADOFHER!SHLEYCLAIMSTO
HAVEBEENGOINGLESSTHANMIH4HECOEFFICIENTOFFRICTIONIS
4OUNDERSTANDWHATITMEANSTORAISEA
NUMBERTOAPOWERTHINKABOUTHOWTHE
EXPONENTISLOCATEDSLIGHTLYHIGHERTHAN
THEBASE
A )F!SHLEYWEREDRIVINGTHESPEEDLIMITBYWHATDISTANCEWOULDSHEHAVE
MISSEDTHEDOG
LœÕÌʙÊvÌ
)SOLATETHERADICALINEACHEQUATION
B )F!SHLEYWEREDRIVINGLESSTHANMIHBYWHATDISTANCEWOULDSHE
HAVEMISSEDTHEDOG
X
LœÕÌÊxnÊvÌ
C 7AS+ODYSPEEDINGORNOT%XPLAINHOWYOUKNOW
34%03IMPLIFY
ANDSOLVE
Sq^
XDSD
ÊÃÊ ÊÊÊ
DÊÊÊÊÚÚÚ
ÎäF
A 3OLVETHEEQUATIONFORDINTERMSOFS
34%02AISE
BOTHSIDESTOTHE
APPROPRIATE
POWER
34%0)SOLATE
THERADICAL
q^
X +ODYSKIDSTOASTOPONASTREETWITHASPEEDLIMITOFMIH(ISSKIDMARKS
MEASUREFTANDTHECOEFFICIENTOFFRICTIONIS+ODYSAYSTHATHEWAS
DRIVINGONLYABOUTMIH+ODYWANTSTOPROVETHATHEWASNOTSPEEDING
q^
X ^^^
q
X
^
q X ÞÊ>Ìʏi>ÃÌÊ£xÊvÌ
#HOOSETHELETTERFORTHEBESTANSWER
q
^X ??
/NABUSYHIGHWAYWITHASPEEDLIMIT
OFMIHATRUCKAHEADOF6ERNAJACK
KNIFESACROSSTHEROAD6ERNASKIDS
TOASTOPFTSHORTOFTHETRUCK(ER
SKIDMARKSMEASUREFT7AS6ERNA
SPEEDING
"ARNEYWASDRIVINGATMIH!CAR
PULLSOUTFTAHEADOFHIM7HICH
STATEMENTISTRUE
! "ARNEYHITSTHECAR
" "ARNEYSTOPSLESSTHANAFOOTFROM
THECAR
^^^
4OWHATPOWERSHOULDBOTHSIDESOFEACHEQUATIONBERAISED
q
^
X /…ˆÀ`Ê«œÜiÀ
^^^
q
X
^^^
q
X
œÕÀ̅ʫœÜiÀ ! 9ESHERSPEEDWASMIH
" 9ESHERSPEEDWASMIH
# .OHERSPEEDWASMIH
# "ARNEYMISSESTHECARBYFT
$ "ARNEYSSKIDMARKSMEASUREFT
q X
Ê ^
Êq
Ê X
ÊÊÎ
Ê ^^^
Ê Ê qÊ XÊÊÓÊ
ÊÊÈ
Ê ^
Êq
Ê X
ÊÊÓ
Ê ^
Êq
Ê X
ÊÊ£n
Ê ^^^
Ê Ê qÊ XÊÊÈÊ
ÊÊÓ
q
^
X -iVœ˜`Ê*œÜiÀ
/…ˆÀ`Ê«œÜiÀ
3OLVETHEFOLLOWINGEQUATIONS
q^
X ^^^
q
X
$ .OHERSPEEDWASONLYMIH
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
AK4up.indd 82
œÌʏ}iLÀ>ÊÓ
XÊÊÈ{
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
82
XÊÊÇ
œÌʏ}iLÀ>ÊÓ
Holt Algebra 2
12/26/05 6:37:16 AM