Class
Date
Name
Reteach
Adding and Subtracting Rational Numbers
To add fractions that have the same denominator:
. Use the common denominator for the sum.
. Add the numerators to get the numerator of the sum.
. Write the sum in simplest form.
To subtract fractions that have the same denominator:
. Use the common denominator for the difference.
. Subtract the numerators.
Subtraction is addition of an opposite.
. Write the difference in simplest form.
1 3 1+3 4'
=6=Z
B'BB
1
-I-
t#
3 t 'l\ 3+1 42
6 \ 6) - 6 =6-=5
Complete to add the fractions.
r.+ *#=
3.-+*(-+)
2.+*(-+)
=
=
Complete to subtract the fractions.
_ 9
o-&-3=
h-_l--l-
"'15
/ 3\
\ 15/-
6 -#-?h)=
To add or subtract decimals, line up the decimal points and then
add or subtract from right to left as usual.
12.83
-
+24.17
37.00
35.78
14.55
21.23
Complete to add the decimals.
8. 44.02 + 8.07 =
7. 14.23 + 3.56 =
9. 1.39 * 13.6 =
Complete to subtract the decimals.
10. 124.33
12. 25.15
-
-
'11.33.47
13.16 =
-
0.6 =
25.06 =
Copynght C by Holt. Rinehart and Winston.
All ilghts reserved.
23
Holt Mathematics
Class
Date
Name
Practice
2. Jacob is replacing the molding
around two sides of a picture frame.
The measurements of the sides of
the frame are 4+ in. and zfr ln.
What length of molding will
Jacob need?
1. Gretchen bought a sweater for
$23.89. ln addition, she had to pay
$1"43 in sales tax. She gave the
sales clerk $30. How much change
did Gretchen receive from her total
purchase?
Use a number line to find each sum.
3. -0.5 + 0.4
-1
-0.8 -0.6 -0.4
-0.2 0 0.2 0.4 0.6 0.8
1
o.-1 *1
Add or subtract. Simplify.
u.***
6.
5
o
-' '18-7 18
-1'10 -'
10.-+
T- 53
"14
7
10
-+
rr.
2
g
_4 -2
-1_6777777
-fr
6
1
A7
8.rs+t
14
+
4
*-**fi
5
16
Evaluate each expression for the given value of the variable'
13.38.1 +
x lorx= -6.1
Copynqnt .] by Holt. Rinehart and Winslon.
All ilghis reserved.
14. 18.7 +
x forx=
2',
8.5
t5.+ * x for v-
_-
4
15
Holt Mathematics
Date
Name
Class
Practice
B
Multiplying Rational Numbers
Multiply. Write each answer in simplest form.
'
8(+)
'-*(*)
'*(-#)
re -+(zf)
-6(* l_j
12)
'-u(*)
3. -el+)
\b/
4.
^ 7 t14\
b'
12\21 /
, -+(#)
' -+(-*)
,o #(-#)
tt.+(-ls;
'-
*('8)
,,
17 {*12\
n"
341
''' 20\
'+(3)
'u -*(r*)
Multiply.
17. *2(*5.2)
18.0.53(0.04)
1e. (-7X-3.e)
20. *2(8.13)
21.0.02(*4.62)
22. 0.5(-7.8)
23. (-0.41X-8.5)
24. (-8X6.3)
25. 15(-0.05)
26. (-3.04X-1.7)
27. 10(-0.0e)
28. (-0.8)(-0.15)
29. Travis painted tor O$ hours. He received $27 an hour for his
work. How much was Travis paid for doing this painting job?
Copyright O by Holt, Rinehatl and Winston
All righis reserved.
29
Holt Mathematics
Class
Date
Name
Reteach
Solving Equations with Rational Numbers
Solving equations with rational numbers is basically the same as
solving equations with integers or whole numbers:
Use inverse operations to isolate the variable.
1
;Z=
4
I
4.7,
e
-^
*lO
= -16.4
Multiply
each side
-64
x*3.5= -17.42
* 3.5 - 3.5 oo
x=
-24.92
7
noo
Y- s =; o
a?
3 ^o+f, 'B
10
v
=
-26t =
317.2
-26t
317.2
Subtract 3.5
from each side.
