# Equations and Inequalities

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1 Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40

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3 Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in mthemticl epressions is not enough. Sometimes the vlue of the vrible is needed. An eqution is used to find the vlue of vrible. Answer these questions, before working through the chpter. I used to think: Wht does it men to sy tht n eqution is liner? Wht does it men to mke vrible the subject of n eqution? Wht re the signs of inequlity? Answer these questions, fter working through the chpter. But now I think: Wht does it men to sy tht n eqution is liner? Wht does it men to mke vrible the subject of n eqution? How re equtions solved? Wht do I know now tht I didn t know before? 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 1

4 Bsics Wht is Liner Eqution? An eqution is mthemticl epression tht hs two sides seprted by n equls sign (=) nd t lest one vrible (or pronumerl ). If the highest power of the vrible is 1 then the eqution is clled liner eqution. For emple, these re ll equtions becuse = ppers in ech of them. + 4 = 1 b = 15 c + = 8 d = 5 e These re lso ll liner becuse the power of (the vrible) is 1 (there is no or etc). 6 7 = 1 How re Equtions Solved? To solve ANY eqution the gol is lwys to get the vrible by itself. But whtever is done to find the vrible by itself, must lso be done to the other side. Ech liner eqution hs only one solution! Finding by itself If you look t the liner eqution + 4 = 1, it s probbly esy to see = 8. Your brin hs ctully simplified the eqution to hve by itself without you even relising it. This is how: Subtrcting 4 from the left side of the eqution leves by itself. But the sme must be done on the right side. + 4 = = 1 4 = 8 For the eqution in b, to find by itself, both sides must be divided by We must do the sme to both sides = 15 = 15 = 5 For the eqution in c, it tkes two steps to get by itself Subtrct from both sides to leve the term with by itself Divide both sides by to leve the by itself + = 8 + = 8 = 6 = 6 = 100% Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

5 Bsics If the vrible is in the numertor of frction, like in d, then both sides must be multiplied by the denomintor. For the eqution in d, we need to multiply the left hnd side by to find by itself Sometimes the vrible will be in frction. Multiply nd divide both sides to get the vrible by itself. In the eqution in e is the numertor of frction # = 5# = 15 Multiply both sides by the denomintor (lwys remove the denomintor first!) Divide both sides by 6 to leve the by itself 6 # 7 = 1# = 84 6 = = 14 Sometimes the left side will hve to be simplified by collecting like terms: Solve for in the following liner eqution Simplify by collecting like terms Divide both sides by 5 to leve by itself + 5+ = = = = 5 5 = = 5 Wht hppens if the Power of the Vrible is? If the highest power of the vrible is (ie. is in the eqution), then the eqution is not liner, but is clled Qudrtic eqution. To get the vrible by itself on one side, the squre root is used to chnge to. However, the squre root of both sides must be found. Ech qudrtic eqution hs two solutions! Solve for in the following liner eqution Find the squre root of both sides + 5 = = 1 5 = 16 = ^! 4h =! 4 So = -4 or = 4 Subtrct 5 from both sides Becuse 4 = 16 nd ^ 4h = % Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC

6 Questions Bsics 1. Choose whether the following equtions re liner or qudrtic by circling the pproprite word (don t solve): + = 6 is liner /qudrtic eqution b = 1 is liner / qudrtic eqution c = 1 is liner / qudrtic eqution d 5 = is liner / qudrtic eqution. Solve these liner equtions: + 4 = 7 b + = 10 c - = 1 d - 6 = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

7 Questions Bsics. Find the vlue of the vrible in ech of these liner equtions: = 8 b 5 = 5 c 4 = 5 d = 8 e = 9 f 5 = 5 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 5

8 Questions Bsics 4. Solve for the vrible in ech liner eqution: 4 + = b - 5 = 9 c 5m + 6 = 1 d n = 4 e -k = 10 f -m + 4 = 6 (Hint: -m = -1m) 6 100% Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

9 Questions Bsics 5. Solve for the vrible in ech liner eqution: y 4 = 4 b 5 4 = 15 c p 9 = 6 d 4d 11 = 8 e 7 = 14 f 10m 4 = 5 6. Solve these qudrtic equtions: y - 9 = 0 b 4 = % Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 7

