Grade 7 Mathematics Textbook PDF

MATHEMATICS Grade 7 Student Textbook Authors, Editors and Reviewers: Gebreyes Hailegeorgis (B.SC.) Basavaraju V.Mohan (Prof./Dr) Evaluators:...

MATHEMATICS Grade 7 Student Textbook Authors, Editors and Reviewers: Gebreyes Hailegeorgis (B.SC.) Basavaraju V.Mohan (Prof./Dr) Evaluators: Menberu Kebede Abdella Mohe Federal Democratic Republic of Ethiopia Ministry of Education Acknowledgements The redesign, printing and distribution of this student textbook has been funded through the General Education Quality Improvement Project (GEQIP), which aims to improve the quality of education for Grades 1–12 students in government schools throughout Ethiopia. The Federal Democratic Republic of Ethiopia received funding for GEQIP through credit/financing from the International Development Associations (IDA), the Fast Track Initiative Catalytic Fund (FTI CF) and other development partners – Finland, Italian Development Cooperation, the Netherlands and UK aid from the Department for International Development (DFID). The Ministry of Education wishes to thank the many individuals, groups and other bodies involved – directly and indirectly – in publishing the textbook and accompanying teacher guide. Every effort has been made to trace the copyright holders of the images and we apologise in advance for any unintentional omission. We would be pleased to insert the appropriate acknowledgement in any subsequent edition of this publication. © Federal Democratic Republic of Ethiopia, Ministry of Education First edition, 2003(E.C.) Developed, printed and distributed for the Federal Democratic Republic of Ethiopia, Ministry of Education by: Al Ghurair Printing and Publishing House CO. (LLC) PO Box 5613 Dubai U.A.E. In collaboration with Kuraz International Publisher P.L.C P.O. Box 100767 Addis Ababa Ethiopia ISBN 978-99944-2-054-4 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means (including electronic, mechanical, photocopying, recording or otherwise) either prior written permission of the copyright owner or a license permitting restricted copying in Ethiopia by the Federal Democratic Republic of Ethiopia, Federal Negarit Gazeta ,Proclamation No. 410/2004 Copyright and Neighbouring Rights Protection Proclamation, 10th year, No. 55, Addis Ababa, 19 July 2004. Disclaimer Every effort has been made to trace the copyright owners of material used in this document. We apologise in advance for any unintentional omissions. We would be pleased to insert the appropriate acknowledgement in any future edition.  Ta T bllee ooff C ab nttss ntteen Coon UNIT 1: RATIONAL NUMBERS 1.1 The Concept of Rational Number.......................................................................................... 2 1.2 Comparing and Ordering Rational Numbers........................................................................ 19 1.3 Operation on Rational Numbers.......................................................................................... 25 Summary............................................................................................................................ 45 Miscellaneous Exercise...................................................................................................... 46 UNIT 2: LINEAR EQUATIONS AND INEQUALITIES 2.1. Solving Linear Equations................................................................................................. 49 2.2. Solving Linear Inequalities................................................................................................ 66 Summary........................................................................................................................... 75 Miscellaneous Exercise...................................................................................................... 77 UNIT 3: RATIO, PROPORTION AND PERCENTAGE 3.1. Ratio and Proportion....................................................................................................... 80 3.2. Further on Percentage...................................................................................................... 91 3.3. Application of Percentage in Calculation........................................................................... 99 Summary.......................................................................................................................... 109 Miscellaneous Exercise.................................................................................................... 111 UNIT 4: DATA HANDLING 4.1. Collecting Data Using Tally Mark.................................................................................... 114 4.2. Construction and Interpretation of Line Graphs and Pie charts................................................................................................................ 120 4.3. The Mean, Mode, Median and Range of Data................................................................ 132 Summary........................................................................................................................ 140 Miscellaneous Exercise.................................................................................................. 141 UNIT 5: GEOMETRIC FIGURES AND MEASUREMENT 5.1. Quadrilaterals, Polygons and Circles.............................................................................. 144 5.2. Theorems of Triangles................................................................................................... 166 5.3.Measurement.................................................................................................................. 182 Summary........................................................................................................................ 215 Miscellaneous Exercise.................................................................................................. 220 UNIT RATIONAL 1 N U MB E RS Unit Outcomes: After Completing this unit, you should be able to:  define and represent rational numbers as fractons.  show the relationship among , and.  order rational numbers.  Perform operation with rational numbers. Introduction In the previous grades you had already learnt about fractions and decimals. These numbers together with integers form a bigger set of numbers known as the set of rational numbers. In this unit you will learn about rational unmbes and their basic properties. You will also learn how to perform the four fundamental operations on rational numbers. 1 1 Rational Numbers 1.1. The Concept of Rational Numbers Group work 1.1 Discuss with your friends/ partners. The Venn diagram below shows the Integers relationships between the sets of …,-3, -2, -1 Natural numbers, whole numbers and Integers. Whole numbers 1. Define the set of: 0 a. Natural numbrs Natural b. Whole numbers numbers 1,2,3,... c. Integers 2. Name six natural numbers. 3. Name six whole numbers. Figure 1.1 Relationship among 4. Name six positive integers. 5. Name six negative integers. 