PATTERNS AND ALGEBRA. zoology. In this chapter, we will learn the techniques involved in solving equations and inequalities.

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1 PATTERNS AND ALGEBRA One of the most common ways to solve comple practical problems is to use equations and inequalities. By relating the various aspects of a problem using variables, we can often fi nd the best solution, or set of solutions. Equations and inequalities are used in numerous professions and trades, from accountancy to 1 1 zoology. In this chapter, we will learn the techniques involved in solving equations and inequalities

2 In this chapter you will: Wordbank solve simple linear equations using concrete backtracking method a method of solving materials, emphasising the notion of doing the equations by undoing, or performing inverse same things to both sides of the equation (opposite) operations in reverse order solve linear equations using strategies such as balancing method a method of solving guess, check and improve, 678 and backtracking equations by using the same operations on both (reverse fl ow charts) sides of the equation solve equations using algebraic methods that equation a statement that two quantities are involve 678 up to and including three 901 steps in the equal, involving algebraic epressions and an solution process equals sign (=) check solutions to equations by substituting guess, 8901 check and improve a method of solving translate a word problem into an equation, solve equations and inequalities by making a guess and the equation and translate the solution into an improving on it answer to the problem inequality a statement that two quantities are solve equations arising from substitution into not equal, involving algebraic epressions and an formulas inequality sign (,,, or ) fi nd a range of values that satisfy an inequality inspection a method of solving equations or using guess and check inequalities by looking at a question and mentally solve simple inequalities and represent solutions fi nding the answer using our knowledge of on the number line numbers (Stage ) change the direction of an inequality solve to fi nd the answer to a problem. To solve sign when multiplying or dividing by a negative an equation or inequality, the value (or values) number of the unknown variable needs to be found. The answer is called the solution

3 1-01 Brainstarters 1 Skillsheet 1-0 Integers Skillsheet -0 Algebra using diagrams 1-0 Just give me a sign Start up 1 Find the number represented by each time: a + = 10 b = 8 c 1 = d = 9 e 9 + = 1 f 7 = 16 g 6 = 0 h = 8 i + = 0 j = 0 k 1 -- = l = Find the value of each of the following: a 8 b -8 + c (-) d -6 e - + f - 9 g - (-9) h -1 (-) i - 7 j 80 (-10) k 6 1 l -8 9 Write each of the following without grouping symbols: a (m + ) b ( ) c (k + 7) d (d 7) e 6(a ) f (b + 8) g (q 6) h 9( j + 1) i -(p + ) j -(n ) a If m = k, evaluate m if: i k = ii k = - iii k = -- b If y =, evaluate y if: i = 7 ii = -1 iii =. c If p =, q = - and r = --, evaluate: i p + q + r ii pq iii qr iv r 1 v p vi q vii 6p + 17 viii p State whether each of the following is true (T) or false (F): a 8 b c 1 d 16 1 e 1 1 f 0 g h a What is the opposite operation to adding? b What is the opposite operation to subtracting? c What is the opposite operation to multiplying? d What is the opposite operation to dividing? 1-0 Guess and check 1-0 Equations Equations An equation is a statement involving a variable (such as ), numbers and an equals (=) sign. For eample + = 11 is an equation. Usually the value of the variable that makes the equation true is unknown. When we solve the equation, we find that value. The value is called the solution to the equation. There are different approaches to solving equations. Two simple methods are: solving by inspection solving using the guess, check and improve method. 86 NEW CENTURY MATHS 8 STAGE ISBN:

4 Eample 1 Solve each of these equations by inspection: a + = 0 b m = 11 a + = 0 = Because we know that + = 0. b m = 11 m = 1 Because we know that 1 = 11. Inspection means looking at an equation and mentally finding the number that makes the equation true. Eample Solve the equation + = 7 for, using guess, check and improve. Guess Check Comment = + = 17 Smaller than 7. Try a bigger number. = = Still smaller than 7. Try a bigger number. = = 8 Much bigger than 7. Try a number between 10 and 0. = = 61 Still too big. Try a number between 10 and 1. = = 7 Correct. Answer: = 1 Eercise Solve each of these by inspection: a + = 6 b = 6 c a + 1 = 17 d m = 1 e b + 1 = 11 f c 6 = 0 g k + = h d 0 = 100 i m + = 10 j p 1 = 0 k m + 7 = l q 17 = 9 Solve each of these by inspection: m a = 1 b --- n = 1 c q = 0 d -- = 6 e 10f = 0 w f --- = 9 g p = h -- a = 8 i y = 18 j -- = 8 Use the guess, check and improve method to solve these equations: a + = 1 b p = 16 c k + 6 = 6 d 9 = 11 m e = 0 f 8 0 = g --- = h -- a + 1 = 18 i -- r + =. j -- d = 7 E 1 E ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 87