-26 -
T=
-26
f,
to
each side.
-2
_{
lE- - t4
I
ooO
Divide each
side by -26.
-12.2
Tell what you would do to isolate the variable.
1.x-1.4=7.82
$olve each equation.
4. 14x = *129.5
7.x*
53.8= -1.2
Copyright @ by Holt, Rinehart and Winston,
All rights reserved.
''I*
Y*+
3.32 =
s.lv*zt
6.265.2
s.25=*k
Lm
56
5
z
=-221
23
e-tr
UV
Holt Mathematics
Beteach
and Subtracting
Reteach
To add lractions that have the same denomlnator:
s with Rational Numbers
ir-riiF]=l-T-fl--]
, Use the common denominalor for the sum.
. Add the numerators to gei the numerator ol the sum
' Write the sum in srmplest form.
1 3 1+3 4
8'BI -B-2
|.-T1I-.T'II
tt I I
i1l
To subkact fractions that have the same denominator:
. Use the common denominator for the ditference.
. Subtract the numerators.
3
Subtraction is addition 01 an opposite.
:-l--l
b
. Write the ditference in simplest form.
/ 1\
1
/-)
'',-f,_
rt, Vz = -rc, 4" ). Multrply
eacl-r side 'l
4.
by
\-.
>./
],= -ro
4
3+1
b/ =-=;=;b
\
Solving equalions wilh rallonal numbers is tlasically the same as
solvrnq equalions lvith inlegers or whole numbers:
lJse inverse opercilons lo i'olale the vaiable.
2
o
'!(
Complete to add the fraclions.
?l
aa:i jI :+
" 14 14
i?
s.-**r-*l= ji
s?
$f
s]
o.-f-r-ht= A="F
-26t =
Tell what you would do
1. x
-
1.4
=
E.jd
4. 1 4x = -1
7. x
Complete to add lhe decimals.
44.02 + e.oz
1.s
z.\
+ y = J;
--
\
-./
'--\__.:'
3.32=5
ll!1'ull
i
,{$$-Ltt3
=
5' :;y = Zt
29.5
+ 53.8 =
e
g.
--
urvrde eacfr -!
\
srde bY -26.
to isolaie the variable'
7.82
-
6.265.2 =
Sj-
1 -t.ts
---{
21.23
1? ?$
.- "'->.-)-/
,x
|
Solve each equalion,
-14.55
=
9. 1.39 + 13.6 =
- --)
3l
:-
35.78
7.14.2g + s.so
317.2
-zor =
a
16 "
I= -12.2
To add or subtract decimals, line up the decimal points and then
add or subtract lrom right to left as usual.
12.83
+24.17
37.00
v
^.- ^
tt
;;
- I / 3\
' 15 \ 15/
.{'
t
*5
= :-s
Complete to subtEcl the fractions.
2
_8
4,---=
3l
c3,.1-!\=*t'r=
- 10 \ 10/
I
=
,.-./
v- 36 7
-3
,8 -,83 -,-,
-.
e.'-3=3
g. e5 =
-t*'tk
-1.2
x?$ ...
--5$
r;:'
itB$l).$?
!+e
t
....
*J
'. 't
t
It
$?.$$
1$.$$
Complete to sublract the decimals.
10. 124.3g
12.
2s;5
-
-
11.33.47-oo=
13.16 =
25 s6
=
$t.$?
$.llS
Holl Mathematic$
-,i&
Practice
Practice B
xidinc u;d suiliractins naitiona|iiii'r,bers
1. Gretch€n bought a sweater
lor
$i23.89, ln addrtion. she had to pay
$1.43 in sales tax. She gave lhe
sales clerk $30. How nluch cllange
did Gretchen receive lrom her lotal
purchase?
2. Jacob is replacing the moldrng
around two sides of a piclure lrame
The nreasurements oi the sides of
in.
the lrame are 4+ in. and
What length
01
each answer in simplest lorm.
2.-6(#
*:i
molding will
Jacob fleed?
- 5,8\
5'rB\15/
,$l
s..1 iir. t.....................
to find each sutn.
