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$$. *=_${ [email protected]&i.F'mBnadw'Br- 4A orange, cherry, and mint. There are I more lime candies than orange, and 36 candies in all. How many lime candies are there if, in one draw, the probabihly ot getting lsl add. = --;;t 5x 12 Write and solve an equation lor each problem. $ )+7419 To undo multiplication, Toundosubtract;on, nJmber of sJccesses 1. A box has lour kinds ol candies: lime, 3( subtract. Rdtl $ So, there are 20 blue marbles in the bag. =11 rz Check:31+ 7 = To undo addition, then +? gomplcl€ to 6elve end €heok ceeh cquali€n' 19 '-l -] 3l = !* 3t+ $ =lt+ 3 and Let x = the number of blue marbles. 14-3111 8+3111 $ $isit!*:S bV $ A bag contains marbles of four colors: red, white, blue, and yellow. There are 3 more blue marbles than red, and 48 marbles in all. How many blue marbles are-there rl. in one draw, the probability ol getting a blue marble is To undo division' Check: Substitute 28 lor 28-.r liv . t work tt x=28 x + 3.211 6. 3t + 7 = rirst N$l:i$v result;. *3 An equation may be used to solve a problem involving probability. isolate the variable, work backward using inverse operations fhe variable is multiplied by 2 then 3 is added. the variable and then the ? ;s $ublraslr{$ $ivi$$ |}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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