6. To which set(s) of numbers does each of the following numbers belong? a. 0 b. 25 c. -102 d. 7. Put < or > in stead of the box between each pairs of numbers given below to make it true. a. -76 600 b. -1200 -800 c. 0 -10,000 1.1.1. Revision on Integers In grades 5 and 6 mathematics lessons you have already learnt several facts about the sets of natural numbers ( ), whole numbers, ( ) and integers ( ). In this subsection you will revise some important facts about the set of integers. Activity 1.1 Discuss with your friends/ partners 1. For each of the following statements write “true” if the statement is correct or “false” other wise. ( Hint: = union and = intersection). a. The set {0, 1, 2, 3, …} describes the set of natural numbers. b. The set {…, -2, -1, 0, 1, 2, …} describes the set of integers. c.  = {0, 1, 2, 3, …}. d.  = {1, 2, 3, 4, …}. e. 126 is a natural number. 2  1 Rational Numbers 2. a. Is every natural number a whole number? If it is so, can you say ? b. Is every natural number an integer? If so, can you say ? c. Is every whole number an integer? If so, can you say ? Note: The set of numbers consisting of whole numbers and negative numbes is called the set of integers. The set of integers is dentoed Z = {…, -3, -2, -1, 0, 1, 2, 3 ,…}. -4 -3 -2 -1 0 1 2 3 4 Figure 1.2 Number line  Anders Celsius the Swedish astronomer who lived between 1701 and 1744 A.D. He devised away of measuring temperature which was adjusted and improved after his death. Figure 1.3 Anders Celsius Directed numbers are used in telling the temperature in degree Celsicus’ (oC). Thus if the temperature is 20 degree Celsius above zero, you can read as positive twenty degree Celsius (+20 oC) and the temperature is (-20) degree Celsius below zero you can read also negative twenty degree celsius (-20 oC). Example 1: Give the directed number describing each of the following temperatures. a. Seventy five above zero. b. Forty below zero. c. Twenty five below zero. d. Twenty one above zero. Figure 1.4 Thermometer 3 1 Rational Numbers Solution a. Positive seventy five (+75). c. Negative twenty five (-25). b. Negative forty (-40). d. Positive twenty one (+21). From Grad 5 and 6 Mathematics lesson recall that:  The set of natural numbers, denoted by is described by = {1, 2, 3, …}.  The set of whole numbers, denoted by is described by = {0,1,2,3,…}.  The set of integers, denoted by is described by = {…,-2, -1, 0, 1, 2, …}.  The sum of two natural numbers is always a natural number.  The product of two natural numbers is always a natural number.  The difference and quotient of two natural numbers are not always natural numbers.  The sum of two whole numbers is always a whole numbers.  The product of two whole numbers is always a whole number.  The difference and quotient of two whole numbers are not always whole number.  The sum of two integers is always an integer.  The product of two integers is always an integer.  The difference of two integers is always an integer.  The qoutient of two integers is not always an integer. 1.1.2. Revision of Fractions From grade 5 and 6 mathematics lessons, you have learnet about definition of fractions, operations on fractions and types of fractions. Recall the following: Note: Fractions are numbers of the form where = a ÷ b when a and b are whole numbers and b is not equal to zero ( b ≠ 0). In the fraction ,the numerator is ‘a’ and the denominator is ‘b’. 4  1 Rational Numbers Example 2: Examples of fractions in Figure 1.5 below. This foot ball pitch has This DVD has eight This chess board has 64 two halves equal Sectors equal small squares One part is one sixty – One part is one half or 7 1 The shaded part is of fourth or 64 of the of the pitch 8 chess board the DVD Figure 1.5 Examples of fractions Example 3: Let two fivths or of the parking spaces be occupied, then find the numerator and denominator. Figure 1.6 Cars Solution: The top number is called the numerator. The bottom number is called the denominator. Note: Based on the numerator and denominator, you can classify a given fraction into two types. These are: i. Proper fraction ii. Improper fraction  If the numerator of a fraction is less than its denominator, then the fraction is a proper fraction. That is the fraction is called proper fraction, if a < b. 5 1 Rational Numbers  If the numerator of a fraction is greater than or equal to its denominator, then the fraction is an improper fraction. That is the fraction is called improper fraction, if a ≥ b.  If an improper fraction is expressed as a whole number and proper fraction, then it is called Mixed number. Activity 1.2. Discuss with your teacher orally 1. Name the numerator and denominator of each fraction. b. c. 3 d. where b ≠ 0 2. Give examples of your own for proper fractions, improper fractions and mixed numbers. 3. Change these improper fractions to mixed numbers. b. c. d. 4. Change these mixed numbers to improper fractions. a. 3 b. 4 c. 3 d. 1 1.1.3. Revision on Equivalent Fractions Activity 1.3 Discuss with your teacher orally 1. Copy and complete each set of equivalent fractions.. 2. Consider the given fractions , are they equivalent fractions? An interesting property of rational numbers is that infinitely many different fractions may be used to represent the same rational numbers. Figure 1.7 below 6  1 Rational Numbers shows that a point on the number line can be represented by infinitely many different fractions. For example all represent the same point. 0 1 Finguer 1.7 Number line Note: From the above Figure 1.7 we get: i. That is are equivalent fractions. ii. that is are equivalent fractions. iii. that is are equivalent fractions. iv. that is are equivalent fractions. v. that is are equivalent fractions. Therefore, using the above discussion, we define equivalent fractions as follows: Definition 1.1. Fractions that represent the same point on the number line are called Equivalent Fractions. 7 1 Rational Numbers 1 3 9 27 81 Example 4: , , , and are equivalent fraction. You may observe 3 9 27 81 243 that: 3 3 × 1 9 9 × 1 27 27 × 1 81 81 × 1 = , = , = and =. 9 3 × 3 27 9 × 3 81 27 × 3 243 81 × 3 Further more the above example 4 can be generalized by the fundamental properties of fraction as follows: Fundamental properties of fraction: For any fraction if m is any number other than zero,. Therefore, are equivalent fractions. Note: Two fractions and , b, d ≠ 0 are equivalent if and only if a×d = b×c. Equivalenetly = if and only if a×d = b×c. Look at the following example very carefully. Example 5. Show that are equivalent fractions. Solution: let then 5 × 18 = 6 × 15 = 90. This is another method for checking the equivalenc of two fractions. 1.1.4. Rational Numbers In sub-section 1.1.1. you have revised important ideas about integers. Integers are represented on a number line as shown below in Figure 1.8. -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 Figure 1.8 Number line Consider the number , it is greater than 1 but less than 2. So it belongs to the interval between 1 and 2 as shown in Figure 1.8. is not a natural number or a whole number and also it is not an integer. It is called a rational number. 