10 Knowing More Wht hppens if the Vrible is on BOTH sides of the Eqution? If the vribles pper on both sides of the eqution, one side needs to be chnged to hve no vribles in it. REMEMBER! Whtever is done to one side must be done to the other side too. Solve this liner eqution: cn be subtrcted from both sides. Now the right hnd side will hve no vribles. 8 4 = = = = 6+ 4 Add 4 to both sides so tht 5 is by itself Divide both sides by 5 so tht is by itself 5 = 10 5 = = Solve this liner eqution: 4p cn be dded to both sides. Now the right hnd side will hve no vribles. p 1= 16 4p p 1+ 4p = 16 4p+ 4p 7p 1 = 16 7p 1+ 1 = Add 1 to both sides so tht 7p is by itself Divide both sides by 7 so tht p is by itself 7p = 8 7p = p = 4 If the eqution hs brckets, they must be epnded first. Then just solve the eqution the sme wy s before. Solve the following liner eqution: Epnd both brckets ^ h= ^ h 9 = = 4 cn be subtrcted from both sides so tht the right side will hve no vribles Add 9 to both sides so tht is by itself 9+ 9 = 4+ 9 = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

11 Knowing More Equtions with Frctions Some shortcuts cn be used when the vrible is prt of frction in the eqution. Type 1: One frction in the eqution Multiply both sides of the eqution by the denomintor. Solve the liner eqution: (multiply both sides by the denomintor) = The denomintor ofthe frction is Multiply both sides by the denomintor # ` j= # + 1 = 18 = 18 1 = 6 Here is nother emple with the vrible in frction: Solve the liner eqution: (multiply both sides by the denomintor) ^5 h = Multiply both sides by the denomintor The denomintor ofthe frction is 5 # c m= # 5 = = 4 4 = 0 = Type : More thn one frction in the eqution Multiply both sides of the eqution by the LCD of the frctions. Solve the liner eqution: (Find the LCD) + 5 = The LCD of the two frctions is 1 Multiply both sides by the LCD 1 # ` + 5 j= 1 # ` j = = = = 1 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 9

12 Knowing More Word Problems Word problems cn be trnsformed into equtions which cn be solved. Trnsforming the words into n eqution is the most difficult prt. The equtions cn be solved using one of the methods lerned in this chpter. When number is hlved the nswer is 5, wht is this number? Let the number be According to the word problem Multiply both sides by the denomintor = 5 # = # 5 = 10 In the bove emple = 5 is the eqution represented by the word problem. To find n eqution from word problem, let vrible equl the missing vlue nd use the informtion in the word problem to crete n eqution. Dniel is 0 cm tller thn Philip nd the sum of their height is 0 cm. How tll is Philip? Let Philip s height be cm So Dniel s height must be ( +0) cm Philip s Height + Dniel s Height = 0 cm + ^+ 0h= = 0 = 0 0 = 00 = 00 = 150 So Phillip is 150 cm tll (nd Dniel is 170 cm tll) Luren is 5 yers older thn her sister Mri. If the sum of their ges is 5, how old is Luren? Let Luren s ge be (Mri) (Luren) So Mri s ge is - 5 ^ 5h+ = 5 5 = 5 = 5+ 5 = 0 = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

13 Questions Knowing More 1. In these liner equtions the vrible ppers on boths sides. Solve for the missing vlue: u- 10 = u b 7-18 = + 10 c ^ + h=- d 4y+ 18 = 1- y e 10^n- 6h= ^10 + nh f 6m- 4 =- 5^m+ h g 8^k-4h- 5k+ = 4 h 5^y-1h-6^y- h+ = 6 i 8t-^t- 18h=-1 j ^+ h- ^+ 4h=-10. Find Ivn's mistke when he tried to solve this eqution? ^h+ h= ^h+ 1h+ 5 h+ = h+ + 5 h+ = h+ 7 h+ h = h+ 7 h h = 5 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 11

14 Questions Knowing More. Solve these liner equtions which contin frctions: 1 8 = 4 b n + 4 = 5n c b + 4 = 5 4 d c + c = 1 e 16r + = 10 5 f m + m = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

15 Questions Knowing More 4. Solve these liner equtions which contin frctions: 7q 5 q = 6 5u+ 5 u b = c g g 1 g + = (Hint: Find LCD ofall frctions) d = 6 (Hint: Multiply both sidesbythe denomintor) e 6 8d = 1 f k 45 k = 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 1

