chapter 6 resource masters - math class · 2019-02-12 · use the three triangle inequalities for...

Chapter 6Resource Masters

© Glencoe/McGraw-Hill A2 Glencoe Algebra 1

Stu

dy

Gu

ide

and I

nte

rven

tion

Sol

ving

Ineq

ualit

ies

by A

dditi

on a

nd S

ubtr

actio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-1

6-1

©G

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Lesson 6-1

Solv

e In

equa

litie

s by

Add

itio

nA

ddit

ion

can

be u

sed

to s

olve

ineq

ualit

ies.

If a

nynu

mbe

r is

add

ed t

o ea

ch s

ide

of a

tru

e in

equa

lity,

the

resu

ltin

g in

equa

lity

is a

lso

true

.

Add

ition

Pro

pert

y of

Ineq

ualit

ies

For

all

num

bers

a, b

, and

c, i

f a!

b, th

en a

"c

!b

"c,

an

d if

a#

b, th

en a

"c

#b

"c.

The

pro

pert

y is

als

o tr

ue w

hen

!an

d #

are

repl

aced

wit

h $

and

%.

Sol

ve x

!8

"!

6.T

hen

gra

ph

it

on a

nu

mbe

r li

ne.

x&

8 %

&6

Orig

inal

ineq

ualit

y

x&

8 "

8 %

&6

"8

Add

8 to

eac

h si

de.

x%

2S

impl

ify.

The

sol

utio

n in

set

-bui

lder

not

atio

n is

{x

|x%

2}.

Num

ber

line

grap

h:

!4

!3

!2

!1

01

23

4

Sol

ve 4

!2a

#!

a.T

hen

grap

h i

t on

a n

um

ber

lin

e.

4 &

2a!

&a

Orig

inal

ineq

ualit

y

4 &

2a"

2a!

&a

"2a

Add

2a

to e

ach

side

.

4 !

aS

impl

ify.

a#

44

!a

is th

e sa

me

as a

#4.

The

sol

utio

n in

set

-bui

lder

not

atio

n is

{a|a

#4}

.N

umbe

r lin

e gr

aph:

!2

!1

01

23

45

6

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sol

ve e

ach

in

equ

alit

y.T

hen

ch

eck

you

r so

luti

on,a

nd

gra

ph

it

on a

nu

mbe

r li

ne.

1.t

&12

$16

{t⏐t

$28

}2.

n&

12 #

6{n

⏐n%

18}

3.6

%g

&3

{g⏐g

$9}

4.n

&8

#&

13{n

⏐n%

!5}

5.&

12 !

&12

"y

{y⏐y

%0}

6.&

6 !

s&

8{s

⏐s%

2}

Sol

ve e

ach

in

equ

alit

y.T

hen

ch

eck

you

r so

luti

on.

7.&

3x%

8 &

4x8.

0.6n

$12

&0.

4n9.

&8k

&12

#&

9k{x

⏐x"

8}{n

⏐n$

12}

{k⏐k

%12

}

10.&

y&

10 !

15 &

2y11

.z&

%12

.&2b

!&

4 &

3b

{y⏐y

#25

}!z ⏐

z"

1"

{b⏐b

#!

4}

Def

ine

a va

riab

le,w

rite

an

in

equ

alit

y,an

d s

olve

eac

h p

robl

em.T

hen

ch

eck

you

rso

luti

on.

13–1

5.S

ampl

e an

swer

:Let

n&

the

num

ber.

13.A

num

ber

decr

ease

d by

4 is

less

tha

n 14

.n

!4

%14

;{n ⏐

n%

18}

14.T

he d

iffe

renc

e of

tw

o nu

mbe

rs is

mor

e th

an 1

2,an

d on

e of

the

num

bers

is 3

.n

!3

#12

;{n ⏐

n#

15}

15.F

orty

is n

o gr

eate

r th

an t

he d

iffe

renc

e of

a n

umbe

r an

d 2.

40 "

n!

2;{n

⏐n$

42}

2 ' 34 ' 31 ' 3

!4

!2

!1

01

23

!3

4!

3!

4!

2!

10

12

34

!9

!10

!8

!7

!6

!5

!4

!3

!2

78

910

1112

1314

1514

1512

1316

1718

1920

2627

2829

3031

3233

34

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4G

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Solv

e In

equa

litie

s by

Sub

trac

tion

Subt

ract

ion

can

be u

sed

to s

olve

ineq

ualit

ies.

If a

nynu

mbe

r is

sub

trac

ted

from

eac

h si

de o

f a t

rue

ineq

ualit

y,th

e re

sult

ing

ineq

ualit

y is

als

o tr

ue.

Sub

trac

tion

Pro

pert

y of

Ineq

ualit

ies

For

all

num

bers

a, b

, and

c, i

f a!

b, th

en a

&c

!b

&c,

an

d if

a#

b, th

en a

&c

#b

&c.

The

pro

pert

y is

als

o tr

ue w

hen

!an

d #

are

repl

aced

wit

h $

and

%.

Sol

ve 3

a(

5 #

4 (

2a.T

hen

gra

ph

it

on a

nu

mbe

r li

ne.

3a"

5 !

4 "

2aO

rigin

al in

equa

lity

3a"

5 &

2a!

4 "

2a&

2aS

ubtr

act 2

afr

om e

ach

side

.a

"5

!4

Sim

plify

.a

"5

&5

!4

&5

Sub

trac

t 5 fr

om e

ach

side

.a

!&

1S

impl

ify.

The

sol

utio

n is

{a⏐a

!&

1}.

Num

ber

line

grap

h:

Sol

ve e

ach

in

equ

alit

y.T

hen

ch

eck

you

r so

luti

on,a

nd

gra

ph

it

on a

nu

mbe

r li

ne.

1.t

"12

$8

2.n

"12

!&

123.

16 %

h"

9{t

⏐t$

!4}

{n⏐n

#!

24}

{h⏐h

$7}

4.y

"4

!&

25.

3r"

6 !

4r6.

q&

5 $

q

{y⏐y

#!

6}{r

⏐r%

6}{q

⏐q$

5}

Sol

ve e

ach

in

equ

alit

y.T

hen

ch

eck

you

r so

luti

on.

7.4p

$3p

"0.

78.

r"

!9.

9k"

12 !

8k

{p⏐p

$0.

7}!r ⏐

r#

"{k

⏐k#

!12

}

10.&

1.2

!2.

4 "

y11

.4y

#5y

"14

12.3

n"

17 #

4n{y

⏐y%

!3.

6}{y

⏐y#

!14

}{n

⏐n#

17}

Def

ine

a va

riab

le,w

rite

an

in

equ

alit

y,an

d s

olve

eac

h p

robl

em.T

hen

ch

eck

you

rso

luti

on.

13–1

5.S

ampl

e an

swer

:Let

n&

the

num

ber.

13.T

he s

um o

f a

num

ber

and

8 is

less

tha

n 12

.n

(8

%12

;{n ⏐

n%

4}

14.T

he s

um o

f tw

o nu

mbe

rs is

at

mos

t 6,

and

one

of t

he n

umbe

r is

&2.

n(

(!2)

"6;

{n⏐n

"8}

15.T

he s

um o

f a n

umbe

r an

d 6

is g

reat

er t

han

or e

qual

to

&4.

n(

6 $

!4;

{n⏐n

$!

10}

1 ' 83 ' 81 ' 4

21

03

45

67

82

34

56

79

18

!8

!7

!6

!5

!4

!3

!2

!1

0

1 ' 23 ' 2

56

78

910

1112

13!

26!

25!

24!

23!

22!

21!

6!

5!

4!

3!

2!

10

12

!4

!3

!2

!1

01

23

4

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Sol

ving

Ineq

ualit

ies

by A

dditi

on a

nd S

ubtr

actio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-1

6-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises

Answers (Lesson 6-1)

©G

lencoe/McG

raw-H

illA

3G

lencoe Algebra 1

Answers

Skills PracticeSolving Inequalities by Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

© Glencoe/McGraw-Hill 345 Glencoe Algebra 1

Less

on 6

-1

Match each inequality with its corresponding graph.

1. x " 11 ! 16 c a.

2. x & 6 # 1 e b.

3. x " 2 % &3 a c.

4. x " 3 $ 1 b d.

5. x & 1 # &7 d e.

Solve each inequality. Then check your solution, and graph it on a number line.

6. d & 5 % 1 {d⏐d " 6} 7. s " 9 # 8 {s⏐s % !1}

8. a & 7 ! &13 {a⏐a # !6} 9. w & 1 # 4 {w⏐w % 5}

10. 4 $ k " 3 {k⏐k " 1} 11. &9 % b & 4 {b⏐b $ !5}

12. &2 $ x " 4 {x⏐x " !6} 13. 2y # y " 2 {y⏐y % 2}

Define a variable, write an inequality, and solve each problem. Then check yoursolution. 14–18. Sample answer: Let n & the number.

14. A number decreased by 10 is greater than &5. n ! 10 # !5; {n⏐n # 5}

15. A number increased by 1 is less than 9. n ( 1 % 9; {n⏐n % 8}

16. Seven more than a number is less than or equal to &18. n ( 7 " !18; {n⏐n " !25}

17. Twenty less than a number is at least 15. n ! 20 $ 15; {n⏐n $ 35}

18. A number plus 2 is at most 1. n ( 2 " 1; {n⏐n " !1}

!2 !1!4 !3 0 1 2 3 4!6 !5!8 !7 !4 !3 !2 !1 0

!8 !7 !6 !5 !4 !3 !2 !1 0!2 !1!4 !3 0 1 2 3 4

2 30 1 4 5 6 7 8!8 !7 !6 !5 !4 !3 !2 !1 0

!2 !1!4 !3 0 1 2 3 42 30 1 4 5 6 7 8

0 1 2 3 4 5 6 7 8

!8 !7 !6 !5 !4 !3 !2 !1 0

876543210

43210!1!2!3!4

!8 !7 !6 !5 !4 !3 !2 !1 0


Match each inequality with its corresponding graph.

1. &8 $ x & 15 b a.

2. 4x " 3 # 5x d b.

3. 8x ! 7x & 4 a c.

4. 12 " x % 9 c d.

Solve each inequality. Then check your solution, and graph it on a number line.