8  1 Rational Numbers Using the above discussion, we define the set of rational numbers as follows: Definition 1.2: Any number that can be written in the form where a and b are integers and b ≠ 0, is called a rational number. Note: i) The set of rational numbers is denoted by such that =. ii) Any integer ‘a’ can be written in the form of where b = 1, it follows that any integer is a rational number. - - Example 6. and 11 are rational numbers. The integer 11 is a rational number since it can be written as. 1.1.5. Representing Rational Numbers on a Number Line Example 7. Sketch a number line and mark the location of each fraction. b. c. d. e. Solution: First draw a number line and mark the location of each fraction. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 Figure 1.9 Number line a) The number is located half way between 1 and 2. b) The number is located between 0 and 1. c) The number is located half way between -2 and -3. d) The number is located half way between -1 and -2. e) The number is located at the point labeled 4,. This description of rational numbers on the number line leads to the following property. 9 1 Rational Numbers Property of rational numbers Every rational number corresponds to some unique point on a number line. In Figure 1.9 above, we will see the following Notes: i. The new numbers marked on the number line to the left of the zero point and fractions between them are called Negative rational numbers. The set of negative rational numbers are denoted by ″ -″ ii. The new numbers marked on the number line to the right of the zero point and fractions between them are called positive rational numbers. The set of positive rational numbers are denoted by ″ + ″. iii. The union of the set of positive rational numbers, set containing zero and the set of negative rational numbers is called the set of rational numbers. Example 8. Calculate 3 7 19 13 −2 8 3 2 a. + b. − c. × d. ÷ 5 10 18 9 7 11 13 5 Solution: 3 7 3× 2 7 6 7 6 + 7 13 a. + = + = + = =. 5 10 5 × 2 10 10 10 10 10 b. 19 − 13 = 19 − 13 × 2 = 19 − 26 = 19 − 26 = −7. 18 9 18 9× 2 18 18 18 18 −2 8 −2 × 8 −16 c. × = =. 7 11 7 × 11 77 3 2 3 5 15 d. ÷ = × =. 13 5 13 2 26 From this example you can easily see that the sum, difference, product and quotient of two rational numbers are also rational numbers. In grade 6 mathematics you had learnt how to convert a given terminating decimal to a fraction. ? Do you remember how you did that? 10  1 Rational Numbers Look at the following examples carefully. Example 9. Convert each decimal given below to a fraction. a. 0.25 b. 2.4 c. 1.28 Solution: 100 0.25 × 100 25 1 × 25 1 1 a. 0.25 = 0.25 × = = = =. Thus 0.25 =. 100 100 100 4 × 25 4 4 10 2.4 × 10 24 12 × 2 12 12 b. 2.4 = 2.4 × = = = =. Thus 2.4 =. 10 10 10 5× 2 5 5 100 1.28 ×100 128 32 × 4 32 32 c. 1.28 = 1.28 × = = = =. Thus 1.28 =. 100 100 100 25 × 4 25 25 As you can see from example 9 above, terminating decimals can be expressed as fractions. So we can say that terminating decimals are rational numbers. Exercise 1A 1. Compute the following ploblems in. a. 270 + 80 d. 23.9 + 28.9 g. b. 320 – 90 e. 49.72 – 58.87 h. c. 2.7 + 2.8 f. i. 2. Are the following pairs of fractions equivalent? Give the reasons to your answer. a. b. c. d. 3. Find at least four equivalent fractions for each fraction. a. b. 4. All of the following expressions represent rational numbers. Rewrite each of them in the form where a and b are integers and b ≠ 0. a. 6 c. -23 e. -0.83 g. 4 b. 1 d. -4.33 f. 8763.2 h. 2 11 1 Rational Numbers 5. Draw a number line and represent the following rational numbers on a number line. a. 5 c. e. -8 g. i. b. 3 d. f. h. Challenge Problems 6. There ae 28 people on a martial arts course. 13 are female and 15 are male. What fraction of the people are: a. Male b. Female 7. Represent the following fact by using a numeral and + and – signs. a. A loss of Birr 100. d. Five minutes Late. o b. A rise of 10 C temperature. e. 28o below zero. c. A walk of 5km forward. f. 46oC above zero. 1.1.6. Relationship Among Group work 1.2 Discuss with your friends/Group The Venn diagram below shows the relationships between the set of Natural numbers, Whole numbers, Integers and Rational numbers. 1. List three numbers that are rationals Rational numbers but not integers. 2. List three numbers that are integers 0.54 Integers 1.28 but not whole numbers. …,-3,-2, -1 3. List three numbers that are integers Whole but not natural numbers. numbers 0 4. What relations have you observed between the sets of natural numbers, Natural numbers whole numbers, integers and 1, 2, 3, … rational numbers. Figure 1.10 5. What is the intersection of the set of integers and rational numbers? 6. What is the union of the set of whole numbers and the set of rational numbers. 12  1 Rational Numbers In Figure 1.10 above, we will see the following facts listed as follows: Note: The set of whole numbers includes the natural numbers. Therefore, every natural number is also a whole number.  The set of integers includes the set of whole numbers. Therefore, every whole number is also an integer.  The set of rational numbers includes the set of integers. Therefore, every integers is also a rational number.  The relationship among the elements of natural numbers, whole numbers, integers and rational numbers is shown in Figure 1.10 above.  The set of whole numbers is a subset of the set of integers and the set of integers is the subset of the set of rational numbers or. 1.1.7. Opposite of a Rational Number Activity1.4 Discuss with your teacher orally 1. Find the opposite of each integer given below. a. 70 b. -23 c. -170 d. 0 2. Can you give the opposite of each rational number given below? c. e. 4.5 g. 3 d. -4.5 f. -0.6 Each point on the number line has another point opposite to it with respect to the point corresponding to zero. The numbers corresponding to these two points are called opposites of each other. A number and its opposite are always found at the same distance from zero as shown in Figure 1.11 below. Opposite Opposite Opposite Opposite -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 Figure 1.11 opposite numbers 13  1 Rational Numbers From the above discussion, we define opposite of rational numbers as follows: Definition 1.3: Two rational numbers whose corresponding points on the number line that are found at the same distance from the origin but on opposite sides of the origin are called opposite numbers. Note: i) As a special case, we will agree that 0 is its own opposite. ii) In general the opposite of a rational number ‘a’ is denoted by ‘-a’. Thus the opposite of –a is –(-a) = a, a ≠ 0. iii) Every rational number has an opposite. Example 10. Find the opposite of each integer given below. A. a. -10 b. -15 c. 60 d. 25 B. Solution: a. -10 is the opposite of 10 c. 60 is the opposite of -60 b. -15 is the opposite of 15 d. 25 is the opposite of -25 Note: On the number line the points corresponding to the integers in each pair above are found on opposite sides but the same distance from the origin. Example 11. Find the opposite of 8 with the help of a number line. Solution: First draw a number line and start from the origin move 8 units to the positive direction, next start from the origin and move 8 units to the left. 8 units 8 units -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 Figure 1.12 number line 14  1 Rational Numbers Hence