16 Questions Knowing More 5. Three times number is 45. Wht is the number? 6. Clire, Lenne nd Lindsy re sisters. Clire is two yers older thn Lenne nd Lenne is 4 yers older thn Lindsy. The sum of ll their ges is 54. How old is ech sister? 7. Chrlie hs been collecting stmps which he keeps in two seprte books. The second book hs 7 more thn triple the stmps of the first book. If he hs 5 stmps in totl (from both books) then: How mny stmps re in the first book? b How mny stmps re in the second book? 8. Victor hs bg filled with c nd 5c coins. He hs more coins worth c thn the coins worth 5c. How mny c coins does Victor hve if ll his coins sum to 88c? % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

17 Using Our Knowledge Wht re Liner Inequlities? An inequlity is mthemticl epression with two sides seprted by one of these inequlity signs: greterthn \$ greter thn or equlto 1 less thn # less thn or equlto For emple 1 4 nd 8. If there is vrible in the inequlity nd its highest power is 1, then the inequlity is liner inequlity (unless the vrible is in denomintor). These epressions re ll liner inequlities becuse n inequlity ppers in ech of them nd the highest power of the vrible is b \$ 18 c \$ 18 d e ^ 4 h# 16 How re Inequlities Solved? Just like equtions, the im is to simplify the inequlity to get the vrible by itself on one side. Whtever is done to one side must be done to both sides. Solve these inequlities y + \$ 10 b m y + \$ 10 m y \$ 7 m 1 18 m 1 18 m 1 6 Multiplying or Dividing by Negtive Number Everyone knows tht 1 5 is true. If both sides re multiplied by -1 then This is NOT true. If both sides of n inequlity re multiplied by negtive number then the inequlity sign must be reversed. So 5. Solve these inequlities 5 # # 5 10 b 5 # # 17 5 # \$ 4 Inequlity sign is reversed fter multiplying both sides by -1 Inequlity sign is reversed fter multiplying both sides by - 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 15

18 Using Our Knowledge Other thn reversing the inequlity sign when multiplying or dividing by negtive number, inequlities re solved in the sme wy s equtions. Vrible is on both sides 5+ 5 \$ \$ \$ 1 \$ \$ 1 7 b m+ 1 5m 5 m+ 1 5m 5 m 5m 1 5m 8 5m m 1 8 m 4 Inequlity sign is reversed fter dividing both sides by - Inequlities with brckets ^q + h # 7 b ^y+ 5h ^y+ 5h q + 9 # 7 q # 7 9 q # 18 # 6 y+ 10 y+ 15 y y y y y+ 5 y y 5 y 1 5 Inequlity sign is reversed fter dividing both sides by -1 Inequlities with frctions (multiply by the LCD of ALL the frctions in the inequlity) 6k 8 \$ 8 5 6k 8 5 # \$ 8# 5 5 6k 8+ 8 \$ k \$ k \$ 8 b t t # # t t ` 4 j# 10 # 1 LCD of the frctions t 8t # 10 5t # 10 t \$ 4 Inequlity sign is reversed fter dividing both sides by % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

19 Using Our Knowledge Grphing Inequlities Solutions to inequlities cn be represented on number line. For emple, look t the inequlity >. This mens cn be ny number greter thn, but not equl to. On number line > looks like this: On number line looks like this: Cn you spot the difference in the grphs bove? In the first grph is not included in the inequlity (>), so the circle on the number line is hollow. In the second grph the inequlity includes the number ( ) so the circle is solid. Here re some more emples: < - 1 b c > - 4 d < e - < < 4 f g - < 4 h 0 < i < - or 0 j 0 or % Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 17

20 Questions Using Our Knowledge 1. Identify if the following re true or flse: 6 b c 8 d 1 5 e 4 4 f Solve these inequlities: b 4 \$ 5 c m + 7 \$ 4 d p 10 # 8 e 5q # 5 f 4h + 51 g h ^ 1h# ^1 h 4h 8 i 1 j \$ % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

21 Questions Using Our Knowledge. Grph these inequlities: b \$ c # 0 d e 5 # 1 5 f 1 # g 1 0nd \$ h 4nd # Write down the inequlity represented by ech of the following grphs: b c d e f 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 19

22 Questions Using Our Knowledge 5. Solve these more complicted liner inequlities, then grph their solution: ^+ h b 7y 4^y+ 4h 5 4 d c # 8 d b b 5 8 # e \$ f 6c c % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