5 Find the value of in these equations by using the method of guess, check and improve : a = b ( + 1) = 6 c = 1 10 d ( ) = Just for the record Albert Einstein ( ) Albert Einstein is regarded as one of the greatest scientists of all time but, as a child, he hated his school in Munich and often did not attend classes. At the age of 1 he even failed his eams. But, after moving to Switzerland and receiving special coaching in mathematics, his school work improved and he graduated in physics in Einstein is best known for the theory of relativity and his famous equation E = mc which shows how energy (E), mass (m) and the speed of light (c) are connected. Einstein did not believe in war and campaigned strongly against the use of the atomic bomb, which his scientific theories had helped to create. Einstein was also a good musician. Find out what musical instrument he played. 1-0 Solving equations algebraically There are two algebraic methods for solving equations: the balancing method (performing the same operation on both sides) the backtracking method (undoing each operation by performing the inverse [opposite] operation). Eample Skillsheet 1-01 Solving equations by balancing Solve the equation + = 9. Method 1: The balancing method Let represent an unknown number of coins in an envelope, and let represent one coin. We can represent the equation + = 9 using balance scales. The two sides of the balance are equal. To find the value of we remove four coins from both sides. We can see is equal to. Using algebra, we have: + = 9 + = 9 Subtracting from both sides. = Check: + = NEW CENTURY MATHS 8 STAGE ISBN:

6 Method : The backtracking method First consider how we get from to +. Using a flowchart: + + In this equation, + = 9. To get back (backtrack) to, we need to undo the operation add. To do this, we subtract from 9. Using a reverse flowchart: + + = = Undoing what has been done to. 9 So we have: + = 9 = 9 Undoing by subtracting. = Skillsheet 1-0 Solving equations by backtracking Skillsheet 1-0 Solving equations using diagrams 1-0 Equations Backtracking Eample Solve p = 1. Method 1: The balancing method We can represent the equation p = 1 using balance scales. p Place the 1 coins in three equal rows. p p To find the value of p we divide both sides by. p So p is equal to. Using algebra, we have: p = 1 p = Dividing both sides by. y = Check: = 1 ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 89

7 Method : The backtracking method First consider how we get from p to p. Using a flowchart: p p = = 1 In this equation, p = 1. To get back to p, we needed to undo the operation multiply by, so we divided 1 by using a reverse flowchart. Algebraically: p = 1 1 p = Undoing by dividing by. p = Eample Solve -- k =. 7 Using the backtracking method: -- k = 7 -- k 7 = 7 Multiplying both sides by k = 1 Check: = 7! Operation + Inverse operation + Eercise 1-0 E 1 Solve each of these equations, showing the working: a w + 10 = b + 1 = 18 c m + = 19 d p + 1 = 0 e + 16 = f k + 11 = 0 g + m = 1 h 6 + y = i + d = NEW CENTURY MATHS 8 STAGE ISBN:

8 Solve each of these, showing the working: a p = 8 b m 11 = c = 1 d y 0 = 0 e k 1 = f n = 18 g 7 = d 11 h y = 1 i = m 90 Solve each of these, showing the working: a m = 18 b n = 0 c k = d c = e 9 = 81 f = -7 g 7d = 8 h 6h = - i 10a = 10 Solve each of these, showing the working: m a --- = b -- d = c -- = 8. d -- k = 1 e = 100 f -- a = -6 g 1 = -- k n h -11 = E E Solve each of the following: a m = 16 b + p = 11 c = p + d n + 7 = -1 e = 0 f n = 0 g = -1 h + y = - i -- h n = 1 j - = -6 k 6 + k = - l -- = Two-step equations 1-0 Eample 6 Solve the equation + 7 = 9. Method 1: The balancing method We can represent the equation + 7 = 9 using balance scales. Guess and check 1-0 Equations Backtracking First subtract 7 coins from both sides: Place the remaining coins into two equal rows. Then divide both sides by. We can see is equal to 1. ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 91

9 Using algebra we have: + 7 = = 9 7 Subtracting 7 from both sides. = = -- Dividing both sides by. = 1 Check: ( 1) + 7 = 9 Method : The backtracking method We use a flowchart to go from to = = In this equation, + 7 = 9. To get back to, we need to undo the operations multiply by and add 7, in the reverse order: subtract 7and divide by using a reverse flowchart. Using algebra, we have: + 7 = 9 = 9 7 Subtracting 7. = = -- Dividing by. = 1 Check: ( 1) + 7 = 9 Note that inverse operations are performed in reverse order. For eample, to undo putting on our socks, and shoes, we take off our shoes first, then take off our socks. Eample 7 M Solve =. Method 1: The balancing method M = M = Subtracting from both sides. M ---- = M ---- = Multiplying both sides by. M = 9 Check: = + = Correct. 9 NEW CENTURY MATHS 8 STAGE ISBN:

10 Method : The backtracking method M M + M + = = 9 Algebraically: M = M ---- = M ---- = M = M = 9 When solving an equation, aim to rewrite it to get the variable on its own, that is: =.! Eample 8 Solve = 9. By the balancing method: = 9 = 9 Subtracting from both sides. - = = Dividing both sides by (-). - - = - Check: (-) = + 6 = 9 Correct. ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 9

11 Eercise 1-0 E 6 E 7 E 8 1 The following is a student s solution for 7 + = 1 using backtracking. 7 + = 1 7 = 1 (Line 1) 7 = 8 (Line ) 8 = -- 7 (Line ) 1 = (Line ) In which of the following is the error made? Select A, B, C or D. A Line 1 B Line C Line D Line Solve these equations showing all steps. (Remember to check your answers.) a + 1 = 11 b + 1 = 10 c + = 1 d + = 1 e + = 1 f + = -7 g = h = 1 i = -1 j -7 + = -17 k = 9 l = Solve these equations showing all steps: a = 9 b 1 = 1 c = 1 d = 18 e 6 = 8 f 19 = -9 g 7 = 8 h 17 = 1 i = -1 j = 0 k = 17 l 8 1 = 6 n Which of the following is the solution for -- 8 = 7? Select A, B, C or D. A B 0 C 6 D 60 Solve these equations showing all steps: a -- m + = 9 b = 9 c -- k + = 6 d -- a n + = 8 e = 1 f -- h + = - 7 m g = - h -- k 7 = - i -- 6 = -10 j -- a 1 = k -- h = -7 l -- = 6 Solve these equations, showing all steps: a 6 = 9 b 9 8 = c 7 = 11 d 6 m = e p = - f q = -9 g 1 j = 6 h = 188 i = - 7 Eplain why the equations in this Eercise are called two-step equations. 9 NEW CENTURY MATHS 8 STAGE ISBN:

12 1-0 More fraction equations Eample 9 H Solve =. By balancing: H = H = Multiplying both sides by. H = 0 H = Dividing both sides by. H = 10 Check: = = Correct. By backtracking: H H H = = 10 0 or: H = H = H = 0 0 H = H = 10 Eample 10 Solve = 7. By backtracking: = Check: = = 7 Correct. + 1 = or: = = = 1 = 1 1 = 0 0 = = 10 ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 9

13 By balancing: = = 7 Multiplying both sides by. + 1 = = 1 1 Subtracting 1 from both sides. = = Dividing both sides by. = 10 Eercise 1-0 E 9 E 10 1 Show all the steps you use to solve the following equations: a = 9 b = 1 c N = 8 d = 6N e B = 18 f = -10 g = - h = - 7 i = -1 j = -9 k = -6 l = 6 8 m m = 11 n = 8 o = 1 p = N q N = 6 r k = 1 s = 0 t m = k Which of the following is the solution to = 9? Select A, B, C or D. 6 A 11 B C 66 D 16 Show all the steps you use to solve the following equations: a = b N = c = 1 d N = N e = 6 f k = - g m = - h = 11 i = j = 10 k = 7 l = 19 7 m = 6 n = - o = -7 p = -1 q = 6 r = -1 s = 6 t = NEW CENTURY MATHS 8 STAGE ISBN:

14 Mental skills 1 Maths without calculators Multiplying decimals 1 Eamine these eamples: a 8 =, so 0.8 =. 0 dp + 1 dp = 1 dp (dp = decimal places) The number of decimal places in the answer is equal to the total number of decimal places in the question. Also, the answer sounds reasonable because, by estimation: = (. ) b 6 = 0, so = 0.0 = 0. 1 dp + 1 dp = dp 1 1 By estimation, = = -- = 0. (0. 0.) c 7 = 1, so = 0.01 dp + 1 dp = dp 1 By estimation, ( ) Now find these products: a 0.7 b 1 0. c d (0.6) e f g 0.0 h i j k 9 0. l Eamine these eamples (given that 1 = ): a 1.. =. 1 dp + 1 dp = dp (Estimate: 1.. = ) b = = =. 10 =. 0 dp + dp = dp 1 (Estimate: = = 0) c = =. 100 = dp + 0 dp = dp 1 (Estimate: = = 60) Given that 9 17 = 66, find: a.9 17 b c d e f g h i j k l ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 97