5. r & (&5) ! &2 {r⏐r # !7} 6. 3x " 8 $ 4x {x⏐x " 8}

7. n & 2.5 $ &5 {n⏐n $ !2.5} 8. 1.5 # y " 1 {y⏐y # 0.5}

9. z " 3 ! !z⏐z # !2 " 10. % c & !c⏐c $ 1 "


11. The sum of a number and 17 is no less than 26.n ( 17 $ 26; {n⏐n $ 9}

12. Twice a number minus 4 is less than three times the number.2n ! 4 % 3n; {n⏐n # !4}

13. Twelve is at most a number decreased by 7.12 " n ! 7; {n⏐n $ 19}

14. Eight plus four times a number is greater than five times the number.8 ( 4n # 5n; {n⏐n % 8}

15. ATMOSPHERIC SCIENCE The troposphere extends from the earth’s surface to a heightof 6–12 miles, depending on the location and the season. If a plane is flying at analtitude of 5.8 miles, and the troposphere is 8.6 miles deep in that area, how muchhigher can the plane go without leaving the troposphere? no more than 2.8 mi

16. EARTH SCIENCE Mature soil is composed of three layers, the uppermost being topsoil.Jamal is planting a bush that needs a hole 18 centimeters deep for the roots. Theinstructions suggest an additional 8 centimeters depth for a cushion. If Jamal wants toadd even more cushion, and the topsoil in his yard is 30 centimeters deep, how muchmore cushion can he add and still remain in the topsoil layer? no more than 4 cm

!4 !3 !2 !1 0 1 2 3 4!4 !3 !2 !1 0 1 2 3 4

1'4

3'4

1'2

1'3

2'3

!4 !3 !2 !1 0 1 2 3 4!4 !3 !2 !1 0 1 2 3 4

4 52 3 6 7 8 9 10!8 !7 !6 !5 !4 !3 !2 !1 0

2 3 4 5 6 7 810

!8 !7 !6 !5 !4 !3 !2 !1 0

876543210

210!1!2!3!4!5!6

Practice (Average)

Solving Inequalities by Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

Answ

ers(Lesson 6-1)


Rea

din

g t

o L

earn

Math

emati

csS

olvi

ng In

equa

litie

s by

Add

ition

and

Sub

trac

tion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-1

6-1

©G

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7G

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ra 1

Lesson 6-1

Pre-

Act

ivit

yH

ow a

re i

neq

ual

itie

s u

sed

to

des

crib

e sc

hoo

l sp

orts

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 6-

1 at

the

top

of

page

318

in y

our

text

book

.

•U

se t

he in

form

atio

n in

the

gra

ph t

o w

rite

an

ineq

ualit

y st

atem

ent

abou

tpa

rtic

ipat

ion

in t

wo

spor

ts.

Sam

ple

answ

er:

For

soft

ball

and

trac

k an

d fie

ld,1

3,00

9 %

14,5

87

•R

ewri

te y

our

ineq

ualit

y st

atem

ent

to s

how

tha

t 40

sch

ools

add

ed b

oth

ofth

e sp

orts

.Is

the

stat

emen

t st

ill t

rue?

Sam

ple

answ

er:1

3,04

9 %

14,6

27;y

es

Read

ing

the

Less

onW

rite

th

e le

tter

of

the

grap

h t

hat

mat

ches

eac

h i

neq

ual

ity.

1.x

%&

1a.

2.x

$&

1b.

3.x

#&

1c.

4.x

!&

1d.

5.U

se t

he c

hart

to

wri

te a

sen

tenc

e th

at c

ould

be

desc

ribe

d by

the

ineq

ualit

y 3n

$2n

"7.

The

n so

lve

the

ineq

ualit

y.

Ineq

ualit

ies

%#

"$

less

than

grea

ter

than

at m

ost

at le

ast

few

er th

anm

ore

than

no m

ore

than

no le

ss th

anle

ss th

an o

r eq

ual t

ogr

eate

r th

an o

r eq

ual t

o

Sam

ple

answ

er:T

hree

tim

es a

num

ber

is a

t le

ast

two

times

the

num

ber

plus

7; n

$7

Hel

ping

You

Rem

embe

r

6.Te

achi

ng s

omeo

ne e

lse

can

help

you

rem

embe

r so

met

hing

.Exp

lain

how

you

wou

ld t

each

anot

her

stud

ent

who

mis

sed

clas

s to

sol

ve t

he in

equa

lity

2x"

4 %

3x.

Sub

trac

t 2x

from

eac

h si

de.S

impl

ify.

32

10

!1

!2

!3

c3

21

0!

1!

2!

3a

!3

!2

!1

01

23

d!

3!

2!

10

12

3b

©G

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8G

lenc

oe A

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ra 1

Tria

ngle

Ineq

ualit

ies

Rec

all t

hat

a lin

e se

gmen

t ca

n be

nam

ed b

y th

e le

tter

s of

its

endp

oint

s.L

ine

segm

ent

AB

(wri

tten

as

A !B!

) ha

s po

ints

Aan

d B

for

endp

oint

s.T

he l

engt

hof

AB

is w

ritt

en w

itho

ut t

he b

ar a

s A

B.

AB

#B

Cm

!A

#m

!B

The

sta

tem

ent

on t

he le

ft a

bove

sho

ws

that

A !B!

is s

hort

er t

han

B!C!

.T

he s

tate

men

t on

the

rig

ht a

bove

sho

ws

that

the

mea

sure

of

angl

e A

is le

ss t

han

that

of

angl

e B

.

The

se t

hree

ineq

ualit

ies

are

true

for

any

tri

angl

e A

BC

,no

mat

ter

how

long

the

sid

es.

a.A

B"

BC

!A

Cb.

If A

B!

AC

,the

n m

!C

!m

!B

.c.

If m

!C

!m

!B

,the

n A

B!

AC

.

Use

th

e th

ree

tria

ngl

e in

equ

alit

ies

for

thes

e p

robl

ems.

1.L

ist

the

side

s of

tri

angl

e D

EF

in o

rder

of

incr

easi

ng le

ngth

.

D#F#,

D#E#,

E#F#

2.In

the

fig

ure

at t

he r

ight

,whi

ch li

ne s

egm

ent

is t

he s

hort

est?

L#M#

3.E

xpla

in w

hy t

he le

ngth

s 5

cm,1

0 cm

,and

20

cm c

ould

not

be

used

to m

ake

a tr

iang

le.

5 (

10 is

not

gre

ater

tha

n 20

.

4.T

wo

side

s of

a t

rian

gle

mea

sure

3 in

.and

7 in

.Bet

wee

n w

hich

tw

ova

lues

mus

t th

e th

ird

side

be?

4 in

.and

10

in.

5.In

tri

angl

e X

YZ

,XY

(15

,YZ

(12

,and

XZ

(9.

Whi

ch is

the

grea

test

ang

le?

Whi

ch is

the

leas

t?!

Z;!

Y

6.L

ist

the

angl

es !

A,!

C,!

AB

C,a

nd !

AB

D,i

n or

der

of in

crea

sing

siz

e.

!A

BD

,!A

,!A

BC

,!C

C ADB

13 1512

5 9

JM

KL

65)

60)

65)

55)

65)

50)

D

FE

60)

35)

85)

A

BC

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-1

6-1



Ans

wer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Sol

ving

Ineq

ualit

ies

by M

ultip

licat

ion

and

Div

isio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-2

6-2

©G

lenc

oe/M

cGra

w-H

ill34

9G

lenc

oe A

lgeb

ra 1

Lesson 6-2

Solv

e In

equa

litie

s by

Mul

tipl

icat

ion

If e

ach

side

of

an in

equa

lity

is m

ulti

plie

d by

the

sam

e po

siti

ve n

umbe

r,th

e re

sult

ing

ineq

ualit

y is

als

o tr

ue.H

owev

er,i

f ea

ch s

ide

of a

nin

equa

lity

is m

ulti

plie

d by

the

sam

e ne

gati

ve n

umbe

r,th

e di

rect

ion

of t

he in

equa

lity

mus

tbe

rev

erse

d fo

r th

e re

sult

ing

ineq

ualit

y to

be

true

.

For

all

num

bers

a, b

, and

c, w

ith c

)0,

1.if

cis

pos

itive

and

a!

b, th

en a

c!

bc;

Mul

tiplic

atio

n P

rope

rty

of In

equa

litie

sif

cis

pos

itive

and

a#

b, th

en a

c#

bc;

2.if

cis

neg

ativ

e an

d a

!b,

then

ac

#bc

;if

cis

neg

ativ

e an

d a

#b,

then

ac

!bc

.

The

pro

pert

y is

als

o tr

ue w

hen

!an

d #

are

repl

aced

wit

h $

and

%.

Sol

ve !

$12

.

&$

12O

rigin

al e

quat

ion

(&8)"&

#%(&

8)12

Mul

tiply

eac

h si

de b

y &

8; c

hang

e $

to %

.

y%

&96

Sim

plify

.

The

sol

utio

n is

{y ⏐

y%

&96

}.

y ' 8y ' 8

y ' 8S

olve

k

%15

.

k#

15O

rigin

al e

quat

ion

"#k

#"#

15M

ultip

ly e

ach

side

by

.

k#

20S

impl

ify.

The

sol

utio

n is

{k⏐k

#20

}.

4 ' 34 ' 3

3 ' 44 ' 3

3 ' 4

3 ' 4Ex

ampl

e1Ex

ampl

e1Ex

ampl

e2Ex

ampl

e2

Exer

cises

Exer

cises

Sol

ve e

ach

in

equ

alit

y.T

hen

ch

eck

you

r so

luti

on.

1.%

22.

&!

223.

h$

&3

4.&

#&

6

{y⏐y

"12

}{n

⏐n%

!11

00}

{h⏐h

$!

5}{p

⏐p#

36}

5.n

$10

6.&

b#

7.#

&8.

&2.

51 %

&

{n⏐n

$40

}!b ⏐

b#

!"

!m⏐m

%!

"{h

⏐h"

5.02

}

9.$

&2

10.&

!&

11.

$5.

412

.$

&6

{g⏐g

$!

10}

!p ⏐p

#"

{n⏐n

$54

}{a

⏐a$

!21

}

Def

ine

a va

riab

le,w

rite

an

in

equ

alit

y,an

d s

olve

eac

h p

robl

em.T

hen

ch

eck

you

rso

luti

on.

13–1

5.S

ampl

e an

swer

:Let

n&

the

num

ber.

13.H

alf

of a

num

ber

is a

t le

ast

14.

n$

14;{

n ⏐n

$28

}

14.T

he o

ppos

ite

of o

ne-t

hird

a n

umbe

r is

gre

ater

tha

n 9.

!n

#9;

{n⏐n

%!

27}

15.O

ne f

ifth

of

a nu

mbe

r is

at

mos

t 30

.n

"30

;{n ⏐

n"

150}

1 ' 5

1 ' 3

1 ' 25 ' 12

2a ' 7n ' 10

9p ' 53 ' 4

g ' 5

1 ' 41 ' 2

2h ' 43 ' 20

3m '5

1 ' 32 ' 3

1 ' 4

p ' 63 ' 5

n ' 50y ' 6

©G

lenc

oe/M

cGra

w-H

ill35

0G

lenc

oe A

lgeb

ra 1

Solv

e In

equa

litie

s by

Div

isio

nIf

eac

h si

de o

f a

true

ineq

ualit

y is

div

ided

by

the

sam

e po

siti

ve n

umbe

r,th

e re

sult

ing

ineq

ualit

y is

als

o tr

ue.H

owev

er,i

f ea

ch s

ide

of a

nin

equa

lity

is d

ivid

ed b

y th

e sa

me

nega

tive

num

ber,

the

dire

ctio

n of

the

ineq

ualit

y sy

mbo

lm

ust

be r

ever

sed

for

the

resu

ltin

g in

equa

lity

to b

e tr

ue.

For

all

num

bers

a, b

, and

cw

ith c

)0,

Div

isio

n P

rope

rty

1.if

cis

pos

itive

and

a!

b, th

en

!; i

f cis

pos

itive

and

a#

b, th

en

#;

of In

equa

litie

s2.

if c

is n

egat

ive

and

a!

b, th

en

#; i

f cis

neg

ativ

e an

d a

#b,

then

!

.

The

pro

pert

y is

als

o tr

ue w

hen

!an

d #

are

repl

aced

wit

h $

and

%.

Sol

ve !