,
,
-0.8 -O 6 -0.4
^
b'71-2r/
7114\
ri
*.!
3. -0.5 + 0.4
-1
B
*
$rt,$s
Use a number line
JMultiply. Wrile
,.'(?)
-02 A 0.2 0.4 0.6 0.8
+utr,
ls,
t. #1-#)
s
'!$
$
..-'(*)
o.-u(-#
-Tl
s$
, -t{?) '-+(-;)
J:]
$31
,,
+(-t;
,
,,
f,'$
u"^
.._..i...
'l]
1
rr.
u
+t{
t,
rs. el/31
r.. -t^t4t
1
r....!....
-"$$
..1t
u.(rurl
ie
"i
--
1...:...
'tt
Multiply.
$.
-...
17,
?..-....._
,.+-+
+.
-r.. '-a**
' +
I ..
ii
,
Add or sqbtract, Simplify.
u.
rl
-i -.
1 ......
rr.-f+S
'-iE-+ n.-+-+
1!t
r
$
l
_$
8.5
-6.1
t.t,t
:
xtolx= -+
4
f
18.
'r$.$
21.
0.02(-4.62)
25.
15(-0.05)
*r].?$
0.53(0.04)
$.$tlt
22.
0.51-7.q
*$.$
*$.$$t$
rz.$*$
Evaluate each expressiqn lor the given value ol the variable'
.13.38.1 +x torx=
15.+ +
14. 1a.7 + x iorx=
:2
+-+
,ti-. ,
... ll
u
-2(-5.2)
26.
(-3.04)(-1.7)
$.'l$$
19. (-7X-3.9)
t?.$
23. (-0.41)(-8.5)
i.$t}s
27. 10(-0.09)
* $.$
20. -2(8.13)
*t$ t$
24. (-8)(6.3)
* $$.-l
28. (-0.8)(-0.15)
$.1?
29. Travis painted lor of nours. He received $27 an hour lor his
work. How much was Travis paid for doing this painting job?
t-
\ t$il
Holt Mathemailcl
Holl Malhemalics
Copyright O by Holt, Rinehart and Winston
Class
Date
Name
Reteach
Solving Two-Step Equations
To solve an equation, it is important to first note how it is formed.
Then, work backward to undo each operation.
4z+3=15
Z
z+3
a
.+
+
9'l
-q--z
The variable is divided
by 4 and then 3 is
The variable is
multiplied by 4 and
then 3 is added. subtracted.
To solve, first
subtract 3 and
then divide by 4.
f
3 is added to the
variable and then the
result is divided by 4.
--
To solve, first add 3
and then multiply by 4.
To solve, multiply
by 4 and then
subtract 3.
Describe how each equation is formed.
Then, tell the steps needed to solve.
1.3x-5=7
and then
The variable is
and then
To solve, first
z.t*5=7
and then
The variable is
and then
To solve, first
s.+g=z
and then the result is
5is
and then
To solve, first
4. 10 *
-3x-2
and then
The variable is
and then
To solve, first
5. 10
=#
2is
To solve, first
Copyright a) by Hoil, Flinehart and Winston.
Ali riglrts reserved.
the variable and then the result is
and then
64
Holl Malhematics
Class
Date
Name
Reteach
Solving Two-Step Equations (continued)
To isolate the variable, work backward using inverse operations.
The variable is multiPlied bY 2 and
then 3 is added.
2x
f - S = tr
+3+ 3
11 To undo addition,
2x + 3 =
-3
The variable is divided bY 2 and
then 3 is subtracted.
-3
8
2x-
Z
2
subtract 3.
+-3:.
11
=11
14
multiply by 2
Check: Substitute 28lor x.
gagj11
11
To undo division,
2
c.I
-2
Check: Substitute 4lor x.
2{4)+31.
add3.
x
To undo multiPlication,
divide by 2.
To undo subtraction,
11
14_tj11
t/
11
=tr1
t/
Eemplete to selve end €heek ceeh cquatigR,
6.3f+7=19
Check:3f*7=19
3(_) 17 z 19
To undo addition,
subtract.
To undo multiplication,
3f
3f+
_+7l
Substitute for
f'
19
divide.
t
7.t-7
To undo subtraction,
L
J
add.