the opposite of 8 is -8. Note: Properties of Opposites i) If a is positive, then its opposite – a is negative. ii) Number zero is the opposite of itself. iii) If a is negative, then its opposite +a is positive. Example 12. Find the opposite of each rational number. a. a. If a = 65, then –a = -65, is the opposite of a. b. b. If a = , then –a = , is the opposite of a. c. c. If a = -75, then –a = -(-75) = 75, is the opposite of a. d. d. If a = then –a = , the opposite of a. e. −12 12 12 Note: = = − are different representations of the same 23 − 23 23 12 number that is the opposite of. 23 Exercise 1B 1. Which of the following statements are true and which are false? - a. d. g. -0.67 j. -5 b. -70 e. 0 h. -3.25 k. 0.668 + c. 0 f. 0.5 i. 0.2 2. Which of the following statements are true and which are false? a. d. g. j. b. e. h. k. c. f. i. 3. Find the opposite of each rational numbers. a. 0.823 d. 3 g. 8.797 b. -26.72 e. h. 20 c. -24.278 f. i. 36 15 1 Rational Numbers 4. Determine the value of x. a. x = -(-28) c. –x = 0 b. –x = 3 d. –x = -(-70) 5. Write a number, the opposite of which is a. Positive b. Negative c. neither positive nor negative Challenge Problem 6. Use your own Venn diagrams to show all the possible relationhips among , and. 1.1.8. The Absolute Value of a Rational Number Activity 1.5 Write each of the following with out the absolute value sign: a) |8| b) |-8| c) |0| d) The absolute value of a rational number can be defined as the distance from zero on the number line. The symbol for the absolute value of a number ‘x’ is |x|. Since points coprreponding to 12 and -12 are at the same distance from the points corresponding to 0, we have, -. −12= 12 12= 12 -12 0 12 Figure 1.13 Number line From the above discussion we have the following true or valid statements. a) If x is a positive number, then = x. Example 13. a) = 5 since the absolute value of a positive rational number is the number itself. b) = 0 since the absolute value of zero is zero. c) If x is a negative number, then − x = -(-x) = x. 16  1 Rational Numbers Example 14: − 5 = 5, since the absolute value of a negative rational number is the opposite of the number. Note: = 12 is read as “the absolute value of (positive) twelve is twelve”. = 12 is read as “the absolute value of (negative) twelve is twelve”. Definition 1.4: The absolute value of a rational number ‘x’ is denoted by the symbol | x | and defined by: x if x > 0 = 0 if x = 0 -x if x < 0 Example 15. Simplifying each of the following absolute value expression. a) 7 − 2 b) 5 − 10 c) 0−π Solution: a) since 7-2 = 5 and 5> 0, we have 7 − 2 = 5 = 5. b) Since 5 – 10 = -5 and – 5 < 0, we have 5 − 10 = − 5 = - ( -5) = 5. c) Since 0-π = -π and -π < 0, we have | 0- π| = |- π | = - (-π ) = π. Equations Involving Absolute Value Geometrically the expression x = 3 means that the point with coordinate x is 3 units form 0 on the number line. Obviously the number line contains two points that are 3 units from the origin. One to the right of the origin and the other to the left. Thus x = 3 has two solution x = 3 and x = -3. Note: The solution of the equation =a For any rational number a, the equation |x| = a has i. two solutions x = a and x = -a if a > 0. ii. one solution, x = 0 if a = 0 and iii. no solution, if a < 0. 17 1 Rational Numbers Example 16. Solve the following absolute value equations. a) x =5 b) x = -70 Solution: a) x =5 If x = 5, then x = 5 or -5 Therefore, the Solution set or S.S= {-5, 5}. b) x = -70 The absolute value of a number can not be negative, hence the solution set is empty set or S.S={ }. Exercise 1C 1. Copy and complete table 1.1 below. x 8 −1 3 6 -9 9 2.6 -3.7 2 −5 2 2 7 2 x 0 5.6 0.92 11 2. Find all rational numbers whose absolute values are given below. 3 c) 2 d) 4 1 e) 3.8 f) 0 a) 8 b) 3.5 5 6 5 3. Evaluate each of the following expression. a) − 7 + 31 − 11 e) − 3 + 10 b) − 18 − − 7 + 5 f) 3 + 30 c) 9 + (−9) g) 4 + − 10 − − 3 d) 4 − 5 h) - 3 + 25 − 21 4. Evaluate each expression. a) -6x + 2 |x – 3|, when x = -3 d) y − x When y = -7 and x = 3 e) ( 9 - y ) 1 b) m -m +3, when m = × ( -1) when y = -5 2 c) x + y when x = -3 and y = -1 f) -2 x - 7 , when x = -3 18  1 Rational Numbers 5. Solve the following absolute value equations. a) x = 2 3 d) 2| x-5| +7 =14 5 e) 4x = 32 b) x = 2.35 c) 1-2 x + 2 =6 f) x - 4 =7 Challenge Problems 6. Solve the following absolute value equations. 2 a) 8 - 12x = 3 5 b) − 3 2x + 10 + 2 = 27 1.2 Comparing and Ordering Rational Numbers Group work 1.3 1. Is there any integer between n and n+1, where n ∈ ? 2. Is there any whole number between n and n + 1 where n ∈ ? 3. Arrange the following integers in ascending order: -70 , - 10, 0, 52 , 43 , 65 , 34 4. Arrange the following integers in a descending order: -5 , - 10 , 0, 16, 70, 100 5. Name all integers which lie between: a) -5 and 2 c) 0 and 3 b) -2 and 10 d) 2 and 5 6. Insert >, = or < to express the corresponding relationship between the following pairs of intigers. a) 0 ______ 500 d) ____ b) 3 ______ - 100 e) 50 ____1023 c) _____ f) -120 ____-120 19 1 Rational Numbers The concept “Less than” for rational numbers is similar to that of integers. Recall that for integers, the smaller of two numbers was to the left of the larger −6 −1 on the number line. As shown in Figure 1.14 below lies to the left of and 5 5 1 6 −6 −1 1 6 lies to the left of. Therefore; < and <. 5 5 5 5 5 5 −6 -7 -6 -5 -4 -3 -2 5 -1 −1 0 1 2 3 4 5 6 7 5 Figure 1.14 Number line All of these fractions have the same denominator, 5 it follows that -6 < - 1 and 1 < 6. Example 17. Consider the number line given in Figure 1.15 below. -8 −15 -7 −13 -6 −11 -5 −9 -4 −7 -3 −5 -2 −1 -1 0 1 1 2 5 3 7 4 9 5 11 6 13 7 15 8 17 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Figure 1.15 Number line −9 As shown in the above number line -5 is to the left of -2; is to the left of 2 −7 −5 −1 1 7 ; is to the left of ; is to the left of 2; and is to the left of 5. 2 2 2 2 2 −9 −7 −5 −1 1 7 There fore - 5 < -2, < , < , < 2 and < 5. 2 2 2 2 2 2 Note: For any two different rational numbers whose corresponding points are marked on the number line, the one located to the left is smaller. Example 18. Compare the following pair of numbers. 5 2 a) – 5 and 5 c) 2 and 3 7 7 2 5 b) -2.5 and -3.5 d) 3 and 2 3 9 20  1 Rational Numbers Solution: 5 2 a) -5 < 5 c) 2 < 3 7 7 2 5 b) - 2.5 > - 3.5 d) 3 > 2 3 9 Example 19. Draw a number line and represent the following equivalent rational numbers on a number line. a) b) Solution: a) All have the same point therefore, -3 -2 -1 0 1 2 3 Figure 1.16 Number line b) All point that corresponding to therefore -3 -2 -1 0 1 2 3 Figure 1.17 Number line From the above fact, it follows that:  Every positive rational number is greater than Zero.  Every negative rational number is less than Zero.  Every positive rational number is greater than every negative rational number.  Among two negative rational numbers, the one with the largest absolute value is smaller than the other. For example, -76 < -7 because -76 > -7. 21  1 Rational Numbers Note: In the system of rational numbers, there are many rational numbers between any two rational numbers. −7 Example 20... − 1.2, − 1.25, − 1.5, − 1.6 and are between -2 and -1. 