12y

$48

.

&12

y$

48O

rigin

al in

equa

lity

%D

ivid

e ea

ch s

ide

by &

12 a

nd c

hang

e $

to %

.

y%

&4

Sim

plify

.

The

sol

utio

n is

{y ⏐

y%

&4}

.

Sol

ve e

ach

in

equ

alit

y.T

hen

ch

eck

you

r so

luti

on.

1.25

g$

&10

02.

&2x

$9

3.&

5c!

24.

&8m

#&

64

{g⏐g

$!

4}!x ⏐

x"

!4

"!c ⏐

c%

!"

{m⏐m

#8}

5.&

6k#

6.18

#&

3b7.

30 #

&3n

8.&

0.24

#0.

6w

!k ⏐k

#!

"{b

⏐b%

!6}

{n⏐n

%!

10}

{w⏐w

#!

0.4}

9.25

$&

2m10

.&30

!&

5p11

.&2n

$6.

212

.&35

#0.

05h

!m⏐m

$!

12"

{p⏐p

#6}

{n⏐n

"!

3.1}

{h⏐h

#!

700}

13.&

40 !

10h

14.&

n$

615

.&3

#

{h⏐h

%!

4}{n

⏐n"

!9}

{p⏐p

#!

12}

Def

ine

a va

riab

le,w

rite

an

in

equ

alit

y,an

d s

olve

eac

h p

robl

em.T

hen

ch

eck

you

rso

luti

on.

16–1

8.S

ampl

e an

swer

:Let

n&

the

num

ber.

16.F

our

tim

es a

num

ber

is n

o m

ore

than

108

.4n

"10

8;{n

⏐n"

27}

17.T

he o

ppos

ite

of t

hree

tim

es a

num

ber

is g

reat

er t

han

12.

!3n

#12

;{n ⏐

n%

!4}

18.N

egat

ive

five

tim

es a

num

ber

is a

t m

ost

100.

!5n

"10

0;{n

⏐n$

!20

}

p ' 42 ' 3

1 ' 2

1 ' 30

1 ' 5

2 ' 51 ' 2

48' &

12&

12y

' &12

b ' ca ' c

b ' ca ' c

b ' ca ' c

b ' ca ' c

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Sol

ving

Ineq

ualit

ies

by M

ultip

licat

ion

and

Div

isio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-2

6-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises


©G

lencoe/McG

raw-H

illA

6G

lencoe Algebra 1

Skills PracticeSolving Inequalities by Multiplication and Division

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2


Less

on 6

-2

Match each inequality with its corresponding statement.

1. 3n # 9 d a. Three times a number is at most nine.

2. n $ 9 f b. One third of a number is no more than nine.

3. 3n % 9 a c. Negative three times a number is more than nine.

4. &3n ! 9 c d. Three times a number is less than nine.

5. n % 9 b e. Negative three times a number is at least nine.

6. &3n $ 9 e f. One third of a number is greater than or equal to nine.

Solve each inequality. Then check your solution.

7. 14g ! 56 8. 11w % 77 9. 20b $ &120 10. &8r # 16

{g⏐g # 4} {w⏐w " 7} {b⏐b $ !6} {r⏐r # !2}

11. &15p % &90 12. # 9 13. $ &15 14. & ! &9

{p⏐p $ 6} {s⏐s % 36} {a⏐a $ !135} {p⏐p % 63}

15. & $ 6 16. 5z # &90 17. &13m ! &26 18. % &17

{t⏐t " !72} {z⏐z % !18} {m⏐m % 2} {k⏐k " !85}

19. &y # 36 20. &16c $ &224 21. & % 2 22. 12 !

{y⏐y # !36} {c⏐c " 14} {h⏐h $ !20} {d⏐d % 144}


23. Four times a number is greater than &48. 4n # !48; {n⏐n # !12}

24. One eighth of a number is less than or equal to 3. n " 3; {n⏐n " 24}

25. Negative twelve times a number is no more than 84. !12n " 84; {n⏐n $ !7}

26. Negative one sixth of a number is less than &9. ! n % !9; {n⏐n # 54}

27. Eight times a number is at least 16. 8n $ 16; {n⏐n $ 2}

1'6

1'8

d'12

h'10

k'5

t'12

p'7

a'9

s'4

1'3

1'3


Match each inequality with its corresponding statement.

1. &4n $ 5 d a. Negative four times a number is less than five.

2. n ! 5 f b. Four fifths of a number is no more than five.

3. 4n % 5 e c. Four times a number is fewer than five.

4. n % 5 b d. Negative four times a number is no less than five.

5. 4n # 5 c e. Four times a number is at most five.

6. &4n # 5 a f. Four fifths of a number is more than five.


7. & # &14 8. &13h % 52 9. $ &6 10. 39 ! 13p

{a⏐a # 70} {h⏐h $ !4} {s⏐s $ !96} {p⏐p % 3}

11. n ! &12 12. & t # 25 13. & m % &6 14. k $ &10

{n⏐n # !18} {t⏐t # !45} {m⏐m $ 10} {k⏐k $ !3}

15. &3b % 0.75 16. &0.9c ! &9 17. 0.1x $ &4 18. &2.3 #

{b⏐b $ !0.25} {c⏐c % 10} {x⏐x $ !40} {j⏐j # !9.2}

19. &15y # 3 20. 2.6v $ &20.8 21. 0 ! &0.5u 22. f % &1

!y⏐y # ! " {v⏐v $ !8} {u⏐u # 0} !f⏐f " ! "Define a variable, write an inequality, and solve each problem. Then check yoursolution. 23!25. Sample answer: Let n & the number.

23. Negative three times a number is at least 57. !3n $ 57; {n⏐n " !19}

24. Two thirds of a number is no more than &10. n " !10; {n⏐n " !15}

25. Negative three fifths of a number is less than &6. ! n % !6; {n⏐n # 10}

26. FLOODING A river is rising at a rate of 3 inches per hour. If the river rises more than 2feet, it will exceed flood stage. How long can the river rise at this rate without exceedingflood stage? no more than 8 h

27. SALES Pet Supplies makes a profit of $5.50 per bag on its line of natural dog food. If thestore wants to make a profit of no less than $5225, how many bags of dog food does itneed to sell? at least 950 bags

3'5

2'3

8'7

1'5

7'8

j'4

10'3

3'5

5'9

2'3

s'16

a'5

4'5

4'5

Practice (Average)

Solving Inequalities by Multiplication and Division

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2

Answ

ers(Lesson 6-2)


Ans

wer

s

Rea

din

g t

o L

earn

Math

emati

csS

olvi

ng In

equa

litie

s by

Mul

tiplic

atio

n an

d D

ivis

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-2

6-2

©G

lenc

oe/M

cGra

w-H

ill35

3G

lenc

oe A

lgeb

ra 1

Lesson 6-2

Pre-

Act

ivit

yW

hy

are

ineq

ual

itie

s im

por

tan

t in

lan

dsc

apin

g?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 6-

2 at

the

top

of

page

325

in y

our

text

book

.

•W

ould

a w

all 6

bri

cks

high

be

low

er t

han

a w

all 6

blo

cks

high

? W

hy?

yes;

6(3)

%6(

12)

•W

ould

a w

all n

bric

ks h

igh

be lo

wer

tha

n a

wal

l nbl

ocks

hig

h? E

xpla

in.

yes;

Whe

n on

e qu

antit

y is

less

tha

n an

othe

r qu

antit

y,m

ultip

lyin

g bo

th q

uant

ities

by

the

sam

e po

sitiv

e nu

mbe

rdo

es n

ot c

hang

e th

e tr

uth

of t

he in

equa

lity.

Read

ing

the

Less

on

1.W

rite

an

ineq

ualit

y th

at d

escr

ibes

eac

h si

tuat

ion.

a.A

num

ber

ndi

vide

d by

8 is

gre

ater

tha

n 5.

n*

8 #

5

b.T

wel

ve t

imes

a n

umbe

r k

is a

t le

ast

7.12

k$

7

c.A

num

ber

xdi

vide

d by

&10

is le

ss t

han

or e

qual

to

50.

x*

(!10

) "

50

d.T

hree

fif

ths

of a

num

ber

nis

at

mos

t 13

.n

"13

e.N

ine

is g

reat

er t

han

or e

qual

to

one

half

of

a qu

anti

ty m

.9

$m

2.U

se w

ords

to

tell

wha

t ea

ch in

equa

lity

says

.

a.12

#6n

12 is

less

tha

n 6

times

a n

umbe

r n.

b.$

14A

num

ber

tdiv

ided

by

!3

is g

reat

er t

han

or e

qual

to

14.

c.11

x%

3211

tim

es a

num

ber

xis

at

mos

t 32

.

Hel

ping

You

Rem

embe

r

3.In

you

r ow

n w

ords

,wri

te a

rul

e fo

r m

ulti

plyi

ng a

nd d

ivid

ing

ineq

ualit

ies

by p

osit

ive

and

nega

tive

num

bers

.

Sam

ple

answ

er:W

hen

you

mul

tiply

or

divi

de e

ach

side

of

a tr

uein

equa

lity

by a

pos

itive

num

ber,

the

resu

lt is

tru

e.W

hen

you

mul

tiply

or

divi

de a

tru

e in

equa

lity

by a

neg

ativ

e nu

mbe

r,yo

u m

ust

reve

rse

the

dire

ctio

n of

the

ineq

ualit

y si

gn.

t' &

3

1 ' 2

3 ' 5

©G

lenc

oe/M

cGra

w-H

ill35

4G

lenc

oe A

lgeb

ra 1

The

May

a 'I'

he M

aya

wer

e a

Nat

ive

Am

eric

an p

eopl

e w

ho li

ved

from

abo

ut15

00 B

.C.t

o ab

out

1500

A.D

.in

the

regi

on t

hat

toda

y en

com

pass

esm

uch

of C

entr

al A

mer

ica

and

sout

hern

Mex

ico.

The

ir m

any

acco

mpl

ishm

ents

incl

ude

exce

ptio

nal a

rchi

tect

ure,

pott

ery,

pain

ting

,and

scu

lptu

re,a

s w

ell a

s si

gnif

ican

t ad

vanc

es in

the

fiel

ds o

f as

tron

omy

and

mat

hem

atic

s.

The

May

a de

velo

ped

a sy

stem

of

num

erat

ion

that

was

bas

ed o

nth

e nu

mbe

r tw

enty

.The

bas

ic s

ymbo

ls o

f th

is s

yste

m a

re s

how

n in

the

tabl

e at

the

rig

ht.T

he p

lace

s in

a M

ayan

num

eral

are

wri

tten

vert

ical

ly—

the

bott

om p

lace

rep

rese

nts

ones

,the

pla

ce a

bove

repr

esen

ts t

wen

ties

,the

pla

ce a

bove

tha

t re

pres

ents

20

*20

,or

four

hun

dred

s,an

d so

on.

For

inst

ance

,thi

s is

how

to

wri

te t

henu

mbe

r 99

7 in

May

an n

umer

als.

←2

*(

800

←9

*(

180

←17

*(

17 997

Eva

luat

e ea

ch e

xpre

ssio

n w

hen

v&

,w&

,x&

,y&

,an

d

z&

.Th

en w

rite

th

e an

swer

in

May

an n

um

eral

s.E

xerc

ise

5 is

don

e fo

r yo

u.

1.2.