23 Thinking More Simultneous Equtions Wht hppens when there re two vribles in equtions? The eqution + y = 7 hs possible solutions: The eqution + y = 18 hs possible solutions: = 1 nd y = 6 = nd y = 9 = nd y= 4 = 10 nd y = ndmnyother possibilities Common Solution The equtions hve the common solution = nd y = 4. This solution solves both equtions simultneously. =- nd y = 8 = nd y = 4 = 6 nd y = = 10 nd y =- 4 ndmnyother possibilities Since two equtions re solved t the sme time they re clled simultneous equtions. There re three methods to solve simultneous equtions. All of them work with ny question. It s up to you to choose the method you think is esiest. Method 1: Substitution In this method, one of the vribles is mde the subject of the formul nd then substituted into the other eqution. It hs three esy steps. Solve the simultneous equtions using substitution + y = y = 18 Step 1: Mke one of the vribles the subject in the eqution, for emple, using 1 y = 7 - Step : Substitute this epression for y into the to mke new eqution, nd solve this new eqution: + (7 - ) = = 18 - = - = Step : Substitute this -vlue into either of the equtions to find the y-vlue: Substitute into 1 + y = 7 y = 7 y = 4 Second eqution ^h + y = 18 y = 1 y = 4 So the solutions re = nd y = 4. As you cn see in Step, it doesn t mtter which originl eqution you substitute the vlue of the first vrible into. The sme nswer is found for both. So choose the eqution you think would be esier to use. 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 1

24 Thinking More Method : Elimintion If the coefficients of one of the pronumerls re the sme in both equtions, then the equtions cn be subtrcted from one nother to eliminte one of the vribles. If the coefficients of both vribles re different in ech eqution we cn mke them the sme by multiplying one of the equtions by n pproprite number. Solve the simultneous equtions using elimintion + y = y = 18 Step 1: Mke sure one of the vribles hs the sme coefficient in both equtions At this stge both nd y hve different coefficients in the two equtions. If 1 is multiplied by then the coefficient of will be the sme s in # + y = 14 1 sme coefficient for s Step : Subtrct equtions with the sme coefficients Subtrct from + y = 18 ^+ y = 14h y = 4 The hs been eliminted nd we only need to solve for y Step : Substitute the vlue of the solved vrible into ny eqution to find the vlue of the vrible which is still unknown Substitute y = 4 into = 7 = So the solutions re = nd y = 4. In Step, y = 4 could hve lso been substituted into choose the eqution you think would be esier to use. nd the sme vlue for would hve been found. So 100% Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

25 Thinking More Method : Grphicl Method If the stright line grphs re drwn for ech liner eqution, then the point where the lines intersect (, y) will be the solution to nd y respectively. Solve the simultneous equtions using the grphicl method + y = y = 18 Step 1: Mke y the subject of the eqution for 1 nd y = y = Step : Drw the grphs of these two equtions on the sme es y Lines intersect Step : Red the point where the lines intersect The lines intersect t the point (, y) = (, 4) So = nd y = 4 Notice tht ll three methods hve the solution = nd y = 4. All three methods work ll the time so just use the one you feel is the esiest. The net pge demonstrtes ll methods with nother emple. 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC

26 Thinking More Find the solution to these simultneous equtions: y + = 1 nd y + 4 = Using substitution, b elimintion nd c grphicl method. Substitution Step 1: Mke one of the vribles the subject of 1 = - y Step : Substitute this epression into nd solve: y+ 4^ yh= y+ 1 8y = 5y = 10 y = Step : Substitute this vlue into 1 or to solve for the remining vrible: ^ h+ = = 4 = 1 So = -1 nd y = b Elimintion Step 1: Mke sure one of the vribles hs the sme coefficient in both equtions 4# 1 = 8y+ 4 = 1 hs the sme coefficient for s Step : Subtrct equtions with the sme coefficients to eliminte vrible So 5y = 10 ` y = 8y+ 4 = 1 ^y+ 4 = h 5y = 10 Step : Substitute the vlue of the solved vrible into ny eqution to find the vlue of the vrible which is still unknown Substitute y = into 1 to obtin: So = -1 nd y = ^ h+ = ` = 4 = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

27 Thinking More c Grphicl Step 1: Mke y the subject of both equtions y = 1 + y = Step : Drw the grphs of these two equtions on the sme es Step : Red the point where the lines intersect The lines intersect t (-1, ) so = -1 nd y = y As you cn see, ll three methods produce the sme solution. Simultneous Equtions Word Problems As with single liner equtions, word problems cn be trnslted into simultneous equtions. Determine which TWO missing vlues re required nd choose vribles to represent these. Write two equtions using these vribles nd then use Substitution, Elimintion or the Grphicl Method to solve the equtions. The sum of two numbers is 1 nd their difference is 6. Find the two numbers: Let nd y represent the numbers. So + y = 1 1 nd - y = 6 These simultneous equtions cn be solved using subsitution, elimintion or the grphicl method. Jun is twelve yers older thn his sister Jmil. In two yers Jmil will be hlf Jun s ge. Find Jun nd Jmil s ge: Let = Jun s ge Let y = Jmil s ge - y = = y These re simultneous equtions which cn be solved using subsitution, elimintion or the grphicl method. 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 5