4 Definition 1.5: (ordering similar fraction) Let and be any fractions with c > 0, then if and only if a < b. −26 −18 2 5 Example 21. a) < because − 26 < −18. c) 1 < 2 because 5 < 11. 7 7 3 3 39 45 b) < because 39 < 45. 17 17 2 3 2 3  To test whether is less than , we change and to equivalent 3 4 3 4 fractions (fractions with the same denominator). 2 2× 4 8 3 3× 3 9 8 9 8 9 = = and = = , then by comparing and , i.e, < 3 3 × 4 12 4 4 × 3 12 12 12 12 12 2 3 Therefore, < (because 2×4 = 8 < 9 = 3×3). 3 4 This example suggests the following definition: Definition 1.6: ordering dissimilar fractions p r If and are rational numbers expressed with positive denominators, q s p r then < if and only if ps < q r. q s 6 10 Example 22. a) < because 6 × 8 < 9× 10 that is 48 < 90. 9 8 −8 2 b) < because - 8× 8 < 10 × 2 that is – 64 < 20. 10 8 4 6 c) > because 4× 7 > 3 × 6 that is 28 > 18. 3 7 22  1 Rational Numbers From the above fact, it follows that: i. Relations Among Numbers: If a and b represent rational numbers, then one and only one of these relations can be true: a is equal to b or a is less than b or a is greater than b or a = b or a < b or a > b. ii. If a ≠ b, then a < b or a > b. Exercise 1D 1. Which of the following statements are true and which are false. 1 a) -3 < − 2.8 e) − 8.6 > 8.6 i) 2 −1 + =1 2 5 6 3 3 − 10 −2 f) < −7 4 b) = 8 5 j) < 7 7 3 3 g) 2 3 > 13 c) 0.2 = 1 5 5 2 −3 k) > 5 3 4 −5 5 3 1 h) < d) > 2 2 4 4 2. Insert ( >, = or < ) to express the corresponding relationship between the following pair of numbers. a) 8 _____ 6 e) − 70 ___70 i) 700 ___ − 700 30 30 − 24 −8 −1 j) 27 – 3 6 b) ____ f) - 0.5 ____ 18 6 2 15 8 g) -0.92 ___ - 0.89 k) 20 + 6 50 + 30 c) ____ 4 3 −3 h) _____ − 0.75 2 2 4 d) 3 _____ 1 3 5 3. From each pair of numbers below which number is to the right of the other? a) 25, 7 2 1 e) -5 , 15 c) 3 ,2 3 6 3 7 2 b) , 5 8 5 −3 f) 1 ,−1.2 d) , 2 8 5 4. Abebe, Almaz and Hailu played Basket balls. The results are shown in table 1.2. 23 1 Rational Numbers First Play Second play Final result Abebe Loss 5 Basket balls Won 7 Basket balls Almaz Won 6 Basket balls Loss 6 Basket balls 0 Hailu Loss 3 Basket balls Loss 2 Basket balls Complete in table 1.2. Who was the winner? Who was the looser of the competition? 5. The five integers x, y, z, n and m are represented on the number line below. 0 y z m x n Figure 1.18 Number line Using < or > fill in the blank spaces. a) z _______ x c) n _______ y b) m ______ x d) z _______ n 6. a, b, c, d, e, f are natural numbers represented on a number line as follows: f e c a b d Figure 1.19 Number line Copy and complete by writing > or − 5 ii) Difference of absolute values: 9- 5 = 4 Hence – 5 + 9 = 4 − 11   5  b) i) sign ( - ) --- due to   >   2  2 ii) Difference of absolute value: 11 5 − 6 − = = −3 2 2 2 Hence: −11 + 5 = −11 + 5 = −6 = −3 2 2 2 2 Exercise 1F 1. Write the Sum. −3 5 a) c ) − 5 +  − 3  e) + 6  13  4 8 1 b) d) 26 + (−0.09) f) 125 + (-75) 5 30  1 Rational Numbers 2. Solve for the value of x and y. x x a) 13 x + 10 = 60 d) + =2 8 6 b) 3x – 7 ( 2x – 13 ) = 3 ( -2x +9) e) -628 + 327 = y c) 8+ y = 9 f) 3x +y = 10 when y = 2 3. Evaluate each expression for the given values of x and y. x 2 −9 a) 18 + for x = −8, 18 c) x +  for x = −3,0. 25 2 3  2  −5 b) y +   for y = ,−2.6 4  8  3 1.3.1.4 Properties of Addition of Rational Numbers The following properties of addition hold true for any rational numbers. For any rational numbers a, b and c a) Commutative property for addition: a+ b = b+ a Example: 5+10 = 10 +5 1 9 9 1 + = + 7 8 8 7 b) Associative property for Addition: a+ ( b+ c) = ( a+ b) + c Example: 3 + (11 + 5) = (3 +11) +5 3 2 6 3 2 6 + +  = + + 5 5 5 5 5 5 c) Properties of 0 a+0=a Example 30 + 0 = 30 3 3 +0= 5 5 d) Property of opposites: a+ ( -a ) = 0 Example 28. Use the associative and commutative properties of addition to simplify these additions. a) 53 + 28 + 47 b) 576 + 637 + 424 + 863 31 1 Rational Numbers Solution: a) 53 + 28 + 47 = (53 + 28 ) + 47 ---- Associative property = (28 + 53) + 47 ---- Commutative property = 28+ (53 + 47) ----- Associative property = 28 + 100 = 128 --- Addition Operation b) 576 + 637 + 424 + 863 = 576 + (637 + 424) + 863 ----- Associative property = 576 + (424 + 637) + 863 ---- Commutative property = (576 + 424) + (637 +863) --- Associative property = 1000 + 1500 = 2500 ---- Addition operation Exercise 1G 1. Copy and complete the following table 1.3 below: a b c a+ b b +a (a + b) + c a+ (b+c) 6 -8 14 -2.3 -5.6 9.6 ¾ -5/7 -2.5 What do you understand from this table? 2. Use the commutative and associative properties to simplify the steps of addition of the following. Mention the property you used in each step. a) 34 + 48+ 66 d) 572+324+176+447+428+253 b) 218 + 125+782+375 e) 3.7+5.8+0.8+0.9 5 c) 59+42+41+36 f) 3.9 + 0.8+0.66+3 2 1.3.2 Subtraction of Rational Numbers Activity 1.6 1. Find the differences -. 2. Find each of the following differences. a) 1 b) c) 1 d) -2 32  1 Rational Numbers  Under this sub topic you will see that subtraction of any rational numbers can be explained as the inverse of addition. You may define subtraction as follows: Subtraction  For any numbers a, b and c , a – b = c, if and only if c + b = a.  c or a – b is the difference obtained by subtracting b from a, a- b is read “a minus b”.  The operation of subtraction is denoted by “-”. Example 29. Find the given difference: a) 5 – 12 b) − 9 −  − 13  2  4  Solution: a) Let 5 – 12 = y, then the value of “ y” has to satisfy y + 12 = 5 Therefore, y = -7 because – 7 + 12 = 5. b) Let − 9 −  − 13  = x, then the value of " x" has to satisfy x +  - 13  = − 9 2  4   4  2 −5 −5 − 13  − 9 Therefore, x = because +  =. 4 4  4  2 Based on the above information, you can formulate the following property for subtraction of rational numbers. Property: For any numbers a and b, a – b = a+ (- b) Subtract add the opposite Note: i) the difference of two rational numbers is always a rational number. ii) addition and subtraction are inverse operations of each other in rational numbers. 33 1 Rational Numbers Example 30. Find the difference by first expressing it as a sum a) – 7 – ( - 6) b) 28 – 7 Solution: a) -7 - (-6) = -7 + (-(-6)) = -7 + 6 = -1 b) 28 – 7 = 28 + ( -7 ) = 21 or 28- 7 = 21----With out using the rule. Exercise 1H 1. Find each of the following differences. a) 18 9 −  − 3  d) − 82.5 − - 82.5 g)`12 − - 7 10  4  1 1 e) 10 − 6.5 h) |15 | − 2.4 b) – 5 − 12 3 6 −3  −7  i) −  c) – 0. 