3.xv

4.vx

y5.

wx

&z

6.vz

"xy

7.w

(v"

x "

z)8.

vwz

9.z(

wx

&x)

Tell

wh

eth

er e

ach

sta

tem

ent

is t

rue

or f

als

e.

10.

"(

"11

.(

12.

(

true

fals

efa

lse

13.(

") "

("

("

)tr

ue

14.H

ow a

re E

xerc

ises

10

and

11 a

like?

How

are

the

y di

ffer

ent?

Bot

h in

volv

e ch

angi

ng t

he o

rder

of

the

sym

bols

. Exe

rcis

e 10

invo

lves

chan

ging

the

ord

er o

f th

e ad

dend

s in

an

addi

tion

prob

lem

. Exe

rcis

e 11

invo

lves

cha

ngin

g th

e or

der

of t

he d

igits

in a

num

eral

.

____

___

___

____

_•

• •

____

___

___

____

_•

• •

• •

•__

___

____

___

___

____

_•

• •

____

___

___

____

_

•__

___

• •

•__

___

____

_

• •

•__

___

____

_

•__

___

• •

•__

___

____

_•

____

_•

____

_•

• •

____

___

___

• •

●●●● •__

___

____

___

___

• •

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• •

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rich

men

t

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ME

____

____

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____

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ATE

____

____

____

PE

RIO

D__

___

6-2

6-2

010

111

212

313

414

515

616

717

818

919

• •

• •

____

___

___

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Stu

dy

Gu

ide

and I

nte

rven

tion

Sol

ving

Mul

ti-S

tep

Ineq

ualit

ies

NA

ME

____

____

____

____

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ATE

____

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____

PE

RIO

D__

___

6-3

6-3

©G

lenc

oe/M

cGra

w-H

ill35

5G

lenc

oe A

lgeb

ra 1

Lesson 6-3

Solv

e M

ulti

-Ste

p In

equa

litie

sTo

sol

ve li

near

ineq

ualit

ies

invo

lvin

g m

ore

than

one

oper

atio

n,un

do t

he o

pera

tion

s in

rev

erse

of

the

orde

r of

ope

rati

ons,

just

as

you

wou

ld s

olve

an e

quat

ion

wit

h m

ore

than

one

ope

rati

on.

Sol

ve 6

x!

4 "

2x(

12.

6x&

4 %

2x"

12O

rigin

al in

equa

lity

6x&

4 &

2x%

2x"

12 &

2xS

ubtr

act 2

xfr

om

each

sid

e.4x

&4

%12

Sim

plify

.4x

&4

"4

%12

"4

Add

4 to

eac

h si

de.

4x%

16S

impl

ify.

%D

ivid

e ea

ch s

ide

by 4

.

x%

4S

impl

ify.

The

sol

utio

n is

{x ⏐

x%

4}.

16 ' 44x ' 4

Sol

ve 3

a!

15 #

4 (

5a.

3a&

15 !

4 "

5aO

rigin

al in

equa

lity

3a&

15 &

5a!

4 "

5a&

5aS

ubtr

act 5

afr

om

each

sid

e.&

2a&

15 !

4S

impl

ify.

&2a

&15

"15

!4

"15

Add

15

to e

ach

side

.&

2a!

19S

impl

ify.

#D

ivid

e ea

ch s

ide

by &

2an

d ch

ange

!to

#.

a#

&9

Sim

plify

.

The

sol

utio

n is

$a⏐a

#&

9%.

1 ' 2

1 ' 2

19 ' &2

&2a

' &2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sol

ve e

ach

in

equ

alit

y.T

hen

ch

eck

you

r so

luti

on.

1.11

y"

13 $

&1

2.8n

&10

#6

&2n

3."

1 !

&5

!y ⏐y

$!

1"

!n ⏐n

%1

"{q

⏐q#

!42

}

4.6n

"12

#8

"8n

5.&

12 &

d!

&12

"4d

6.5r

&6

!8r

&18

{n⏐n

#2}

{d⏐d

%0}

{r⏐r

%4}

7.%

128.

7.3y

&14

.4 !

4.9y

9.&

8m&

3 #

18 &

m

{x⏐x

$!

6}{y

⏐y#

6}{m

⏐m#

!3}

10.&

4y&

10 !

19 &

2y11

.9n

&24

n"

45 !

012

.$

&4

!y ⏐y

%!

14"

{n⏐n

%3}

!x ⏐x

$!

4"

Def

ine

a va

riab

le,w

rite

an

in

equ

alit

y,an

d s

olve

eac

h p

robl

em.T

hen

ch

eck

you

rso

luti

on.

13–1

5.S

ampl

e an

swer

:Let

n&

the

num

ber.

13.N

egat

ive

thre

e ti

mes

a n

umbe

r pl

us f

our

is n

o m

ore

than

the

num

ber

min

us e

ight

.!

3n(

4 "

n!

8;{n

⏐n$

3}

14.O

ne f

ourt

h of

a n

umbe

r de

crea

sed

by t

hree

is a

t le

ast

two.

n!

3 $

2;{n

⏐n$

20}

15.T

he s

um o

f tw

elve

and

a n

umbe

r is

no

grea

ter

than

the

sum

of t

wic

e th

e nu

mbe

r an

d &

8.12

(n

"2n

((!

8);{

n ⏐n

$20

}

1 ' 4

1 ' 21 ' 2

4x&

2'

5

&3x

"6

'' 2

3 ' 53 ' 11

q ' 7

©G

lenc

oe/M

cGra

w-H

ill35

6G

lenc

oe A

lgeb

ra 1

Solv

e In

equa

litie

s In

volv

ing

the

Dis

trib

utiv

e Pr

oper

tyW

hen

solv

ing

ineq

ualit

ies

that

con

tain

gro

upin

g sy

mbo

ls,f

irst

use

the

Dis

trib

utiv

e P

rope

rty

to r

emov

e th

egr

oupi

ng s

ymbo

ls.T

hen

undo

the

ope

rati

ons

in r

ever

se o

f th

e or

der

of o

pera

tion

s,ju

st a

s yo

uw

ould

sol

ve a

n eq

uati

on w

ith

mor

e th

an o

ne o

pera

tion

.

Sol

ve 3

a!

2(6a

!4)

#4

!(4

a(

6).

3a&

2(6a

&4)

!4

&(4

a"

6)O

rigin

al in

equa

lity

3a&

12a

"8

!4

&4a

&6

Dis

trib

utiv

e P

rope

rty

&9a

"8

!&

2 &

4aC

ombi

ne li

ke te

rms.

&9a

"8

"4a

!&

2 &

4a"

4aA

dd 4

ato

eac

h si

de.

&5a

"8

!&

2C

ombi

ne li

ke te

rms.

&5a

"8

&8

!&

2 &

8S

ubtr

act 8

from

eac

h si

de.

&5a

!&

10S

impl

ify.

a#

2D

ivid

e ea

ch s

ide

by &

5 an

d ch

ange

!to

#.

The

sol

utio

n in

set

-bui

lder

not

atio

n is

{a⏐a

#2}

.

Sol

ve e

ach

in

equ

alit

y.T

hen

ch

eck

you

r so

luti

on.

1.2(

t"

3) $

162.

3(d

&2)

&2d

!16

3.4h

&8

#2(

h&

1)

{t⏐t

$5}

{d⏐d

#22

}{h

⏐h%

3}

4.6y

"10

!8

&(y

"14

)5.

4.6(

x&

3.4)

!5.

1x6.

&5x

&(2

x"

3) $

1

!y ⏐y

#!

2"

{x⏐x

%!

31.2

8}!x ⏐

x"

!"

7.3(

2y&

4) &

2(y

"1)

!10

8.8

&2(

b"

1) #

12 &

3b9.

&2(

k&

1) !

8(1"

k)

{y⏐y

#6}

{b⏐b

%6}

!k ⏐k

%!

"10

.0.3

(y&

2) !

0.4(

1 "

y)11

.m"

17 %

&(4

m&

13)

{y⏐y

%!

10}

!m⏐m

"!

"12

.3n

"8

%2(

n&

4) &

2(1

&n)

13.2

(y&

2) !

&4

"2y

{n⏐n

$18

}+

14.k

&17

%&

(17

&k)

15.n

&4

%&

3(2

"n)

{k⏐k

is a

rea

l num

ber}

!n ⏐n

"!

"D

efin

e a

vari

able

,wri

te a

n i

neq

ual

ity,

and

sol

ve e

ach

pro

blem

.Th

en c

hec

k y

our

solu

tion

.16

–18.

Sam

ple

answ

er:L

et n

&th

e nu

mbe

r.16

.Tw

ice

the

sum

of

a nu

mbe

r an

d 4

is le

ss t

han

12.

2(n

(4)

%12

;{n ⏐

n%

2}17

.Thr

ee t

imes

the

sum

of

a nu

mbe

r an

d si

x is

gre

ater

tha

n fo

ur t

imes

the

num

ber

decr

ease

d by

tw

o.3(

n(

6) #

4n!

2;{n

⏐n%

20}

18.T

wic

e th

e di

ffer

ence

of

a nu

mbe

r an

d fo

ur is

less

tha

n th

e su

m o

f th

e nu

mbe

r an

d fi

ve.

2(n

!4)

%n

(5;

{n⏐n

%13

}

1 ' 24 ' 5

3 ' 54 ' 72 ' 7

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Sol

ving

Mul

ti-S

tep

Ineq

ualit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-3

6-3

Exam

ple

Exam

ple

Exer

cises

Exer

cises


©G

lencoe/McG

raw-H

illA

9G

lencoe Algebra 1

Answers

Skills PracticeSolving Multi-Step Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3


Less

on 6

-3

Justify each indicated step.

1. t & 3 $ &15

t & 3 " 3 $ &15 " 3 a.

t $ &12

" #t $ (&12) b.

t $ &16

a. Add 3 to each side.b. Multiply each side by .

2. 5(k " 8) & 7 % 235k " 40 & 7 % 23 a.

5k " 33 % 235k " 33 & 33 % 23 & 33 b.

5k % &10

% c.

k % &2

a. Distributive Propertyb. Subtract 33 from each side.c. Divide each side by 5.

?&10'5

5k'5

?

?

4'3

?4'3

3'4

4'3

3'4

?3'4

3'4


3. &2b " 4 ! &6 4. 3x " 15 % 21 5. & 1 $ 3

{b⏐b % 5} {x⏐x " 2} {d⏐d $ 8}

6. a & 4 # 2 7. & " 7 ! &4 8. j & 10 $ 5

{a⏐a % 15} {t⏐t % 55} {j⏐j $ 20}

9. & f " 3 # &9 10. 2p " 5 $ 3p & 10 11. 4k " 15 ! &2k " 3

{f⏐f # 18} {p⏐p " 15} {k⏐k # !2}

12. 2(&3m & 5) $ &28 13. &6(w " 1) # 2(w " 5) 14. 2(q & 3) " 6 % &10

{m⏐m " 3} {w⏐w # !2} {q⏐q " !5}


15. Four more than the quotient of a number and three is at least nine. ( 4 $ 9;{n⏐n $ 15}

16. The sum of a number and fourteen is less than or equal to three times the number.n ( 14 " 3n; {n⏐n $ 7}

17. Negative three times a number increased by seven is less than negative eleven.!3n ( 7 % !11; {n⏐n # 6}

18. Five times a number decreased by eight is at most ten more than twice the number.5n ! 8 " 2n ( 10; {n⏐n " 6}

19. Seven more than five sixths of a number is more than negative three. n ( 7 # !3;{n⏐n # !12}

20. Four times the sum of a number and two increased by three is at least twenty-seven.4(n ( 2) ( 3 $ 27; {n⏐n $ 4}

5'6

n'3

2'3

3'4

t'5

2'5

d'2


Justify each indicated step.