15 1-0 Equations with variables on both sides For equations with variables on both sides, such as + = + 6, we can only use the balancing method, not the backtracking method.! For equations with variables on both sides, do the same thing to both sides of the equation to: get all the variables onto one side of the equation get all the numbers onto the other side of the equation. Eample 11 Solve + = + 6. Let represent the number of paperclips in a cup and let represent one paperclip. = Take two cups from each side. = Take three paperclips from each side. = Algebraically, we can write: + = = + 6 Subtracting from both sides. + = 6 + = 6 Subtracting from both sides. = Check: LHS = + = 9 + = 1 RHS = + 6 = = 1 = LHS Correct. That is, = 98 NEW CENTURY MATHS 8 STAGE ISBN:

16 Eample 1 Solve each of the following: a N = N + 6 b K + = -8K 7 a N = N + 6 N N = N + 6 N Subtracting N from both sides. N = 6 Now this is a two-step equation. N = 6 Adding to both sides. N = 9 N = -- 9 Dividing both sides by. N = b K + = -8K 7 K + 8K = -8K 7 8K Adding 8K to both sides. 1K + = -7 Now this is a two-step equation. 1K + = -7 Subtracting from both sides. 1K = K -1 = Dividing both sides by K = -1 Eercise Solve these equations showing all the steps you use: a a + 6 = a + 18 b k + = k + 8 c + 6 = + 9 d 6p + = p + 19 e 6n + = n + 17 f q + 6 = q + 1 g y + 1 = y + 1 h a + 7 = a + 1 i r + 9 = r + 1 The following is a student s solution for = = = 8 (Line 1) = 8 (Line ) = 8 (Line ) = (Line ) = -- (Line ) In which of the following were the errors made? Select A, B, C or D. A Lines 1 and B Lines and C Lines 1 and D Lines and E 11 ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 99

17 1-06 Equations! E 1 Solve these equations, showing all steps: a a = a + 6 b 6 = + 6 c a 6 = a d 8p 1 = p 10 e m + = -m + 6 f + 6 = g + = h + 18 = i y 1 = -y + 6 Solve these equations, showing all steps: a d + = d b 10k = 1 8k c p = p + 9 d 7 + = 6 + e 6k 11 = k f m + 0 = 7m g t = 1 t h 8j 17 = 10j i 6 q = 8 q 1-06 Equations with grouping symbols For equations with grouping symbols, epand the epressions and then proceed as usual Equations (etension) Skillsheet -0 Algebra using diagrams Eample 1 Solve ( + ) = 9. ( + ) = = 9 Epanding the epression to make it a two-step equation = 9 1 Subtracting 1 from both sides. = -6 Can you think of a way to solve this equation = Dividing both sides by. without epanding? = - Check: (- + ) = = 9 (Correct) Eample 1 Solve ( ) = 8. ( ) = 8 6 = 8 Epanding the brackets. 6 6 = 8 6 Subtracting 6 from both sides. - = = Dividing both sides by (-). - - = -1 Check: [ (-1)] = = 8 (Correct) 00 NEW CENTURY MATHS 8 STAGE ISBN:

18 Eercise Solve these equations, showing all steps: a ( + ) = 8 b (m + ) = 18 c (k + 1) = d (p + ) = e 10(j + ) = 0 f 7(d + ) = 6 g ( ) = 1 h ( ) = 1 i ( 1) = 10 j 8( ) = k (k ) = 8 l (q 7) = 6 Solve these equations, showing all steps: a 6 = ( 6) b = (p ) c (6 ) = 8 d (16 ) = 1 e = 8(q + 1) f = (j + ) g (6 m) = 0 h ( y) = E 1 E 1 Solve these equations, showing all steps: a ( + ) = -1 b ( 1) = -0 c -0 = ( + ) d -8 = ( + ) e ( ) = -8 f ( 16) = -1 g -10 = ( ) h -( ) = i 16(16 ) = 0 j 0 = ( ) k -( ) = 8 l (6 ) = -1 m 7 = -( + 1) n -( 1) = 7 Solve: a ( + 1) = b (p ) = p c ( ) = ( + 7) d 9(k + ) = (k ) e y + = 6(y ) f q 7 = (q ) Using technology Guess, check and improve A spreadsheet can be used to solve equations. Use the link provided to go to an activity involving spreadsheets and the guess, check and improve method. Spreadsheet 1-01 Guess, check and improve 1-0 Guess and check ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 01

19 1-07 Solving problems using equations Many mathematical problems are stated in words. We can solve these problems using equations by translating the problems into mathematical symbols. Eample 1 Five times a number is the same as si more than three times the number. What is the number? Let represent the number. = + 6 That is: = + 6 = + 6 Subtracting from both sides. = = -- 6 Dividing both sides by. = The number is. (Check that this answer solves the problem.) Eample 16 Tom is 8 years older than Susi. If the sum of their ages is, find their ages. Let the age of Susi be n years. So: Tom s age is (n + 8) years. Because Tom is 8 years older. Also: Susi s age + Tom s age = We are told the sum is. n + (n + 8) = n + 8 = Adding like terms. Solving: n = 8 Subtracting 8 from both sides. n = n 1 = Dividing both sides by. n = 7 Therefore Susi is 7 years old and Tom is (7 + 8) = 1 years old. Check: = (Correct) 0 NEW CENTURY MATHS 8 STAGE ISBN:

20 For word problems involving equations: i choose your pronumeral ii translate the words into an equation iii solve the equation iv write a sentence that answers the problem.! Eercise Solve each of the following problems by writing an equation and then solving it. You may use diagrams to help you think about the information. a Five adult tickets for a film cost $. How much does each ticket cost? (Let P represent the price of a ticket.) b Ten oranges cost $.00. How much does each orange cost? (Let R represent the cost of one orange.) c A number is doubled and the result is 110. What is the number? (Let N represent the unknown number.) d A number has subtracted from it and the result is 6. Find the number. (Let N represent the unknown number.) Choose the correct equation (A, B, C or D) for each of the following word problems. Then use the equation to solve the problem. a Jarrad has collected 179 beetles. This is 6 times as many beetles as Lisa has in her collection. How many beetles does Lisa have? (Let N represent the number of beetles that Lisa has.) A 179 N = 6 B 6N = 179 C N = D N + 6 = 179 b Kurt mied 90 millilitres of white paint with some blue paint. He mied 1.7 litres of paint altogether. How much blue paint did he use? A N + 90 = 170 B N 90 = 170 C N = 170 D 90N = c Fourteen packets of chocolate biscuits are packed in a bo. The supermarket sold 6 packets of biscuits. How many boes were sold? (Let N represent the number of boes.) A 1N = 6 B 6 N = 1 C N + 1 = 6 D N = 6 1 d When a certain number is subtracted from 100, the result is 7. What is the number? (Let N represent the number.) A N 100 = 7 B = N C N 7 = 100 D 100 N = 7 E 1 ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 0

21 E 16 Translate each of the following problems into an equation, then solve the equation to solve the problem. a The Student Representative Council is holding a school disco to raise $00. Each ticket is $ and the costs of the evening total $10. How many tickets must be sold to make the required profit? (Let n stand for the number of tickets sold.) b In eight years Kelly s age will be twice what it is now. How old is Kelly now? (Let m stand for Kelly s age now.) c Eight sheep have the same mass as three sheep and one cow. If the cow s mass is 00 kg, what is the mass of the sheep? (Let s stand for the mass of one sheep.) d I pick a number. If I take the number, multiply it by and add 1, I get the same answer as I would if I multiplied the number by and took away 11. What is the number? (Let represent the number.) e Your maths teacher says, I will not tell you my age but I will tell you this: If you add 1 to my age and multiply by 7, the answer is 9. How old is your maths teacher? (Let T stand for the teacher s age.) f The perimeter of a rectangle is 100 cm and its width is 17 cm. What is the length? (Let l represent the length of the rectangle.) g I think of a number. If I take away 1, multiply by 6 and add, the answer is 9. What is the number? (Let y represent the number.) h The area of a rhombus is calculated by multiplying the diagonals and dividing by. If a rhombus has an area of cm and one diagonal is 11 cm, what is the length of the other diagonal? (Let d represent the length of the diagonal.) i A salesperson, working on commission, gets paid $00 per week plus one-fifth of the value of her sales for that week. If she is paid $70 for one week, what is the value of her sales for that week? (Let stand for the value of her sales.) The relationship between degrees Celsius and degrees Fahrenheit is given by the equation: 9C F = If the temperature is 98 F, what is this in C? (Round your answer to the nearest degree.) Working mathematically Applying strategies Make your own equation Here are two equations that have the same solution, = 6: 1 = 9 and = 10 1 For each solution below, make up three equations that have that solution. 1 a = b = -- c = 1 d = 1. e = 0 f = - Compare your answers with those of other students. Check that each equation is correct. 0 NEW CENTURY MATHS 8 STAGE ISBN:

22 1-08 Equations and formulas A formula is a rule written in algebraic form. It shows the relationship between variables. For eample: the formula A = lb gives the area, A, of a rectangle with length l and breadth b. the formula D = ST gives the distance, D, travelled by a car at a speed S for time T. Solving mathematical problems often involves substituting values in formulas and solving equations Equation problems 1-09 Working with formulas Eample 17 A repairman charges for fiing washing machines using the formula: C = h + where C is the charge in dollars and h is the number of hours the job takes. Find: a the charge for a job that takes hours. b the number of hours worked if the charge is $0. a Substitute h = into the formula. C = + = 11 The charge is $11. b Substitute C = 0 into the formula. 0 = h + 0 = h = h h = = h The time taken is hours. Eample 18 The formula for the perimeter, P, of a rectangle of length l and width w is given by P = (l + w). Find: a l if P = 0 cm and w = 8 cm b w if P = 100 cm and l = cm. a 0 = (l + 8) b 100 = ( + w) 0 = l = 70 + w 0 16 = l = 70 + w 70 = l 0 = w l 0 w = = = l 1 = w l = 1 w = 1 ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 0