5 – ( -0.2) 4  4  f) 8 - - 6 2. Copy and complete in table 1.4 below. a 2 -10 0 14 28 2.8 b -6 -8 -12 10 a +b 40 3.8 a-b 3. Evaluate each expression: a) 4( 1+x) , When x = 2 d) 2 − ( 4- t) when t = 1 b) x – ( 3 −8) +4 When x = 10 e) 12 – ( −x) -5, when x = -2 c) – x− ( 7+6) +2 When x = 9 f) -9− ( -13) – p When p = -7 4. Show the difference 5- 2 = 5 + ( -2 ) on a number line. 1.3.3. Multiplication of Rational Numbers Activity 1.7 1. Multiply 4 2 −7 −4 5  4  − 4  a) × c) × e) × ×  16 15  3  g) 5 7 8 9 3 5 2 1 − 31  − 16 2  b) × d) 4 × 5 f) × ×  7 11 7 6 32  7 62  34  1 Rational Numbers When you multiply rational numbers use the following fact. Note:  The product of a negative rational number and a positive rational number is a negative rational number. Example 31. Find the product 2× ( -3) by using a number line. Solution: 2 ×(-3) = -6 -3 -3 -7 -6 -5 -4 -3 -2 -1 0 1 2 Therefore, 2× ( -3) = -6 Figure 1.28 Number line Note: You can find the product of two rational numbers with different signs in three steps: i. Decide the sign of the product, it is “ - ”. ii. Take the product of the absolute value of the numbers. iii. Put the sign infront of the product. Example 32. Find the product: a) -3× 5 b) 3 ×  − 5  8  7  Solution: a) -3×5 b) 3 ×  − 5  i) Sign ( - ) 8  7  ii) multiply Absolute value or i) Sign ( - ) = ( 3× 5) = 15 ii) Absolutevalue or = 3 ×  − 5  8  7  Hence, -3× 5 = - 15 3 5 = ×  8 7 −15 = 56 3  − 5  − 15 Hence, ×   = 8  7  56 35 1 Rational Numbers Note: The product of two negative rational numbers is a positive rational number. 2  1 Example 33. Multiply − 4 ×  − 3  7  4 Solution: First note that the product is positive, then work out with positive numbers only. 2 1 30 13 195 13 4 ×3 = × = or 13 7 4 7 4 14 14 2  1 13 Since the product is positive -4 ×  − 3  = 13 7  4 14 Note: You can find the product of two negative rational numbers in two steps: i) Decide the sign of the product, it is “ +”. ii) take the absolute values of the numbers and multiply them. Example 34. Find the product: a) − 3 ×  − 5  b) – 4.8 × ( -7.8) 7  11  Solution: −3  − 5 a) ×  7  11  b. – 4. 8 ×( -7.8) i) sign ( + ) i) sign ( +) −3  − 5 −3 − 5 ii) -4.8 × (-7.8) × ii) = × 7  11  7 11 = - 4.8 × − 7.8 3 5 = × = 4.8 × 7.8 7 11 = 37.44 3×5 = 7 × 11 15 = 77 − 3  − 5  15 Hence × = 7  11  77 36  1 Rational Numbers The following table 1.5 summarizes the facts about product of rational numbers. The two factors The product Example Both positive Positive 3×5 = 15 Both negative Positive -3 × ( -5) = 15 Of opposite sign Negative -3× 5 = -15 One or both 0 Zero -3× 0 = 0 Exercise 1I 1. Express each sum as product. a) 0+0+0 c) 5+5+5+5 e) 8+8+8+8 b) 3+3+3+3 d) 6+6+6+6 f) 50+50+50 2. Express each of the following products as a sum. a) 5×1 b) 4×0 c) 5× 5 d) 3×3 3. 5 is added to a number. The result is multiplied by 4 and gave the product 32. What was the original number? 4. A number is added to 12. The result is multiplied by 5 and gave the product 105. What was the original number? 5. Adding 6 to a number and then multiplying the result by 7 gives 56. What is the number? 6. Squaring a number and then multiplying the result by 4 gives 1.What is the number? Challenge Problems 7. Multiply a) 1.3.3.1 Properties of Multiplication of Rational Numbers Activity 1.8 Which of the following statements are true or false? a) 4 ( 3+ 2) = ( 4×3) + ( 4×2) b) 5 37  1 Rational Numbers c) 2× ( 10× 5) = ( 2×10) ×5 d) 2 e) The following properties of multiplication hold true for any rational numbers. For any rational numbers a, b and c: 1. Commutative property for multiplication: a × b = b × a Example: 5× 70 = 70 × 5 3 2 2 3 × = × 11 9 9 11 2. Associative property for multiplication: a × ( b× c) = (a×b) ×c Example: 5 × ( 7× 12 ) = ( 5× 7) × 12 3 7 8 3 7 8 × ×  =  × × 5 5 5 5 5 5 3. Distributive property of multiplication over addition: a × (b + c) = (a × b) + (a × c) Example: 5 × ( 7 + 16) = (5 × 7) + ( 5 × 16) 2  13 1   2 13   2 1  × +  =  ×  +  ×  3  8 9 3 8  3 9 4. Properties of 0 and 1: a × 0 = 0, a × 1 = a Examples: 6 × 0 = 0 , 6 × 1 = 6 Example 35. Use the above property to find the following products. 3 6 a) × d) (0.67 × 0.8) ×0 7 11 − 3  2  − 4  5 e) ×  ×  b) 2 × 1 16 15  3  6 − 11  − 8 2  c) 4 ( 2+3) f) × ×  32  7 33  Solution: 3 6 3 × 6 18 a) × = = 7 11 7 × 11 77 5 17 17 b) 2 × 1 = × 1 = 6 6 6 c) 4 ( 2+3) = 4×2 +4×3 ---- Distributive property = 8 + 12 = 20 38  1 Rational Numbers d) ( 0. 67 × 0.8 ) × 0 = 0 ……. Property of zero − 3  2  − 4  − 3  2 × (−4)  e) ×  ×  = × 16 15  3  16  15 × 3  = − 3 ×  − 8  16  45  24 = 720 − 11  − 8 2   − 11  − 8  2 f) × × = ×  × - - - - Associative property 32  7 33   32  7  33 =  11 × 8  × 2  32 × 7  33 88 2 176 = × = 224 33 7392 Example 36. Simplify each of the following using the properties of rational numbers. a) 3x + 2 (7x + 5) b) 3x – 7 ( 2x + 10) c) 2(x + 2y) + 3y Solution: a) 3x + 2 ( 7x+5) = 3x + [ (2 × 7x) + (2 × 5)] --- Distributive property = 3x + [(2+7) x + 2 (5)] --- Associative property of multiplication = 3x + [14x+10] --- Computation = [3x + 14x] + 10 ---- Associative property of addition = [3+ 14] x + 10 --- Distributive property = 17x +10 --- Computation b) 3x – 7 ( 2x+10) = 3x + (-7)(2x + 10) = 3x + [-7(2x) + (-7)(10)] ….. Distributive property = [3x + -14x] + (-70) ….. Associative property = (3x – 14x) – 70 …. Computation = (3 – 14)x – 70 ….. Factor out x = -11x – 70 …. Computation c) 2 ( x + 2y) + 3y = 2x + 4y + 3y … Distributive property = 2x + (4y + 3y) … Associative property of addition = 2x + (4 +3) y … Distributive property = 2x + 7y … Computation 39 1 Rational Numbers The following properties can be helpful in simplifying products with three or more factors 1. The product of an even number of negative factors is positive. 2. The product of an odd number of negative factors is negative. 3. A product of rational number with at least one factor 0 is zero. 4. If you multiply a rational number ‘a’ by -1, then you get the opposite of a, (i.e – a). Therefore, you can write -1 × a = -a. 5. When you multiply a number by a variable, you can omit the multiplication sign and keep the number in front of the variable. Example 37. Find the Products below: 5 −5 a) - 4 ×( -7) = 28 d) -1× = 2 2 b) -7 × 3 = -21 e) a × 30 = 30 a ( but not a 30) 5 5 c) -7× 0 = 0 f) ×b = b 2 2 Exercise 1J 1. Simplify each of the following using the properties of rational numbers.  − 4 x2  a) -5 + 2 ( 3x + 40) e) 2x2 +   + 3y 2  2  − 4y  x b) 5 ( x + y) +3 (2x +y) f) -2x +  +  2  2  − 2x 2   + 2x + 10 2 c) 6 (x + 2y) + 2 ( 3x +y) g) 0 +   2  d) 4 (3+2 (x + 5) 2. State the properties, in order that are used in these simplifications. a) 7x + 5x = ( 7+5)x = 12 x b) 20x + 6x = (20 + 6)x = 26 x c) 5a + 3b + 2a = 5a + ( 3b +2a) = 5a + (2a+ 3b) = (5a + 2a) + 3b = 7a + 3b 40  1 Rational Numbers d) 4 [ 3+2 (x+5)] = 12 + 8 (x+5) = 12 + (8x + 40) = 12 + (40 + 8 x) = (12 +40) + 8x = 52 + 8x e) x (x +3) +2 ( x+5) = x2 + x×3 + (2x + 10) = (x2 + 3x) + (2x + 10) = [x2+ (3x + 2x)] + 10 = [x2 + (3+2) x] + 10 = x2 + 5x + 10 f) – 5 + 2 ( 3x + 4) = -5 + [2 (3x) + 2 (4)] _________ = -5 + [6x + 8] ______________ = 6x + [-5+8] _______________ = 6x + 3 ___________________ 1.3.4 Division of Rational Numbers Activity 1.9 Divide and write each answer in lowest terms. 3 5 3 1 3 5 c) ÷ e) ÷ a) ÷ 8 3 8 6 7 8 4 4 2 11 1 d) 9÷ f) ÷ b) ÷ 5 9 8 5 5 Multiplication and division are inverse operations of each other in the set of non- zero rational numbers. To divide 12 by 3 is to find a number, which gives the product 12 when multiplied by 3. This number is 4. Thus 12÷3 = 4 because 4×3 = 12.  The symbol ″÷″ denotes the operation of division and it is read as divided by so, 12÷ 3 is read as 12 divided by 3.  In the division 12÷ 3 = 4, 12 is called the dividend, 3 is called the divisor and 4 which is the result of the division is called the quotient. You may define division as follows. 