Practice (Average)

Solving Multi-Step Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3

1. x !

8x ! (8) a.

8x ! 5x & 128x & 5x ! 5x & 12 & 5x b.

3x ! &12

! c.

x ! &4a. Multiply each side by 8.b. Subtract 5x from each side.c. Divide each side by 3.

2. 2(2h " 2) # 2(3h " 5) & 124h " 4 # 6h " 10 & 12 a.4h " 4 # 6h & 2

4h " 4 & 6h # 6h & 2 & 6h b.&2h " 4 # &2

&2h " 4 & 4 # &2 & 4 c.&2h # &6

! d.

h ! 3a. Distributive Propertyb. Subtract 6h from each side.c. Subtract 4 from each side.d. Divide each side by !2 and

change % to #.

?&6'&2

&2h'&2

?

?

?

?&12'3

3x'3

?

?5x & 12'8

5x & 12'8


3. &5 & $ &9 4. 4u & 6 $ 6u & 20 5. 13 ! a & 1

{t⏐t " 24} {u⏐u " 7} {a⏐a % 21}6. # &8 {w⏐w % !19} 7. ! 7 {f⏐f # 15}

8. h % {h⏐h $ !3} 9. 3(z " 1) " 11 # &2(z " 13) {z⏐z % !8}

10. 3e " 2(4e " 2) % 2(6e " 1) {e⏐e $ 2} 11. 5n & 3(n & 6) $ 0 {n⏐n $ !9}

Define a variable, write an inequality, and solve each problem. Then check yoursolution. 12–13. Sample answer: Let n & the number.12. A number is less than one fourth the sum of three times the number and four.

n % ; {n⏐n % 4}13. Two times the sum of a number and four is no more than three times the sum of the

number and seven decreased by four. 2(n ( 4) " 3(n ( 7) ! 4; {n⏐n $ !9}

14. GEOMETRY The area of a triangular garden can be no more than 120 square feet. Thebase of the triangle is 16 feet. What is the height of the triangle? no more than 15 ft

15. MUSIC PRACTICE Nabuko practices the violin at least 12 hours per week. Shepractices for three fourths of an hour each session. If Nabuko has already practiced 3 hours in one week, how many sessions remain to meet or exceed her weekly practice goal? at least 12 sessions

3n ( 4'

4

6h " 3'5

3f & 10'5

w " 3'2

2'3

t'6

Answ

ers(Lesson 6-3)


Rea

din

g t

o L

earn

Math

emati

csS

olvi

ng M

ulti-

Ste

p In

equa

litie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-3

6-3

©G

lenc

oe/M

cGra

w-H

ill35

9G

lenc

oe A

lgeb

ra 1

Lesson 6-3

Pre-

Act

ivit

yH

ow a

re l

inea

r in

equ

alit

ies

use

d i

n s

cien

ce?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 6-

3 at

the

top

of

page

332

in y

our

text

book

.T

hen

wri

te a

n in

equa

lity

that

cou

ld b

e us

ed t

o fi

nd t

he t

empe

ratu

res

inde

gree

s C

elsi

us f

or w

hich

eac

h su

bsta

nce

is a

gas

.

Arg

on:

C(

32 #

!30

3B

rom

ine:

C(

32 #

138

Read

ing

the

Less

on

1.W

hat

does

the

phr

ase

“und

oing

the

ope

rati

ons

in r

ever

se o

f the

ord

er o

f ope

rati

ons”

mea

n?

Sam

ple

answ

er:F

irst

add

or

subt

ract

to

undo

sub

trac

tion

or a

dditi

on,

then

mul

tiply

or

divi

de t

o un

do d

ivis

ion

or m

ultip

licat

ion.

2.D

escr

ibe

how

che

ckin

g th

e so

luti

on o

f an

ineq

ualit

y is

dif

fere

nt f

rom

che

ckin

g th

eso

luti

on o

f an

equ

atio

n.S

ampl

e an

swer

:Ins

tead

of

subs

titut

ing

one

valu

e fo

r th

e va

riab

le,t

here

are

infin

itely

man

y va

lues

tha

t ca

n be

use

d to

che

ck.I

t is

a g

ood

idea

to

use

a va

lue

that

is le

ss t

han,

the

valu

e eq

ual t

o,an

d a

valu

e gr

eate

r th

anth

e nu

mbe

r in

the

sol

utio

n to

che

ck a

n in

equa

lity.

3.D

escr

ibe

how

the

Dis

trib

utiv

e P

rope

rty

can

be u

sed

to r

emov

e th

e gr

oupi

ng s

ymbo

ls in

the

ineq

ualit

y 4x

&7(

2x"

8) %

3x&

5.

Mul

tiply

!7

by b

oth

2xan

d 8.

4.Is

it p

ossi

ble

to h

ave

no s

olut

ion

whe

n yo

u so

lve

an in

equa

lity?

Exp

lain

you

r an

swer

and

give

an

exam

ple.

Sam

ple

answ

er:Y

es;i

f so

lvin

g re

sults

in a

n in

equa

lity

that

is n

ever

tru

e(a

nd t

he s

igns

hav

e be

en r

ever

sed

if ne

cess

ary)

,the

n th

ere

is n

oso

lutio

n.E

xam

ple:

3(t!

4) !

8 #

3(t(

4) !

8

Hel

ping

You

Rem

embe

r

5.M

ake

a ch

eckl

ist

of s

teps

you

can

use

whe

n so

lvin

g in

equa

litie

s.

(1)

Use

the

Dis

trib

utiv

e P

rope

rty

to r

emov

e an

y gr

oupi

ng s

ymbo

ls.

(2)

Com

bine

any

like

ter

ms.

(3)

Add

or

subt

ract

the

sam

e va

riab

le t

erm

s or

con

stan

ts o

n bo

th s

ides

.(4

)M

ultip

ly o

r di

vide

to

undo

ope

ratio

ns.

(5)

Rev

erse

the

dir

ectio

n of

the

ineq

ualit

y sy

mbo

l if

both

sid

es w

ere

mul

tiplie

d or

div

ided

by

a ne

gativ

e nu

mbe

r.(6

)B

e su

re t

he v

aria

ble

is b

y its

elf

on o

ne s

ide

of t

he f

inal

ineq

ualit

y.

9 ' 59 ' 5

©G

lenc

oe/M

cGra

w-H

ill36

0G

lenc

oe A

lgeb

ra 1

Car

los

Mon

tezu

ma

Dur

ing

his

lifet

ime,

Car

los

Mon

tezu

ma

(186

5?–1

923)

was

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ME

____

____

____

____

____

____

____

____

____

____

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ATE

____

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RIO

D__

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6-3

6-3



Ans

wer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Sol

ving

Com

poun

d In

equa

litie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-4

6-4

©G

lenc

oe/M

cGra

w-H

ill36

1G

lenc

oe A

lgeb

ra 1

Lesson 6-4

Ineq

ualit

ies

Cont

aini

ng a

ndA

com

poun

d in

equa

lity

cont

aini

ng a

ndis

tru

e on

ly if

both

ineq

ualit

ies

are

true

.The

gra

ph o

f a

com

poun

d in

equa

lity

cont

aini

ng a

ndis

the

inte

rsec

tion

of t

he g

raph

s of

the

tw

o in

equa

litie

s.E

very

sol

utio

n of

the

com

poun

din

equa

lity

mus

t be

a s

olut

ion

of b

oth

ineq

ualit

ies.

Gra

ph

th

e so

luti

onse

t of

x%

2 an

d x

$!

1. Gra

ph x

#2.

Gra

ph x

$&

1.

Fin

d th

e in

ters

ectio

n.

The

sol

utio

n se

t is

{x⏐&

1 %

x#

2}.

!2

!1

!3

01

23

!3

!2

!1

01

23

!3

!2

!1

01

23

Sol

ve !

1 %

x(

2 %

3 u

sin

ga

nd

.Th

en g

rap

h t

he

solu

tion

set

.

&1

#x

"2

and

x"

2 #

3&

1 &

2 #

x"

2 &

2x

"2

&2

#3

&2

&3

#x

x#

1

Gra

ph x

!&

3.

Gra

ph x

#1.

Fin

d th

e in

ters

ectio

n.

The

sol

utio

n se

t is

{x ⏐

&3

#x

#1}

.

!2

!1

!4

!3

01

2

!3

!4

!2

!1

01

2

!3

!4

!2

!1

01

2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Gra

ph

th

e so

luti

on s

et o

f ea

ch c

omp

oun

d i

neq

ual

ity.

1.b

!&

1 an

d b

%3

2.2

$q

$&

53.

x!

&3

and

x%

4

4.&

2 %

p#

45.

&3

#d

and

d#2

6.&

1 %

p%

3

Sol

ve e

ach

com

pou

nd

in

equ

alit

y.T

hen

gra

ph

th

e so

luti

on s

et.

7.4

#w

"3

%5

8.&

3 %

p&

5 #

2{w

⏐1 %

w"

2}{p

⏐2 "

p%

7}

9.&

4 #

x"

2 %

&2

10.y

&1#

2 an

d y

"2

$1

{x⏐!

6 %

x"

!4}

{y⏐!

1 "

y%

3}

11.n

&2

!&

3 an

d n

"4

#6

12.d

&3

#6d

"12

#2d

"32

{n⏐!

1 %

n%

2}{d

⏐!3

%d

%5}

!3

!2

!1

01

23

45

!3

!4

!2

!1

01

23

4

!3

!4

!2

!1

01

23

4!

7!

6!

5!

4!

3!

2!

10

1

01

23

45

67

8!

3!

4!

2!

10

12

34

!3

!4

!2

!1

01

23

4!

3!

4!

2!

10

12

34

!3

!2

!1

01

23

45

!4

!3

!2

!1

01

23

4!

4!

3!

6!

5!

2!

10

12

!4

!3

!2

!1

01

23

4

©G

lenc

oe/M

cGra

w-H

ill36

2G

lenc

oe A

lgeb

ra 1

Ineq

ualit

ies

Cont

aini

ng o

rA

com

poun

d in

equa

lity

cont

aini

ng o

ris

tru

e if

one

or

both

of

the

ineq

ualit

ies

are

true

.The

gra

ph o

f a

com

poun

d in

equa

lity

cont

aini

ng o

ris

the

un

ion

of t

he g

raph

s of

the

tw

o in

equa

litie

s.T

he u

nion

can

be

foun

d by

gra

phin

g bo

thin

equa

litie

s on

the

sam

e nu

mbe

r lin

e.A

sol

utio

n of

the

com

poun

d in

equa

lity

is a

sol

utio

n of

eith

er in

equa

lity,

not

nece

ssar

ily b

oth.

Sol

ve 2

a(

1 %

11 o

r a

#3a

(2.

2a"

1 #

11or

a!

3a"

22a

"1

&1

#11

&1

a&

3a!