23 Eercise 1-08 E 17 1 The volume of a rectangular prism is given by V = lbh. Which of the following is the value of l when V = 0, b = and h =? Select A, B, C or D. A 17 B 68 C 8 D a The formula for the number of toothpicks (T) needed to build a pattern of N squares is T = N + 1. How many squares can be built using toothpicks? b The charge, in dollars, for hiring a hall for an event is C = 10 + N, where N stands for the number of people at the event. Find: i the charge when people are at the event ii the number of people at the event when the charge is $9. c The cost, in dollars, of classified advertisements in the local newspaper is C = 0.8N +., where N is the number of words in the advertisement. Find: i the cost of a 1-word advertisement ii the number of words in an advertisement costing $.90 d The members of the school social committee calculate that the profit, in dollars, made on the school disco is given by P = T 900, where T represents the number of tickets sold. Find: i the profit made when 19 tickets are sold ii the number of tickets sold if the profit is $. 9 e The formula F = --C + is used to convert temperatures in degrees Celsius to temperatures in degrees Fahrenheit. Find the temperature in: i degrees Fahrenheit for 7 C ii degrees Fahrenheit for - C iii degrees Celsius for 10 F iv degrees Celsius for 1 F 06 NEW CENTURY MATHS 8 STAGE ISBN:

24 f The number of hours (H) of sleep that children need depends on their age (A) in A years and is given by the formula H = Find: i the hours of sleep needed when a child is 1 years old ii how old the child is who needs 8 hours sleep. a Using the perimeter formula, P = (l + w), find: i P when l = 1, and w = ii l when P = 6, and w = 7 iii w when P = 8, and l = b The area of a trapezium is given by A = -- ( a + b)h. Find: i A when h = 8, a = 0, and b = 1 ii h when A = 90, a =, and b = 8 iii a when A = 6, h =, and b = 17 iv b when A = 10, h = 1, and a = c The volume of a pyramid is given by Ah V = , where A is the area of the base and h is the height of the pyramid. Find: i V when A =, and h = 7 ii A when V = 100, and h = 10 iii h when V =, and A = 7. d Interest (I) earned on $P invested for PRT T years at R% p.a. is given by I = Find each of the following. (Give your answers correct to two decimal places.) i Find I, when P = 10, R = and T =. ii Find P, when I = 60, R = and T =. iii Find R, when I = 8, P = 700 and T =. iv Find T, when I = 0, P = 160 and R =. E 18 ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 07

25 Working mathematically Communicating and reasoning Understanding inequalities 1 Consider the inequality. a Is = a solution? b Is = 1 a solution? c Is = - a solution? d Which of these values of are solutions: 1 6,, 0, -, 8,, -1, --,.9,.99,.01? e Can you list all of the values for that make the inequality true? Why? Consider the inequality <. a Is = 7 a solution? b Is = -7 a solution? c Is = 0 a solution? d Which of these values of are solutions? 1 1-1, 1,, 8,, --, -, 1 --,.01, 1.999, e List some more values of that satisfy <. f Can you list all of the values for that make the inequality true? Why? Consider the inequality a Which of these values of are solutions:, -, 10, -10, 6, 0, 11, 1, 1.? b List some more values of that satisfy c Can you write the solution to 9 17 as a simpler inequality, of the form? Using what you have found, describe how inequalities are different from equations. 1-0 Just give me a sign 1-09 Inequalities An inequality looks like an equation ecept that the equals sign (=) is replaced by an inequality symbol.! Inequality symbols: is greater than is greater than or equal to is less than is less than or equal to 9 = 17 is an equation. There is only one value of that makes it true is an inequality. There are many values of that make it true. Since inequalities have many values that make them true, one easy way to represent them is on a number line. 08 NEW CENTURY MATHS 8 STAGE ISBN:

26 Eample 19 Graph each of the following inequalities on a number line: a 1 b a 1 means can be any number greater than 1 or equal to The filled circle at 1 means we include 1. b means can be any number less than, but not including The open circle on means that is not included Eercise Graph each of the following inequalities on a separate number line: a 1 b c -1 d - e f 0 g - h 1 E 19 Which of the following inequalities is graphed below? Select A, B, C or D A -. B -. C -. D -. Write the inequality represented by each of these number lines: a b c d e f g h i j ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 09