41 1 Rational Numbers Division For any numbers a, b and c where b ≠ 0, a ÷ b = c, if and only if c× b = a.  c or a ÷ b is the quotient obtained by dividing a by b.  a ÷ b is read as a is divided by b.  In the division a ÷ b = c the number ‘a’ is called the dividend.  ‘b’ is called the divisor and ‘c’ is called quotient.  The quotient a ÷ b is also denoted by or a/b. Based on the above information, you can easily find out rules for the division of rational numbers analogous to those of multiplication. Rule: The rules for division of two rational numbers: 1. To determine the sign of the quotient: a) If the sign of the dividend and the divisor are the same, the sign of the quotient is ( + ). b) If the sign of the dividend and the divisor are different, the sign of the quotient is ( - ). 2. Determination of the values of the quotient: Divide the absolute value of the dividend by the divisor. Example 38. Look at Table 1.6 below: Problem Divisor and Absolute value Quotient dividend a 28÷4 Both positive ( + ) 28 ÷4 = 7 7 b -2.8 ÷ -0.2 Both negative ( - ) 2.8 ÷0.2 = 14 14 c -10÷2 One negative and 10÷2 = 5 -5 one positive d 4.8 ÷ (-4) One positive and one 4.8 ÷4 = 1.2 -1.2 negative e 0÷10 Dividend 0 0÷10= 0 0 f 0÷(-10) Dividend 0 0÷10 = 0 0 42  1 Rational Numbers Note: Division by zero is not defined under the set of rational numbers. CAUTION is okay but is not okay (undefined)! Figure 1.29 For any rational numbers a and b, = a ÷ b (a, b ∈ , b ≠ 0) 3 4 Remember that, is the reciprocal of. 4 3 1 Remember that, is the reciprocal of 5. 5 −7 −11 Remember that, is the reciprocal of 11 7. Dividing a given rational number (the dividend) by another non- zero rational number (the divisor) means multiplying the dividend by the reciprocal of the divisor. Note: For any two rational numbers, and where b, c and d ≠ 0; = = where is the reciprocal of 9 5 11 -7 Example 39. Compute: a) ÷ b) − 6 ÷ c) ÷ −2 10 7 13 9 Solution: a) 9 ÷ 5 = 9 × 7 = 63 c) − 7 ÷ (−2) = − 7 × − 1 = 7 10 7 10 5 50 9 9 2 18 b) − 6 ÷ = −6 × 13 = − 78 11 13 11 11 43 1 Rational Numbers Exercise 1k 1. Divide a) 3 ÷  − 6  c) − 4 ÷  − 4  e)  − 16  −8÷  g) 0÷ 3 5  15  11  11   21  5 −4 3 −5 11 −14 −27 b) ÷ d) ÷ f) ÷ (− 7 ) h) ÷0 7 14 16 8 15 3 2. Compute a) 4. 6 ÷ (-6) d) 90 × ( -8) + 100 ÷( -50) b) 12 × 4 ÷6× ( -8) e) – 0.2× ( -0.3) + (0. 8×(-0.7) c) 9 × (-8) ÷72(-2) 3. Reduce to the lowest term if possible: a) −54 c) − 48 e) 245 ÷ 10 72 − 120 f) 79.2 ÷ 10 50 − 2a 2 b 2 d) (b ≠ 0) b) 80 − b 4. Solve the following equations. 1 0 a) 2y × ( - 28) = 48 c) y = −8 e) – 2x = 2 10 2 −2 b) 3y ÷ ( -2) = 24 d) 5x + 10 = - 30 f) x= 3 27 5. Simplify  − 18 9   − 3   2 2 1 a)  ÷ ×  c) 1 3 × 4 3  ÷ 6 9  5 35   7     − 12  − 5   − 9   1 3  5 b)  ×  ÷   d) 5 ÷ 6  × 7   25  7   14   16 4  9 Challenge Problems 6. Find the quotient. Think of a simpler problem and use the pattern to solve the problem:. 7. Does (56 ÷ 8) ÷ 2 equal 56 ÷ (8 ÷ 2)? Is division associative. 8. Find the quotient of (8x2 + 20xy) ÷ 4x. 44  1 Rational Numbers Summary For Unit 1 1. The sum and product of two whole numbers are always a whole number. 2. The sum, difference and product of two or more integers are always an integer. 3. The set of rational numbers is defined as: = 4. = + {0} - 5. = +  { 0}  - 6. The absolute value of a rational number X is denoted by the symbol |x| and defined as: /X/ = - 7. Subtraction of any rational number can be treated as the inverse operation of addition. 8. The sum of two opposite rational numbers is 0. 9. Rules of signs for Addition: Let a and b be rational numbers: a) Negative plus negative equals negative:- a+(-b) = - ( a +b). b) Positive plus negative equals positive if a > b : a + (-b) = a- b is positive. c) Positive plus negative equals negative if a < b : a+ (-b) = - ( b- a) is negative. 10. Rules of signs for Multiplication Let a and b be rational numbers: a) Positive times negative equals negative: a× (-b) = - (a×b). b) Negative times positive equals negative :– a × b = - (a× b). c) Negative times negative equals positive -a× (-b) = a×b. 11. Rules of signs for Division Let a and b be rational numbers: a) Positive divided by negative equals negative: a ÷ -b = - (a÷b). b) Negative divided by positive equals negative: -a ÷ b = - ( a ÷ b). c) Negative divided by negative equals positive: – a ÷ (-b) = a ÷ b. 45  1 Rational Numbers Miscellaneous Exercise 1 1. Decide whether each of the following is true or false. a) 0 > - 100 c) – 10, 000 > 10, 000 e) - 2.9 > 2.6 1 0 f) 98.6 = − 98.6 b) 3 < d) 2.6 < 2.6 2 10 2. Evaluate: a) -4 ( 5 – ( 36 ÷4)) c) 3 1 +  − 7  5  8  b) 10 – (5 – (4 – (8 - 2)) −1  −5 d) +  4  9  3. Simplify by combining Like terms. a) 3k – 2k d) 2x2 + 5x – 4x2 + x – x2 b) 5x2 – 10x – 8x2 + x e) ( 3x+y) + x c) – ( m +n) +2 ( m- 3n) f) 2 ( 5 +x ) + 4 ( 5+x) 4. Simplify each of the following expression. a) 3x + 2 ( 7x +5) d) -7 ( -2 ( 3x+1) + 4 ) +9 b) -5 + 2 (3x+4) e) 3x2 +2 ( 5x +3x2) c) -2 ( -3) + ( 4 ( -3) + 5 (2)) f) 3 (y + 2) − 1 (y − 2) 8 4 5. Find the simplified form of  1 + 1  × 1  ÷  2 + 3  ÷ 6 .  2 3  4   5 4  12  6. Find the simplified form of  − 24 × 15  ÷  6 × − 12 .  5 16   4 8  7. Simplify the following expression. 1  1 1 ÷ −   3 1 5  7 7 a) 8 4 3 c) 2 + 4 × 1  ÷ 7 ÷ 10  4 1 1 3  4 8 11  8 20  + ÷ 3 2  2 2  15  4 23 5  5 2 1 + − − 37  15 30 12  11  3 5  d) b) 1  14 1  ÷ 7  5 7  3 3  + 1+ 2  6  46  1 Rational Numbers 8. Solve each of the following absolute value equations. a) 2y - 4 = 12 c) 3 4x - 1 − 5 = 10 b) 3x + 2 = 7 d) 2x + 15 = −10 x − 3y 9. If x = - 6 and y = 10, then find. xy 10. Evaluate the following expression. x a) + 11 when x = 10 2 1 b) 7x – 4y when x = 10 and y = 2 c) 3x2 + 6y2 when x = 0 and y = 2 n 7 11. Solve for n: =. 18 9 12. In the expression 8÷ 2 = 4 the dividend is ? the divisor is ? and the quaint is ?. 13. Multiply  3  1   1  3  3  − 16  a) 4 +  1  × 6 +  5  c) 4 ×  × (− 3.25 )  4  2   8  8  4  15  5 4  − 4  b) (2.01 + ( −3.17)) × ( −4.2 + 17.8) d) 16 ×  15 ×  3     14. Adding 3 to some number, then multiplying the result by 7 gives 28. What was the original number? 15. Some number is added to itself. The result is multiplied by 5 and the product is 15. What was the number? 47 UNIT LINEAR EQUATIONS AND 2 INEQUALITIES Unit outcomes: After completing this unit, you should be able to:  solve linear equations using transformation rules.  solve linear inequalities using transformation rules. Introduction Based on your knowledge of working with variables and solving one step of linear equations and inequalities. You will learn more about solving linear equations and inequalities involving more than one steps. When you do this you will apply the rules of equivalent transformations of equations and inequalities appropriately. 48  2 Linear Equations and Inequalities 2.1. Solving Linear Equations Group Work 2.1 Discuss with your friends 1. Explain each of the following key terms, and give your own example. a. Term, like terms or similar terms. b. Coefficient of a term. d. Equation. c. Algebraic expressions. e. Equivalent equation. 2. Give examples of your own for: a. like terms or similar term c. equation b. unlike terms d. algebraic expressions 3. What are the numerical coefficients of x and –y3? 