3a&

3a"

22a

#10

&2a

!2

##

a#

5a

#&

1

Gra

ph a

#5.

Gra

ph a

#&

1.

Fin

d th

e un

ion.

The

sol

utio

n se

t is

{a⏐a

#5}

.

Gra

ph

th

e so

luti

on s

et o

f ea

ch c

omp

oun

d i

neq

ual

ity.

1.b

!2

or b

%&

32.

3 $

qor

q%

13.

y%

&4

or y

!0

4.4

%p

or p

#8

5.&

3 #

dor

d#

26.

&2

%x

or 3

%x

Sol

ve e

ach

com

pou

nd

in

equ

alit

y.T

hen

gra

ph

th

e so

luti

on s

et.

7.3

#3w

or 3

w$

98.

&3p

"1

%&

11 o

r p

#2

{w⏐1

%w

}{p

⏐p$

4 or

p%

2}

9.2x

"4

%6

or x

$2x

&4

10.2

y"

2 #

12 o

r y

&3

$2y

{x⏐x

"4}

{y⏐y

%5}

11.

n!

&2

or 2

n&

2 #

6 "

n12

.3a

"2

$5

or 7

"3a

#2a

"6

{n⏐n

is a

rea

l num

ber}

{a⏐a

%!

1 or

a$

1}

0!

1!

2!

3!

41

23

40

!1

!2

!3

!4

12

34

1 ' 2

01

23

45

67

8!

2!

10

12

34

56

01

23

45

67

8!

3!

4!

2!

10

12

34

!3

!4

!2

!1

01

23

40

!1

!2

!3

!4

12

34

0!

1!

21

23

45

6

!3

!4

!5

!2

!1

01

23

!3

!4

!2

!1

01

23

4!

3!

4!

2!

10

12

34

!2

!1

01

23

45

6

!2

!1

01

23

45

6

!2

!1

01

23

45

6

2' &

2&

2a' &

210 ' 2

2a ' 2

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Sol

ving

Com

poun

d In

equa

litie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-4

6-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises


©G

lencoe/McG

raw-H

illA

12G

lencoe Algebra 1

Skills PracticeSolving Compound Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4


Less

on 6

-4

Graph the solution set of each compound inequality.

1. b ! 3 or b % 0 2. z % 3 and z $ &2

3. k ! 1 and k ! 5 4. y # &1 or y $ 1

Write a compound inequality for each graph.

5. 6.

!3 % x " 3 1 " x " 4

7. 8.

x % !2 or x $ 1 x % !1 or x # 2

Solve each compound inequality. Then graph the solution set.

9. m " 3 $ 5 and m " 3 # 7 10. y & 5 # &4 or y & 5 $ 1{m⏐2 " m % 4} {y⏐y % 1 or y $ 6}

11. 4 # f " 6 and f " 6 # 5 12. w " 3 % 0 or w " 7 $ 9{f⏐!2 % f % !1} {w⏐w " !3 or w $ 2}

13. &6 # b & 4 # 2 14. p & 2 % &2 or p & 2 ! 1{b⏐!2 % b % 6} {p⏐p " 0 or p # 3}


15. A number plus one is greater than negative five and less than three.!5 % n ( 1 % 3; {n⏐!6 % n % 2}

16. A number decreased by two is at most four or at least nine.n ! 2 " 4 or n ! 2 $ 9; {n⏐n " 6 or n $ 11}

17. The sum of a number and three is no more than eight or is more than twelve.n ( 3 " 8 or n ( 3 # 12; {n⏐n " 5 or n # 9}

!3!4 !2 !1 0 1 2 3 4!2 !1 0 1 2 3 4 5 6

!3!4 !2 !1 0 1 2 3 4!4 !3 !2 !1 0 1 2 3 4

!2 !1 0 1 2 3 4 5 6!2 !1 0 1 2 3 4 5 6

!4 !3 !2 !1 0 1 2 3 4!4 !3 !2 !1 0 1 2 3 4

!2 !1 0 1 2 3 4 5 6!2 !1!4 !3 0 1 2 3 4

!3!4 !2 !1 0 1 2 3 40 1 2 3 4 5 6 7 8

!4 !3 !2 !1 0 1 2 3 4!3!4 !2 !1 0 1 2 3 4


Graph the solution set of each compound inequality.

1. &4 % e % 1 2. x ! 0 or x # 3

3. g # &3 or g $ 4 4. &4 % p % 4

Write a compound inequality for each graph.

5. 6.

x " !3 or x $ 3 x % 2 or x $ 37. 8.

0 " x % 5 !5 % x % 0

Solve each compound inequality. Then graph the solution set.

9. k & 3 # &7 or k " 5 $ 8 10. &n # 2 or 2n & 3 ! 5{k⏐k % !4 or k $ 3} {n⏐n # !2}

11. 5 # 3h " 2 % 11 12. 2c & 4 ! &6 and 3c " 1 # 13{h⏐1 % h " 3} {c⏐!1 % c % 4}

Define a variable, write an inequality, and solve each problem. Then check yoursolution. 13–14. Sample answer: Let n & the number.13. Two times a number plus one is greater than five and less than seven.

5 % 2n ( 1 % 7; {n⏐2 % n % 3}14. A number minus one is at most nine, or two times the number is at least twenty-four.

n ! 1 " 9 or 2n $ 24; {n⏐n " 10 or n $ 12}

METEOROLOGY For Exercises 15 and 16, use the following information.Strong winds called the prevailing westerlies blow from west to east in a belt from 40° to60° latitude in both the Northern and Southern Hemispheres.

15. Write an inequality to represent the latitude of the prevailing westerlies.{w⏐40 " w " 60}

16. Write an inequality to represent the latitudes where the prevailing westerlies are not located. {w⏐w % 40 or w # 60}

17. NUTRITION A cookie contains 9 grams of fat. If you eat no fewer than 4 and no more than7 cookies, how many grams of fat will you consume? between 36 g and 63 g inclusive

!2 !1 0 1 2 3 4 5 6!2 !1!4 !3 0 1 2 3 4

!4 !3 !2 !1 0 1 2 3 4!3!4 !2 !1 0 1 2 3 4

!2!3!4!5!6 !1 0 1 2!2 !1 0 1 2 3 4 5 6

!2 !1 0 1 2 3 4 5 6!4 !3 !2 !1 0 1 2 3 4

!2 !1!4 !3 0 1 2 3 4!3!4 !2 !1 0 1 2 3 4

0!1!2!3!4 1 2 3 4!2 !1!4 !3!6 !5 0 1 2

Practice (Average)

Solving Compound Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4

Answ

ers(Lesson 6-4)


Ans

wer

s

Rea

din

g t

o L

earn

Math

emati

csS

olvi

ng C

ompo

und

Ineq

ualit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-4

6-4

©G

lenc

oe/M

cGra

w-H

ill36

5G

lenc

oe A

lgeb

ra 1

Lesson 6-4

Pre-

Act

ivit

yH

ow a

re c

omp

oun

d i

neq

ual

itie

s u

sed

in

tax

tab

les?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 6-

4 at

the

top

of

page

339

in y

our

text

book

.

•E

xpla

in w

hy it

is p

ossi

ble

that

Mr.

Kel

ly’s

inco

me

is $

41,3

70.

$41,

370

is g

reat

er t

han

or e

qual

to

$41,

350

and

less

th

an $

41,4

00.

•E

xpla

in w

hy it

is n

otpo

ssib

le t

hat

Mr.

Kel

ly’s

inco

me

is $

41,4

00.

$41,

400

is n

ot le

ss t

han

$41,

400.

Read

ing

the

Less

on

1.W

hen

is a

com

poun

d in

equa

lity

cont

aini

ng a

ndtr

ue?

It is

tru

e w

hen

both

ineq

ualit

ies

are

true

.

2.T

he g

raph

of

a co

mpo

und

ineq

ualit

y co

ntai

ning

and

is t

he

of t

hegr

aphs

of

the

two

ineq

ualit

ies.

3.W

hen

is a

com

poun

d in

equa

lity

cont

aini

ng o

rtr

ue?

It is

tru

e w

hen

one

or b

oth

of t

he in

equa

litie

s is

tru

e.

4.T

he g

raph

of

a co

mpo

und

ineq

ualit

y co

ntai

ning

or

is t

he

of t

hegr

aphs

of

the

two

ineq

ualit

ies.

5.Su

ppos

e yo

u us

e ye

llow

to

show

the

gra

ph o

f In

equa

lity

#1 o

n th

e nu

mbe

r lin

e.Yo

u us

ebl

ue t

o sh

ow t

he g

raph

of

Ineq

ualit

y #2

.Wri

te a

ndor

or

in e

ach

blan

k to

com

plet

e th

ese

nten

ce.

a.T

he p

art

that

is g

reen

is t

he g

raph

of

Ineq

ualit

y #1

In

equa

lity

#2.

b.A

ll co

lore

d pa

rts

form

the

gra

ph o

f In

equa

lity

#1

Ineq

ualit

y #2

.

Hel

ping

You

Rem

embe

r

6.O

ne w

ay t

o re

mem

ber

som

ethi

ng is

to

conn

ect

it t

o so

met

hing

tha

t is

fam

iliar

to

you.

Wri

te t

wo

true

com

poun

d st

atem

ents

abo

ut y

ours

elf,

one

usin

g th

e w

ord

and

and

the

othe

r us

ing

the

wor

d or

.

Sam

ple

answ

er:I

am

14

and

I am

a f

resh

man

in h

igh

scho

ol.A

fter

scho

ol,I

will

go

to fo

otba

ll pr

actic

e or

I w

ill g

o ho

me.

oran

d

unio

n

inte

rsec

tion

©G

lenc

oe/M

cGra

w-H

ill36

6G

lenc

oe A

lgeb

ra 1

Som

e P

rope

rtie

s of

Ineq

ualit

ies

The

tw

o ex

pres

sion

s on

eit

her

side

of

an in

equa

lity

sym

bol a

reso

met

imes

cal

led

the

firs

tan

d se

cond

mem

bers

of

the

ineq

ualit

y.

If t

he in

equa

lity

sym

bols

of

two

ineq

ualit

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____

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PE

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D__

___

6-4

6-4


©G

lencoe/McG

raw-H

illA

14G

lencoe Algebra 1

Study Guide and InterventionSolving Open Sentences Involving Absolute Value

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5


Less

on 6

-5

Absolute Value Equations When solving equations that involve absolute value, thereare two cases to consider.Case 1: The value inside the absolute value symbols is positive.Case 2: The value inside the absolute value symbols is negative.

Solve ⏐x ( 4⏐ & 1. Thengraph the solution set.

Write ⏐x " 4 ( 1⏐ as x " 4 ( 1 or x " 4 ( &1.

x " 4 ( 1 or x " 4 ( &1x " 4 & 4 ( 1 & 4 x " 4 ( &1

x ( &3 x " 4 & 4 ( &1& 4x ( &5

The solution set is {&5, &3}.The graph is shown below.

!8 !7 !6 !5 !4 !3 !2 !1 0

Write an inequalityinvolving absolute value for the graph.

Find the point that is the same distancefrom &2 as it is from 4.

The distance from 1 to &2 is 3 units. Thedistance from 1 to 4 is 3 units.So, ⏐x & 1⏐ ( 3.

10!1!2!3 2 3

3 units 3 units

4 5

!3 !2 !1 0 1 2 3 4 5

Example 1Example 1 Example 2Example 2

ExercisesExercises

Solve each open sentence. Then graph the solution set.