27 Working mathematically Reasoning and applying strategies How can we solve inequalities? We have already solved equations by doing the same thing to both sides (keeping the equation balanced ). Will this method work with inequalities, such as + > 10 or 6 1? Eample: 1 Start with an inequality that is true. (Choose one of your own.) Add (or any number you choose) to both sides of the inequality. Is the new inequality true or false? Subtract 9 (or any number you choose) from each side of the original inequality. Is the new inequality true or false? Multiply both sides of the original inequality by (or any positive number you choose). Is the new inequality true or false? Divide both sides of the original inequality by (or any positive number you choose). Is the new inequality true or false? 6 Multiply both sides of the original inequality by (-) (or any negative number you choose). Is the new inequality true or false? 7 Divide both sides of the original inequality by (-) (or any negative number you choose). Is the new inequality true or false? 1 < < < (True) 1 9 < 18 9 < 9 (True) 1 < 18 6 < 7 (True) 1 < 18 7 < 9 (True) 1 (-) < 18 (-) - < - (False) 1 (-) < 18 (-) -. < -. (False) Now try this process using different inequalities and different numbers until you are confident you can answer the questions below correctly. 8 Which of the si operations used in Questions to 7 can be used on inequalities to give a correct (true) answer? 9 Which of the si operations used in Questions to 7 cannot be used with inequalities because they give an incorrect (false) answer? 10 For those operations that cannot be used with inequalities, find an etra rule that could be used in order for them to give correct (true) answers. 10 NEW CENTURY MATHS 8 STAGE ISBN:

28 1-10 Solving inequalities algebraically Inequalities can be solved using some of the same methods used for solving equations as long as we are not multiplying or dividing by a negative number Graphing inequalities Eample 0 Solve each of the following inequalities: a + b y 6 c 18 d -- 7 a + + Subtracting from both sides of the inequality. b y 6 y Adding to both sides of the inequality. y 10 c Dividing both sides by. 6 d Multiplying both sides by. 8 Eample 1 Solve: a b m a Subtracting 7 from both sides Dividing both sides by. -- m b m Adding to both sides. m m Multiplying both sides by. m 18 ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 11

29 Eercise 1-10 E 0 E 1 1 Solve each of the following inequalities: a y + 9 b k c p d t + 78 e n + 7 f m + 11 g t h r + 0 i w j k 7 l p m r 1 1 n m 7 1 o a 1-8 p n 1-1 q k 1-1 r 1.7 s t t n Solve each of the following inequalities: a p 0 b k 6 c 8r 10 d 1j 1 e 6n - f 9k g 17 h 8w -0 i 10d -- 7 j d 9 -- k -- t 8 l -- a m p 6 n -- r n 90 o p -- v - q -- k 1. r a s m t For each of the following inequalities: i solve it ii graph your solution on a number line. a a b 6m c u d -- n 1 e f e g y h q -9 i j p 7 - k y l k 9 Solve each of the following inequalities: a + 1 b c k + 1 > 7 d m e -- d 7 - f a 6 g n -- 0 h i m 8-9 j -- t NEW CENTURY MATHS 8 STAGE ISBN:

30 1-11 Multiplying and dividing inequalities by negative numbers When you multiply or divide both sides of an inequality by a negative number, the inequality is no longer true. For eample, is true but, if you multiply both sides by (-1), - - is false. To make the inequality true, you need to reverse the inequality sign. Stage 1-11 Inequalities etension Eample Solve these inequalities: a -m b a -m -m - - Dividing both sides by (-) and reversing the inequality sign. m -1. b Multiplying both sides by (-) and reversing the inequality sign. - 1 Eercise Solve these inequalities: a -k 1 b -m 17 c -1p -108 d -7q -6 e -j f -8a 0 1 g h -10y 6 Solve these inequalities: a ---- n 6 b ---- d - - c ---- m - d ---- k e f r g a 1-- h ---- t E Solve these inequalities: a 1 11 b 1 8 c 8 d - 11 e - f ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 1

31 Power plus 1 Find the value of in each of the following: a ( ) = b ( + ) + = 0 c ( ) = - d ( + ) = 0 e ( + 1) + ( 1) = 1 f ( + ) ( 1) = 9 g ( 1) ( ) = 6 h ( + ) = ( + 1) i ( + 1) ( + ) = 1 ( + 1) j -( + ) 8 = 1 ( ) ( + 1) ( ) 7 6( k = l = ) ( + ) 6( m = ) n = 16 o = 10 p = 6 10 Write an equation and solve it to find the unknown values in each of the following: a b c + Perimeter = 0 cm Area = 100 cm Each of the following equations has two solutions. Find them. a = b 7 = 9 c = d + 8 = 0 e ( + ) = 9 f + = 1 Diophantus was a famous mathematician of ancient Greece who lived some time between AD100 and AD00. He was the first to abbreviate his mathematical thoughts using symbols (of his own) and is known as the Greek father of algebra. Although we don t know eactly when he lived, we do know how long he lived because of this riddle written by one of his admirers: 1 Diophantus youth lasted -- of his life. He grew a beard after more. After more of his life, Diophantus married; years later he had a son. The son lived 7 1 eactly -- as long as his father, and Diophantus died just years after his son. All this adds to the years Diophantus lived. Write this riddle as an equation and solve it to find how long Diophantus lived. (Hint: Let years equal his life.) Graph each of the following inequalities on a separate number line: a 8 b 1 c - 6 Solve these inequalities: a + 1 b -6 c ( ) < ( + ) ( 0) 1 NEW CENTURY MATHS 8 STAGE ISBN:

32 Chapter 1 review Language of maths backtracking balancing check equation epand flowchart formula greater than guess, check and improve inequality inverse less than number line reverse solution solve substitution undoing unknown variable 1-1 Equations and inequalities crossword 1 What does the word solution mean? Which words in the list mean opposite? What is the difference between an equation and an inequality? Why is the variable in an equation sometimes called an unknown? What is the balancing method for solving equations and why does it have that name? 6 What is backtracking and why do you think it has that name? Topic overview Which parts of this topic did you find easy? What did you already know? Give eamples of some problems that might be solved using equations. Are there any parts of this topic that you still don t understand? Talk to your teacher about them. In what sort of careers would people use equations? Copy and add to this chapter summary. Use colour to highlight key sections. two-step equations variables on both sides grouping symbols ( ) EQUATIONS Solving by balancing Solving by backtracking (undoing) = Word problems Formulas Solve like an equation Number lines INEQUALITIES (many solutions) On a number line Filled circle (the number is included) Open circle (number is not included) ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 1

33 Topic test Chapter 1 Eercise 1-01 Eercise 1-01 Eercise 1-0 Eercise 1-0 Eercise 1-0 Eercise 1-0 Eercise 1-06 Chapter revision 1 Solve each of the following equations by inspection. a k + 6 = 1 b = 8 c a + = 17 d a 1 = 1 e = 1 f 10f = 10 g m --- = 7 h -- = 8 i 1 = + 7 Solve the following equations using the guess, check and improve method. a + 7 = b m 9 = 7 c k = d -- d = 7 Show all the steps in solving each of the following equations: a + 9 = 1 b = 7 c = -16 d -- = -1 e + 77 = 0 f 77 = 0 g - = 1 h ---- = 6 - Show all of the steps needed for solving each of the following equations: a + = 10 b n + 6 = c 7 = d n = e q 6 = -1 f -- + = 8 g m --- m 8 = -9 h = 1 i -- k 10 = 9 Solve each of these equations: a = b = c n m = 10 d = 1 7 e a = -1 f = 11 g k d = h = Show all the steps in solving each of the following equations: a + = + 6 b N = N + 6 c 1P 8 = 8P + d 10 + N = N + 6 e = f + 9 = 7 g 9 = 1 h N = 8N + 7 Show all the steps in solving the following equations: a (n + ) = 18 b ( + 1) = 1 c 10( ) = -10 d ( ) = e ( ) = 1 f -( ) = 0 g ( 1) + 6 = 1 h ( + ) 7 = 16 NEW CENTURY MATHS 8 STAGE ISBN:

34 8 Choose the correct equation for each of the following word problems. Then use the equation to solve the problem. a When a certain number is multiplied by 17, the product is 100. What is the number? (Let N represent the number.) A N + 17 = 100 B 100 N = 17 C 17N = 100 D N = 17 b Kay is paid $ for each jumper she knits. If she earns $70, how many jumpers does she knit? (Let N represent the number.) A 70 = N B N = 70 C N + = 70 D 70 = N 9 a Seven times a certain number is the same as nine more than four times the same number. Find the number. (Let stand for the number.) b A number is doubled and then has added to it. The result is 1. What is the number? (Let N represent the unknown number.) 10 A school organises a student disco. The only cost is $00 for a Music Machine. a If they set the price of tickets at $, write an equation and solve it to find how many tickets they must sell to just cover their costs. (Let N represent the number of tickets.) b To make a profit of $00, how many tickets must they sell? 11 The sum (S) of the interior angles of a polygon is given by S = 180n 60, where n is the number of sides. Find: a the sum of the interior angles when a polygon has 9 sides b the number of sides for a polygon whose angle sum is Graph each of the following on a number line: a 1 b - c 6 d - 1 Solve the following inequalities: a y b q 1 c 6p d -- e k + 7 f a -8 g m 1 h n Solve each of these inequalities: a -m 0 b -p - c ---- k d d e 7 9 f Eercise 1-07 Eercise 1-07 Eercise 1-07 Eercise 1-08 Eercise 1-09 Eercise 1-10 Eercise 1-11 ISBN: CHAPTER 1 EQUATIONS AND INEQUALITIES 17