W
To undo division,
3
w
a-=
3
.12
z-3
2
z-3
.T
z-3
Check:[-t=$
T-7 i5
-7 a5
Substitute.
multiPlY.
w=
_
z-Q
To undo division,
o
g
Check:
t
-
A
multiply.
r 8 substitute.
--3 :"
2
To undo subtraction,
2 =u
?o
add.
Copyrioht (t by Horl Rinehan and Winston
All rights reserved.
65
Holt Mathematics
Class
Date
Name
Practice
Describe the operation performed on both sides of the equation
in steps 2 and 4.
3x=
:-/
XJA
1. 3x+2=11
3x*2-l-11 -2
4
x
4
1*1=-2+1
x
4-
9
3x_9
3 -3
|
off) = 4(-1)
x= -4
v-.)
,/t-J
Solve.
3.2x*3=9
4.t-
7.5y-2*28
8.2x-7 =7
1
^ x+2 = -11^
o'T
5. -3a+4=7
=5
s.ff=-t
10.2r*1=-1
Write and solve a two-step equation to answer the question.
11. Pearson rented a moving van for 1 day. The total rental charge
is $66.00. A daily rental costs $45.00 plus $0.25 per mile.
How many miles did he drive the van?
Copyrighl O by Holl, Rirehart and Winston
All righls reserved.
61
Holl Mathematics
Reieach
._ Practice A
ttiii sornins iwo-ste-p Equati;;s
z'
1. 3x+2=11
3x+z-z=11-2-${l,Vh4!-lf
av=
4-
9.
3l
x=
i;ivirls
bV 3
The variable
multiplied by 4
then 3 is
'
millliSlY rti
4
x= -4
3
Solve,
4.i*1=5
3.2x+3=9
,,.1
s,'
5"
t
..
*3a+4=7 O.{t-a=-a
Describe how each equalion is rormed.
Then, tell the steps needed to solve.
1,3x-S=7
..{.: *s
:1 ..
9. sf?i-1
l$
d..:
8.2x-7=7
7.5y-2=28
.....[.: t]
42+3=15
\l=t
i-t=,
The variable is divided 3 is added to the
is
variable and then the
and by 4 and then 3 is
result is divided by 4
added. subtracted.
To solve, multiply
To solve, first add 3
To solve, first
by 4 and then
and then multiply by 4.
subtract 3 and
subtract 3.
then divide by 4.
a$*1
4(-1t
4(1\l =
is
first
To sotve,
l.
msllisli*rJ
rhe variabre
10.2/+ 1 = -1
iiNd
tsv
$
and then
$
2n6
----$j!-$-t$lE$.lS.L.
thss
tllYide b$ 3
z.!+s=z
1
The variabte
is
first
$i$i$s$ t:v
To sotve,
liN\tr.\ti
stltJ*$ ts
llt* v*tiit[:ir'
Wilte and solve a two-step equalion tg answer lhe question'
I
lirsi note how it is formed
Then, work backward to undo each operation.
i-t=-z
}-r +r=*z+r
Q
-q.I:
Equations
Solving
To solve an equation, it is important to
Describe the operalion performed on bgth sides ol the equalion
in steps 2 and 4.
Pearson rented a moving van for 1 day. l he total rental charge
is $66.00. A daily rental costs $45.0i1 plus $0 25 per mile
How many miies did he dri'/e lhe van?
..of=t
s
is
To sotve,
r**llilllV
first
3
and
and then
llV
$ i$ *d*s$
thgn
then
un6
S
S
]l\$iti$i$
the,u.r|,1r
and rhen
SV S
$isiNstN
il]
S
$$blr${:l $
4.10=-3^-2
{5
- 4i
-i-
*.t5r
* i$ -. .{S
is
first
t*iiltr$licd fuf -
The vanabte
To sotve,
it$$
3
"n6
x-2
e. ru=--E2
is
To sotve.
Reteach
To
and
The variable is divided by 2 and
then 3 is subtracted.
i - z = ll
+3+ 3
To undo multiplication, i
=,0
z'i
=2'14
orvroe oy z.