1. ⏐y⏐ ( 3 {!3, 3} 2. ⏐x & 4⏐ ( 4 {0, 8} 3. ⏐y " 3⏐ ( 2 {!5, !1}

4. ⏐b " 2⏐ ( 3 {!5, 1} 5. ⏐w & 2⏐ ( 5 {!3, 7} 6. ⏐t " 2⏐ ( 4 {!6, 2}

7. ⏐2x⏐ ( 8 {!4, 4} 8. ⏐5y & 2⏐ ( 7 !!1, 1 " 9. ⏐p & 0.2⏐ ( 0.5 {!0.3, 0.7}

10. ⏐d & 100⏐ ( 50 {50, 150}11. ⏐2x & 1⏐ ( 11 {!5, 6} 12. ⏐3x " ⏐ ( 6 !!2 , 1 "

For each graph, write an open sentence involving absolute value.

13. 14. 15.

⏐x⏐ & 4 ⏐x ! 1⏐ & 2 ⏐x ( 3⏐ & 4!3!4!5!6!7 !2 !1 0 1!3!4 !2 !1 0 1 2 3 40!2!4!6!8 2 4 6 8

!2!3 !1 0 1 2 3 4 5!4!6 !2 0 2 4 6 8 1050 100 150 200

5'6

1'6

1'2

!0.8 !0.4 0 0.4 0.8!3!4 !2 !1 0 1 2 3 4!3!4 !2 !1 0 1 2 3 4

4'5

!8 !6 !4 !2 0 2 4 6 8!8 !6 !4 !2 0 2 4 6 8!6 !5 !4 !3 !2 !1 0 1 2

!8 !7 !6 !5 !4 !3 !2 !1 00 1 2 3 4 5 6 7 8!3!4 !2 !1 0 1 2 3 4


Absolute Value Inequalities When solving inequalities that involve absolute value, there are two cases to consider for inequalities involving # (or %) and two cases to consider for inequalities involving ! (or $).

Remember that inequalities with and are related to intersections, while inequalities with or are related to unions.

Solve |3a ( 4| % 10. Then graph the solution set.

Write ⏐3a " 4⏐ # 10 as 3a " 4 # 10 and 3a " 4 ! &10.3a " 4 # 10 and 3a " 4 ! &10

3a " 4 & 4 # 10 & 4 3a " 4 & 4 ! &10 & 43a # 6 3a ! &14

# !

a # 2 a ! &4

The solution set is $a⏐&4 # a # 2%.


1. ⏐c & 2⏐ ! 6 2. ⏐x & 9⏐ # 0 3. ⏐3f " 10⏐ % 4

{c⏐c % !4 or c # 8} + !f⏐!4 " f " !2"

4. ⏐x⏐ % 2 5. ⏐x⏐ $ 3 6. ⏐2x " 1⏐ $ &2

{x⏐!2 " x " 2} {x⏐x " !3 or x $ 3} {x⏐x is a real number}

7. ⏐2d & 1⏐ % 4 8. ⏐3 & (x & 1)⏐ % 8 9. ⏐3r " 2⏐ # &5

!d⏐!1 " d " 2 " {x⏐!4 " x " 12} +


10. 11. 12.

⏐x⏐ # 1 ⏐x ! 2⏐ # 1 ⏐x ! 1⏐ " 3

!2 !1!3 0 1 2 3 4 5!4 !3 !2 !1 0 1 2 3 4!4 !3 !2 !1 0 1 2 3 4

!4 !3 !2 !1 0 1 2 3 4!4 !2 0 2 4 6 8 10 12!4 !3 !2 !1 0 1 2 3 4

1'2

1'2

0!1!2!3!4 1 2 3 40!1!2!3!4 1 2 3 4!4 !3 !2 !1 0 1 2 3 4

!6 !5 !4 !3 !2 !1 0 1 2!4 !3 !2 !1 0 1 2 3 420!2!4!6 4 6 8 10

2'3

2'3

2'3

&14'3

3a'3

6'3

3a'3

If ⏐x⏐ # n, then x ! &n and x # n.If ⏐x⏐ ! n, then x ! n or x # &n.

Study Guide and Intervention (continued)

Solving Open Sentences Involving Absolute Value

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5

ExampleExample

ExercisesExercises

Now graph the solution set.

!2 !1!4!5 !3 0 1 2 3

Answ

ers(Lesson 6-5)

©G

lencoe/McG

raw-H

illA

15G

lencoe Algebra 1

Answers

Skills PracticeSolving Open Sentences Involving Absolute Value

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5


Less

on 6

-5

Match each open sentence with the graph of its solution set.

1. ⏐x⏐ ! 2 c a.

2. ⏐x " 5⏐ ( 3 a b.

3. ⏐x & 2⏐ % 3 d c.

4. ⏐x " 1⏐ # 4 b d.

Express each statement using an inequality involving absolute value. Do not solve.

5. The weatherman predicted that the temperature would be within 3° of 52°F.

⏐t ! 52⏐ " 3

6. Serena will make the B team if she scores within 8 points of the team average of 92.

⏐p ! 92⏐ " 8

7. The dance committee expects attendance to number within 25 of last year’s 87 students.

⏐a ! 87⏐ " 25


8. ⏐s " 1⏐ ( 5 {!6, 4} 9. ⏐c & 3⏐# 1 {c⏐2 % c % 4}

10. ⏐n " 2⏐ $ 1 {n⏐n " !3 or n $ !1} 11. ⏐t " 6⏐ ! 4 {t⏐t % !10 or t # !2}

12. ⏐w & 2⏐ ( 2 {0, 4} 13. ⏐k & 5⏐ % 4 {k⏐1 " k " 9}


14. 15.

⏐x⏐ & 1 ⏐x ( 3⏐ # 2

16. 17.

⏐x ! 4⏐ " 1 ⏐x⏐ $ 4

!2!3!4!5 !1 0 1 2 3 4 5!2!3!4!5 !1 0 1 2 3 4 5

!2!3!4!5!6!7 !1 0 1 2 3!5 !4 !3 !2 !1 0 1 2 3 4 5

0 1 2 3 4 5 6 7 8 9 10!3!4 !2 !1 0 1 2 3 4 5 6

!8 !7!10 !9 !6 !5 !4 !3 !2 !1 0!2 !1!4!5!6 !3 0 1 2 3 4

!3 !1 0 1 2 3 4 5 6!2 7!3!4!5!6 !2 !1 0 1 2 3 4

!2!3!4 !1 0 1 2 3 4 5 6

!2!3!4!5 !1 0 1 2 3 4 5

!2!3!4!5 !1 0 1 2 3 4 5

!8!9!10 !7 !6 !5 !4 !3 !2 !1 0


Match each open sentence with the graph of its solution set.

1. ⏐x " 7⏐ ( 3 c a.

2. ⏐x & 3⏐ $ 1 a b.

3. ⏐2x " 1⏐ # 5 d c.

4. ⏐5 & x⏐ $ 3 b d.

Express each statement using an inequality involving absolute value. Do not solve.

5. The height of the plant must be within 2 inches of the standard 13-inch show size.⏐h ! 13⏐ " 2

6. The majority of grades in Sean’s English class are within 4 points of 85.⏐g ! 85⏐ " 4


7. |2z & 9| % 1 {z⏐ 4 " z " 5} 8. |3 & 2r| ! 7 {r⏐ r % !2 or r # 5}

9. |3t " 6| # 9 {t⏐ !5 % t % 1} 10. |2g & 5| $ 9 {g⏐g " !2 or g $ 7}


11. 12.

⏐x ! 6⏐ % 5 ⏐x ( 4⏐ # 2

13. 14.

⏐x ( 3⏐ $ 4 ⏐x ! 2⏐ " 4

15. FITNESS Taisha uses the elliptical cross-trainer at the gym. Her general goal is to burn280 Calories per workout, but she varies by as much as 25 Calories from this amount onany given day. What is the range of the number of Calories burned for Taisha’s cross-trainer workout? {c⏐255 " c " 305}

16. TEMPERATURE A thermometer is guaranteed to give a temperature no more than1.2°F from the actual temperature. If the thermometer reads 28°F, what is the range forthe actual temperature? {t⏐26.8 " t " 29.2}

!2!3 !1 0 1 2 3 4 5 6 7!2!3!4!5!6!7!8 !1 0 1 2

!2!3!4!5!6!7!8 !1 0 1 21 2 3 4 5 6 7 8 9 10 11

!2 !1 0 1 2 3 4 5 6 7 8!5 !4 !3 !2 !1 0 1 2 3 4 5

!5 !4 !3 !2 !1 0 1 2 3 4 5!5 !4 !3 !2 !1 0 1 2 3 4 5

!2!3!4!5 !1 0 1 2 3 4 5

!8!9!10 !7 !6 !5 !4 !3 !2 !1 0

!2 !1 0 1 2 3 4 5 6 7 8

!2!3!4!5 !1 0 1 2 3 4 5

Practice (Average)

Solving Open Sentences Involving Absolute Value

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5

Answ

ers(Lesson 6-5)


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6-5

6-5


©G

lencoe/McG

raw-H

illA

17G

lencoe Algebra 1

Answers

Study Guide and InterventionGraphing Inequalities in Two Variables

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6


Less

on 6

-6

Graph Linear Inequalities The solution set of an inequality that involves twovariables is graphed by graphing a related linear equation that forms a boundary of a half-plane. The graph of the ordered pairs that make up the solution set of the inequalityfill a region of the coordinate plane on one side of the half-plane.

Graph y " !3x ! 2.

Graph y ( &3x & 2.Since y % &3x & 2 is the same as y # &3x & 2 and y ( &3x & 2,the boundary is included in the solution set and the graph should bedrawn as a solid line.Select a point in each half plane and test it. Choose (0, 0) and (&2, &2).

y % &3x & 2 y % &3x & 20 % &3(0) & 2 &2 % &3(&2) & 20 % &2 is false. &2 % 6 & 2

&2 % 4 is true.The half-plane that contains (&2, &2) contains the solution. Shade that half-plane.

Graph each inequality.

1. y # 4 2. x $ 1 3. 3x % y

4. &x ! y 5. x & y $ 1 6. 2x & 3y % 6

7. y # & x & 3 8. 4x & 3y # 6 9. 3x " 6y $ 12

x

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ExampleExample

ExercisesExercises


Solve Real-World Problems When solving real-life inequalities, the domain andrange of the inequality are often restricted to nonnegative numbers or to whole numbers.

BANKING A bank offers 4.5% annual interest on regular savingsaccounts and 6% annual interest on certificates of deposit (CD). If Marjean wantsto earn at least $300 interest per year, how much money should she deposit ineach type of account?

Let x ( the amount deposited in a regular savings account.Let y ( the amount deposited in a CD.Then 0.045x " 0.06y $ 300 is an open sentence representing this situation.

Solve for y in terms of x.

0.045x " 0.06y $ 300 Original inequality

0.06y $ &0.045x " 300 Subtract 0.045x from each side.

y $ & 0.75x " 5000 Divide each side by 0.06.

Graph y $ & 0.75x " 5000 and test the point (0, 0).Since 0 $ &0.75(0) " 5000 is false, shade the half-plane that does not contain (0, 0).One solution is (4000, 2000). This represents $4000 deposited at 4.5% and $2,000 deposited at 6%.