28 Questions Thinking More 1. Write down possible solutions for the vribles in these equtions: + y = 4 b + b = 6 c - 4y = 10. Solve for the vribles in these simultneous equtions using the substitution method: + y = -1 - y = -4 b p + q = 10 q - 4p = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

29 Questions Thinking More. Use the grphicl method to solve for these equtions: + y = - y = 6 b y - 4 = 4 y + = 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 7

30 Questions Thinking More 4. Solve for the vribles in these simultneous equtions using the elimintion method: - y = -15 y + = 0 b b - 4 = -1 - b = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

31 Questions Thinking More 5. Solve these simultneous equtions using ny method: 8c - 8d = 9c - 8d = 5 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 9

32 Questions Thinking More 6. Solve these simultneous equtions using ny method: 7m = 16-6n n = m % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

33 Questions Thinking More 7. Solve these simultneous equtions using ny method: 6 - b - 10 = 0 + b - 9 = 0 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 1

34 Questions Thinking More 8. Find equtions nd solve them for these word problems (using ny method): The sum of two numbers is 1. The sum of the first number, nd double the second number is 16. Wht re the numbers? b Ari is three yers older thn Eric. In three yers from now, Ari will be twice s old s Eric will be. How old re they now? c A resturunt sells two kinds of mels: pizz nd pst. A pizz costs \$14 nd pst costs \$10. In single dy the resturunt sold 79 mels. If they erned \$994 on this dy, how mny of ech mel ws sold? 100% Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

35 Answers Bsics: 1. Liner b Qudrtic c Liner d Liner 4. Knowing More: q = 4 b u = 11 c g = 0 5 d = e d = 1 f k = 45. = b = 7 5. n 15 = c = 15 d = Lenne is 18, Clire is 0 nd Lindsy is 14. = 4 b = 7 c = 0 d = 4 e = 6 f = 7. 7 b 8 4. d 5 = b = 7 c m = 5 n = 4 e k = 5 f m = 8. Victor hs 1 coins worth of 5c nd 14 coins worth of c. 5. d y = 16 b = 1 c p = 7 d = e = 6 f m = Using Our Knowledge: d True Flse b e True Flse c f Flse True 6. b so y = or y = so = 5or = 5. d 1 1 b \$ 9 c m \$ 11 p # e q # 7 f h 1 Knowing More: g 6 h # 1 i h u = 10 b = 7 c = j \$ 40 d g y = 1 e n = 10 f m = 1 k = 11 h y = 1 i t 5 =. j 4 = b. Ivn epnded the brckets incorrectly Finl nswer should be: h 1 = c. = 40 b n = 1 c b = 8 d c = 8 e r = f m = 6 d 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC

36 Answers. Using Our Knowledge: e f g Thinking More: 1. Anything where nd y dd up to 4 Eg =, y = 1 or =, y = or = 4, y = 0 or = 5, y = -1 or =.5, y = 0.5 etc b Anything where twice, plus b, mkes 6. Eg = 1, b = 4, or =, b = or =, b = 0 or = 4, b = - or = 0.5, b = 5 etc 4. h 1 4 b # 4 c Anything where times the first number, minus three times the second number, mkes 10. Eg = 6, y = 4, or = -, y = -4 or = 0, y = 0 or =, y = -1 or = 1, y = etc c \$-1 d - # 1 4. = -1. nd y = 1.4 e -4 1 # 0 f #- b p = -7 nd q = b b y 7 c d \$ 4 = nd y = - = - nd y = 4 d b \$ 4 4. b = - nd y = 6 = 5 nd b = 8 e # 1 5. c = nd d = 11 4 f c n 5 = 7. = 4, b = % Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

37 Answers Thinking More: 8. b c = 8 nd b = 4 Eric is 0; Ari is. 51 pizzs nd 8 pst mels were sold 100% Equtions & Inequlities Mthletics 100% P Lerning SERIES TOPIC 5

38 Notes 6 100% Equtions & Inequlities SERIES TOPIC Mthletics 100% P Lerning

39

40 Equtions nd Inequlities

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