3 Definition 2.1: A constant (a number), a variable or product of a number and variable is called a term. −3 Example 1: 2, , x, 3x, -4x2 are called terms. 2 Consider Group A and Group B Group A Group B 5x and -20x -10a2b2and 12c2d2 −1 2 2 -80a2b2and ab 20xy and abcd 2 6x2 and 70x2 5ab and 6xy In general how do you see the differences between Group A and Group B? ? Discuss the differences with your teacher orally. Definition 2.2: Like terms or similar terms are terms whose variables and exponents of variables are exactly the same but differ only in their numerical coefficients. 49 2 Linear Equations and Inequalities Note: Terms that are not like terms are called unlike terms. Example 2: Terms like -10a2, 170a2 and a2 are like terms. Because they have the same variables with equal exponents but differ only in their numerical coefficients. Example 3: Terms like -5ab and 7x2y2 are unlike terms. Because they do not have the same variables. Definition 2.3: In the product of a number and variable, the factor which is a numerical constant of a term is called a numerical coefficient. Example 4: In each of the following expression, determine the numerical coefficient. −5 2 2 −1 a. 56b b. a b c. xy d. –x2 2 4 Solution: a. The numerical coefficient of 56b is 56. −5 2 2 −5 b. The numerical coefficient of a b is. 2 2 c. The numerical coefficient of −1 xy is −1. 4 4 d. The numerical coefficient of -x2 is -1. Consider Group C and Group D Group C Group D 2x - 3 2x – 3 = 10 5y 5y = 60 a + 2b + 3c a + 2b + 3c = 100 2(+w) P = 2(+w) Do you observe the differences between Group A and group B? Discuss ? the differences with your teacher. 50  2 Linear Equations and Inequalities Definition 2.4: An equation is a mathematical statement in which two algebraic expressions are joined by equality sign. Therefore, an equation must contain an equal sign,=. Example 5. Some examples of equations are: 1 3 5 c. 3 x − 5 = 10 a. x − 10 = 40 2 2 2 1 1 2 b. 4x + 10 = 3 d. x + x − 10x = 50 2 2 5 Note: Algebraic expressions have only one side. Algebraic expressions are formed by using numbers, letters (variables) and the basic operations of addition, subtraction, multiplication, and division. Example 6. Some examples of algebraic expressions are: a. 2x-4 c. e. 215 b. d. f. 3x Exercise 2A 1. State whether each of the following is an equation or an algebraic expression. a. 2x+10=5x+60 c. 10+3.8= 14.78x-10 b. 2 x + 10 d. 9x + 10=5x 2. In each of the following expressions, determine the numerical coefficient. a. 3 x 4 b. -3 1 x 2 c. −2 x 2 y 2 d. −2 x 5 2 2 3 7 51 2 Linear Equations and Inequalities 3. Identify whether each pair of the following algebraic expressions are like terms or unlike terms. 3 5 2 −5 2 5 a. a b and b a c. -80abc and abc 5 2 5 5 b. 3 xy and 3 x 2 y 2 d. a2b2c2d2 and a4b4c4d4 6 6 Challenge Problems 4. 0.0056x+26=100x+3 is a linear equation. Explain the main reason with your partner. 5. a5b5c5d5and -2(a5b5c5d5) are like terms. State the reason with your teacher orally. 2.1.1 Rules of Transformation for Equation The following are basic rules of equality (=) that are used to get equivalent equations in solving a given equation. Rule 1: For all rational numbers a, b and c a. If an equation a =b is true, then a + c = b + c is true for any rational number c. b. If an equation a =b is true, then a-c =b-c is true for any rational number c. Addition and subtraction properties of equality indicate that adding or subtracting the same quantity to each side of an equation results in an equivalent equation. This is true because if two quantities are increased or decreased by the same amount, then the resulting quantities will also be equal see Figure 2.1 below. a) 60kg = 60kg 52  2 Linear Equations and Inequalities b) 30 kg 30 kg = 60 kg 60 kg Figure 2.1 Balance Rule 2: For all rational numbers a, b and c where c ≠0, and a. If an equation a=b is true, then ac = bc is true for any rational number c. a b b. If an equation a=b is true, then = is true for any rational c c number c. To understand the multiplication property of equality, consider the following example. Suppose you start with a true equation such as 20=20. If both sides of an equations are multiplied by a constant such as 2 the result is also a true statement, see Figure 2.2 below. a) = 20kg 20kg 20 = 20 2×20 = 2×20 40 = 40 b) 2×20kg 2×20kg 20kg 20kg = 20kg 20kg Figure 2.2 balance 53 2 Linear Equations and Inequalities Similarly, if an equation is divided by a non zero real numbers such as 2, the result is also a true statement, see Figure 2.3 below. 20 = 20 a) 20 20 = 20kg = 20kg 2 2 10 = 10 20 kg÷2 kg 20 kg÷2 kg 10 kg = 10 kg b) Figure 2.3 Balance Example 7: To find X from X + 60 = 90 To find X you need: X+60 90 x ? X=? X + 60=90 X+60-60 90 - 60 x 30 X=30 X + 60 - 60=90-60 Taking 60 from each side keeps the balance. Figure 2.4 Balance 54  2 Linear Equations and Inequalities Example 8: Solve each of the following equations by using addition rules. 3 8 a. x + = b. x -6 = -20 5 5 Solution: 3 8 a. x+ =..... Given equation 5 5 −3 x+ 3 +  − 3  = 8 +  − 3 .......... Adding on both sides. 5  5  5  5  5 x+0 = 1……..Simplifying x = 1 ……. x is solved  Check: When x = 1 3 8 x+ = 5 5 3 8 1+ ? 5 5 8 8 =............ True 5 5 8 8 Since = is a true statement, x=1. 5 5 b. x - 6 =-20…….Given equation x - 6 + 6=-20+6…….Adding 6 on both sides. x + 0= -14……Simplifying x = -14 …. x is solved  Check: when x = -14 x-6 = -20 -14 – 6 ? -20 - 20 = -20 ….True Since -20 = -20 is a true statement, x= -14 Example 9: Solve each of the following equations by using multiplication rules. −4 a. 8x = 72 b. x = 10 5 55 2 Linear Equations and Inequalities Solution: a. 8x = 72 …..Given equation 1 1 1 × 8x = × 72............. Multiplying by on both sides 8 8 8 1×x=9………Simplifying x= 9 ……..x is solved  Check: When x= 9 8x= 72 8×9 ? 72 72=72…….True Since 72 = 72 is a true statement, x=9 −4 b. x = 40 …….Given equation 5  −5  − 4   −5 − 5 on both sides.  × x =   × 40 …..Multiplying by  4   5   4  4 1 × x=-50……..Simplifying x = -50 ……..x is solved  Check: When x = -50 −4 x = 40 5 −4   × −50 ? 40  5  40 = 40 ……True Since 40 =40 is a true statement, x= -50. 2.1.2 Linear Equations in One Variable Consider the equation 3x+5 = 0, etc are ? examples of linear equations. Why? Discuss the reason with your teacher in the class. Definition 2.5: A linear equation in one variable x is an equation which can be written in standard form ax + b= 0, where a and b are constant numbers with a ≠ 0. 56  2 Linear Equations and Inequalities From this definition, you can deduce that an equation of a single variable in which the highest exponent of the variable involved is one is called a linear equation. Example 10: Which of the following equations are linear and which are not linear. 5 2 a. 5x+3 = 10 c. 3x2 − 8 x + 10 = 0 6 2 −3 1 x + 20 = 10 − x 2 b. d. 2x +2x=10 2 2 Solution: a and b are linear equations. Because the highest exponent of the variable is one, but c and d are not linear equations. Why? Briefly, all equations have two sides; with respect to the equality sign called left hand sides (L.H.S) and right hand sides (R.H.S) of the equality sign. These two sides are equal to each other like that of a simple balance. Thus equation is just a simple balance