1. SOCIAL EVENTS Tickets for the school play cost $5 per student and $7 per adult. The school wants to earn at least $5,400 on each performance.

a. Write an inequality that represents this situation.5x ( 7y $ 5400

b. Graph the solution set.

c. If 500 adult tickets are sold, what is the minimumnumber of student tickets that must be sold? 380

2. MANUFACTURING An auto parts company can produce 525 four-cylinder engines or270 V-6 engines per day. It wants to produce up to 300,000 engines per year.

a. Write an inequality that represents this situation. 525f ( 270s " 300000

b. Are there restrictions on the domain or range? Neither f nor s is negative.

3. GEOMETRY The perimeter of a rectangular lot is less than 800 feet. Write aninequality that represents the amount of fencing that will enclose the lot.2! ( 2w " 800

Ticket Sales

Student Tickets

Adu

lt T

icke

ts

3000 600 900 x

y

900

600

300

Interest on Accounts

Regular Savings Account ($)

CD A

ccou

nt ($

)

20000 4000 6000 x

y

6000

4000

2000

Study Guide and Intervention (continued)

Graphing Inequalities in Two Variables

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6

ExampleExample

ExercisesExercises

Answ

ers(Lesson 6-6)

©G

lencoe/McG

raw-H

illA

18G

lencoe Algebra 1

Skills PracticeGraphing Inequalities in Two Variables

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6


Less

on 6

-6

Determine which ordered pairs are part of the solution set for each inequality.

1. y ! 3x, {(1, 5), (1, 0), (&1, 0), (5, 1)} {(1, 5), (!1, 0)}

2. y $ x " 3, {(2, &3), (&2, &1), (1, 6), (3, 4)} {(1, 6)}

3. y # x & 1, {(3, 1), (&2, &4), (4, &2), (&3, 3)} {(3, 1), (!2, !4), (4, !2)}

Match each inequality with its graph.

4. y & 2x # 2 b a. b.

5. y % &3x d

6. 2y & x $ 4 a

7. x " y ! 1 cc. d.


8. y # &1 9. y $ x & 5 10. y ! 3x

11. y % 2x " 4 12. y " x ! 3 13. y & x $ 1

x

y

Ox

y

Ox

y

O

x

y

O

x

y

O

x

y

O

x

y

Ox

y

O

x

y

O

x

y

O


Determine which ordered pairs are part of the solution set for each inequality.

1. 3x " y $ 6, {(4, 3), (&2, 4), (&5, &3), (3, &3)} {(4, 3), (3, !3)}

2. y $ x " 3, {(6, 3), (&3, 2), (3, &2), (4, 3)} {(!3, 2)}

3. 3x & 2y # 5, {(4, &4), (3, 5), (5, 2), (&3, 4)} {(3, 5), (!3, 4)}

Match each inequality with its graph.

4. 5y & 2x % 10 d a. b.

5. 3y ! 3x " 9 c

6. y & 2x # 3 b

7. x " 2y $ &6 ac. d.


8. 2y & x # &4 9. 2x & 2y $ 8 10. 3y ! 2x & 3

11. MOVING A moving van has an interior height of 7 feet (84 inches). You have boxes in12 inch and 15 inch heights, and want to stack them as high as possible to fit. Write aninequality that represents this situation. 12x ( 15y " 84

BUDGETING For Exercises 12 and 13, use the following information.

Satchi found a used bookstore that sells pre-owned videos and CDs. Videos cost $9 each, andCDs cost $7 each. Satchi can spend no more than $35.

12. Write an inequality that represents this situation. 9x ( 7y " 35

13. Does Satchi have enough money to buy 2 videos and 3 CDs?No, the purchases will be $39, which is greater than $35.

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

Practice (Average)

Graphing Inequalities in Two Variables

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6

Answ

ers(Lesson 6-6)


Ans

wer

s

Rea

din

g t

o L

earn

Math

emati

csG

raph

ing

Ineq

ualit

ies

in T

wo

Vari

able

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-6

6-6

©G

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7G

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Lesson 6-6

Pre-

Act

ivit

yH

ow a

re i

neq

ual

itie

s u

sed

in

bu

dge

ts?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 6-

6 at

the

top

of

page

352

in y

our

text

book

.

Wha

t do

3 a

nd 4

rep

rese

nt in

the

ter

ms

3xan

d 4y

?th

e av

erag

e am

ount

spe

nt o

n a

cafe

teri

a lu

nch

and

a fa

st-f

ood

lunc

h

Read

ing

the

Less

on

1.C

ompl

ete

the

char

t to

sho

w w

hich

typ

e of

line

is n

eede

d fo

r ea

ch s

ymbo

l.

Sym

bol

Type

of

Line

Bou

ndar

y P

art

of S

olut

ion?

#da

shed

no

!da

shed

no

%so

lidye

s

$so

lidye

s

2.If

a t

est

poin

t re

sult

s in

a f

alse

sta

tem

ent,

wha

t do

you

kno

w a

bout

the

gra

ph?

The

half-

plan

e co

ntai

ning

the

tes

t po

int

is n

ot p

art

of t

he s

olut

ion

and

is n

ot s

hade

d.

3.If

a t

est

poin

t re

sult

s in

a t

rue

stat

emen

t,w

hat

do y

ou k

now

abo

ut t

he g

raph

?

The

half-

plan

e co

ntai

ning

the

tes

t po

int

is p

art

of t

he s

olut

ion

and

is s

hade

d.

4.W

hen

can

the

orig

in n

otbe

use

d as

a t

est

poin

t?

The

orig

in c

anno

t be

use

d as

a t

est

poin

t w

hen

it is

on

the

boun

dary

.

Hel

ping

You

Rem

embe

r

5.T

he t

wo-

vari

able

ineq

ualit

ies

in t

his

less

on c

an b

e so

lved

for

yin

ter

ms

of x

to g

et a

sent

ence

in s

lope

-int

erce

pt f

orm

.It

look

s m

uch

like

a sl

ope-

inte

rcep

t eq

uati

on,b

ut it

has

an in

equa

lity

sym

bol i

nste

ad o

f an

equ

als

sign

.For

exa

mpl

e,4x

"2y

%5

can

be w

ritt

en

as y

%&

2x"

.Exp

lain

how

to

grap

h an

ineq

ualit

y on

ce it

is w

ritt

en in

slo

pe-i

nter

cept

form

.Use

the

idea

tha

t gr

eate

rca

n m

ean

abov

ean

d le

ssca

n m

ean

belo

w.

Dra

w t

he b

ound

ary

line.

If th

e in

equa

lity

sym

bol i

s #

or %

,mak

e th

ebo

unda

ry d

ashe

d.If

the

sym

bol i

s $

or "

,mak

e th

e bo

unda

ry li

ne s

olid

.If

the

sym

bol i

n th

e sl

ope-

inte

rcep

t in

equa

lity

is %

or "

,sha

de b

elow

the

boun

dary

to

indi

cate

sm

alle

r va

lues

of

y.If

the

sym

bol i

s #

or $

,sha

deab

ove

the

boun

dary

to

indi

cate

gre

ater

val

ues

of y

.

5 ' 2

©G

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8G

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Usi

ng E

quat

ions

:Ide

al W

eigh

tYo

u ca

n fi

nd y

our

idea

l wei

ght

as f

ollo

ws.

A w

oman

sho

uld

wei

gh 1

00 p

ound

s fo

r th

e fi

rst

5 fe

et o

f he

ight

and

5

addi

tion

al p

ound

s fo

r ea

ch in

ch o

ver

5 fe

et (

5 fe

et (

60 in

ches

).A

man

sho

uld

wei

gh 1

06 p

ound

s fo

r th

e fi

rst

5 fe

et o

f he

ight

and

6

addi

tion

al p

ound

s fo

r ea

ch in

ch o

ver

5 fe

et.T

hese

for

mul

as a

pply

to

peop

le w

ith

norm

al b

one

stru

ctur

es.

To d

eter

min

e yo

ur b

one

stru

ctur

e,w

rap

your

thu

mb

and

inde

x fi

nger

arou

nd t

he w

rist

of y

our

othe

r ha

nd.I

f the

thu

mb

and

finge

r ju

st t

ouch

,yo

u ha

ve n

orm

al b

one

stru

ctur

e.If

the

y ov

erla

p,yo

u ar

e sm

all-

bone

d.If

the

y do

n’t

over

lap,

you

are

larg

e-bo

ned.

Smal

l-bo

ned

peop

le s

houl

dde

crea

se t

heir

cal

cula

ted

idea

l wei

ght

by 1

0%.L

arge

-bon

ed p

eopl

esh

ould

incr

ease

the

val

ue b

y 10

%.

Cal

cula

te t

he

idea

l w

eigh

ts o

f th

ese

peo

ple

.

1.w

oman

,5 f

t 4

in.,

norm

al-b

oned

2.m

an,5

ft

11 in

.,la

rge-

bone

d12

0 lb

189.

2 lb

3.m

an,6

ft

5 in

.,sm

all-

bone

d4.

you,

if y

ou a

re a

t le

ast

5 ft

tal

l18

7.2

lbA

nsw

ers

will

var

y.

For

Exe

rcis

es 5

–9,u

se t

he

foll

owin

g in

form

atio

n.

Supp

ose

a no

rmal

-bon

ed m

an is

xin

ches

tal

l.If

he

is a

t le

ast

5 fe

etta

ll,th

en x

&60

rep

rese

nts

the

num

ber

of in

ches

thi

s m

an is

ove

r 5

feet

tal

l.Fo

r ea

ch o

f th

ese

inch

es,h

is id

eal w

eigh

t is

incr

ease

d by

6

poun

ds.T

hus,

his

prop

er w

eigh

t (y

) is

giv

en b

y th

e fo

rmul

a y

(6(

x&

60)

"10

6 or

y(

6x&

254.

If t

he m

an is

larg

e-bo

ned,

the

form

ula

beco

mes

y(

6x&

254

"0.

10(6

x&

254)

.

5.W

rite

the

for

mul

a fo

r th

e w

eigh

t of

a la

rge-

bone

d m

an in

slo

pe-

inte

rcep

t fo

rm.

y&

6.6x

!27

9.4

6.D

eriv

e th

e fo

rmul

a fo

r th

e id

eal w

eigh

t (y

) of

a n

orm

al-b

oned

fem

ale

wit

h he

ight

xin

ches

.Wri

te t

he f

orm

ula

in

slop

e-in

terc

ept

form

.y

&5x

!20

0

7.D

eriv

e th

e fo

rmul

a in

slo

pe-i

nter

cept

for

m f

or t

he id

eal w

eigh

t (y

)of

a la

rge-

bone

d fe

mal

e w

ith

heig

ht x

inch

es.

y&

5.5x

!22

0

8.D

eriv

e th

e fo

rmul

a in

slo

pe-i

nter

cept

for

m f

or t

he id

eal w

eigh

t (y

)of

a s

mal

l-bo

ned

mal

e w

ith

heig

ht x

inch

es.

y&

5.4x

!22

8.6

9.F

ind

the

heig

hts

at w

hich

nor

mal

-bon

ed m

ales

and

larg

e-bo

ned

fem

ales

wou

ld w

eigh

the

sam

e.68

in.,

or 5

ft

8 in

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-6

6-6


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