2x + 3 = 11 To undo addition,
-3 -3
2x = 8
2rA
=Z
T
subtract 3.
Check: Substitute 4 for
2(4J
To undo subtraction,
add3.
11
P(DlueJ
multiPlybY2.
x
$.
"
l.*-t
= 5
3
+l +.]
*
= t:
:" .4= t.tz
tt
divide.
"l$
19
Substituteforl
+7319
*
"l$,*
aa =
n.
Check:{-z=S
$-zrs
'i[ - 7 4 s
To undo d;visron.
8. += I
t .-z--9'= g .s
z - 3 = 'J$
.s$ i$
z- +$-
Check:3t'
?3
multiply.
To undo subtraction,
add.
rt
Murtipry by 48.
*
2. A carton has lour kinds ol cookies;
lemon, mint, vanilla, and chocolate
There are 7 fewer mint cookies than
lemon, and 64 cookies in all. How
many mint cookies are there if, in
one draw, the probability ot getting
a lime candy is f,?
a lemon cookie is
f?
Lei .s * *i:illllet
$$$kis$.
$$
x
*
*r:*t**r *i
Slst*
J_=$.
3S$
't$*-l-- 1-" |l$
3S $-'
x* l$
There are \$ lime candies.
substitute.
$='$w
mulriolv.
Toundodivision,
20=x
tu;6;--
$$l.l$i$$.
*=_${
cor@nory&i.F'mBnadw'Br-
4A
orange, cherry, and mint. There are
I more lime candies than orange,
and 36 candies in all. How many
lime candies are there if, in one
draw, the probabihly ot getting
lsl
add.
= --;;t
5x
12
Write and solve an equation lor each problem.
$ )+7419
To undo multiplication,
Toundosubtract;on,
nJmber of sJccesses
1. A box has lour kinds ol candies: lime,
3(
subtract.
Rdtl $
So, there are 20 blue marbles in the bag.
=11 rz
Check:31+ 7 =
To undo addition,
then
+?
gomplcl€ to 6elve end €heok ceeh cquali€n'
19
'-l -]
3l = !*
3t+ $ =lt+ 3
and
Let x = the number of blue marbles.
14-3111
8+3111
$
$isit!*:S bV $
A bag contains marbles of four colors: red, white, blue, and yellow.
There are 3 more blue marbles than red, and 48 marbles in all.
How many blue marbles are-there rl. in one draw, the probability
ol getting a blue marble is
To undo division'
Check: Substitute 28 lor
28-.r
liv
.
t
work tt
x=28
x
+ 3.211
6. 3t + 7 =
rirst N$l:i$v
result;.
*3
An equation may be used to solve a problem involving probability.
isolate the variable, work backward using inverse operations
fhe variable is multiplied by 2
then 3 is added.
the variable and then the
? ;s $ublraslr{$
$ivi$$ |}Y
Challenqe
W
-
Equations (continued)
Solving
h$$t
S$l:}tl$tt$$
then
1Lup
and
t
Tl]$l] J *?
r$$ki{r$.
=g
8
ls*]s$
= $tiilllr*r $t it:ii:t
i,,,. ll$
{ -
lemsil
s$$ki|r$
subslitute
ru
There
are 3ii
mint cookies.
$*Ss
65
Copyright O by Holt. Rineharl and Winston.
All rights reserved.
86
Holt Mathematics
Unit 7 - 8th Grade Readiness
Na
me
Period:
Date:
1.2a+7 =LI
2.
3. 7c-L5=-20
4.5(d+L)=45
5. -3(x+ L)=-27
6. L6+2f=-10
7. g-3 =-t2
8. h+(-7) =25
9. -36 = -6j
LL. 8m = -56
1-0b - 25 = 205
10. 14 k=36
t2.
Z=-[
3
13.
'=-6
3
L4.
ln=9
3
15.38+x=I20
L6. -8Y = 72
L7. x-2.I=2.44
18. x+(-5.8) =L2.7
L9. -5=3x-2
20.5-9w=23
2L. -2d-7 =-5
22.:'-
23. 7 =L+3
24. L+e=-12
3
-)
25.
-!P+1=11
5
8=7
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