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1A_Ch4(1) 4.1 Formulating Inequalities 1A_Ch4(2) Example Inequalities 1. Expressions that involve inequality signs are called inequalities. e.g. x > 2, 3x + 1 4, etc. 2. Inequality signs Inequality sign Meaning Example > < Greater than 9>5 Less than Greater than or equal to 10 < 20 Less than or equal to x3 x5 3. All values that satisfy an inequality are called solutions of that inequality. Index 4.1 Formulating Inequalities 1A_Ch4(3) If a number is smaller than 20, (a) formulate an inequality to represent this fact, (b) write three numbers that satisfy the inequality. (a) Let the number be x. The required inequality is x < 20 . (b) The numbers are 19.7, 15, 2 1 , etc. 2 Index 4.1 Formulating Inequalities 1A_Ch4(4) Last week, Annie worked as a part-time worker in a fast-food restaurant. Her hourly wage was $10. (a) Let the number of hours that Annie worked in that week be x. Write an inequality in x to express the fact that Annie earned at least $500 in that week. (b) Write 2 possible numbers of hours that Annie worked in that week. Index 4.1 Formulating Inequalities 1A_Ch4(5) Back to Question (a) Since the number of hours that Annie worked in that week is x, the required inequality is 10x 500 . (b) By guessing, two possible solutions are 50 and 60 . Fulfill Exercise Objective Applications of inequalities. Index 4.1 Formulating Inequalities 1A_Ch4(6) Mark wants to take some courses in Chinese Culture. He learns that the registration fee is $800 and the tuition fee for each course is $x. (a) If the total amount he has to pay for 3 courses is less than $3 800, write an inequality in x to express this. (b) If Mark pays exactly $3 500 for 3 courses, find the tuition fee for each course. Index 4.1 Formulating Inequalities 1A_Ch4(7) Back to Question (a) $x is the tuition fee for each course. The required inequality is 800 + 3x < 3 800 . (b) According to the question, 800 + 3x = 3 500 3x = 3 500 – 800 3x = 2 700 2 700 x= 3 = 900 Fulfill Exercise Objective Applications of inequalities. ∴ The tuition fee for each course is $900. Key Concept 4.1.1 Index 4.2 Formulas 1A_Ch4(8) Example Formulas ‧ A formula shows the relationship between two or more variables. E.g. The following diagram is a rectangle. Length = l cm Perimeter = P cm Width = w cm Then P = 2(w + l) is a formula. Index 4.2 Formulas 1A_Ch4(9) Example Substitution ‧ The method of replacing a variable by a number, and then finding the value of the other variable is called the method of substitution. Index 4.2 Formulas 1A_Ch4(10) Write down the formulas for the perimeter (P cm) and the area (A cm2) of the following triangle. h cm b cm P=h+b+s hb A= 2 Key Concept 4.2.1 Index 4.2 Formulas 1A_Ch4(11) In the formula S (a) When m = 250, m then S = 2 250 = 2 = 125 m (a) if m = 250, find the value of S, , 2 (b) if S = 83, find the value of m. (b) When S = 83, m then S = 2 m 83 = 2 m = 83 × 2 = 166 Index 4.2 Formulas 1A_Ch4(12) In the formula F = ma, if m = 5 and a = 10, find the value of F. F = ma = 5 × 10 = 50 Fulfill Exercise Objective Find the values of the subjects of formulas by substitution. Index 4.2 Formulas 1A_Ch4(13) In the formula S = 180(n – 2), (a) if S = 540, find the value of n, (b) find the value of n such that the corresponding value of S is triple the value of S given in (a). (a) 540 = 180(n – 2) 540 180(n 2) i.e. = 180 180 3 =n–2 3+2 =n 5 =n i.e. n= 5 Index 4.2 Formulas 1A_Ch4(14) Back to Question (b) 3S = 3 × 540 = 1620 Here 1 620 = 180(n – 2) i.e. 1 620 180(n 2) = 180 180 9=n–2 Fulfill Exercise Objective 9+2 =n Find the values of 11 = n variables of formulas by substitution. i.e. n = 11 Key Concept 4.2.2 Index 4.3 Sequences 1A_Ch4(15) Sequences 1. A chain of numbers is called a sequence. 2. Each number in a sequence is called a term. 3. For a sequence that is arranged in a certain pattern, we can use an algebraic expression to represent the sequence. The algebraic expression is called the general term of the sequence. Index 4.3 Sequences 1A_Ch4(16) Example Sequences 4. We can use symbols a1, a2, ..., an to represent the first term, the second term, ..., the general term (i.e. the n th term) of a sequence respectively. E.g. 1, 2, 3, 4, 5, ..., n is a sequence where a1 = 1, a2 = 2, a3 = 3, ..., an = n. Index 4.3 Sequences 1A_Ch4(17) Suppose there are n terms in each of the following sequences. Guess the general term in terms of n. (a) –2, –4, –6, –8, ... (a) First term: Second term: Third term: Fourth term: (b) 5, 10, 15, 20, ... –2 = –2 × 1 –4 = –2 × 2 –6 = –2 × 3 –8 = –2 × 4 … … … The nth term: –2n = –2 × n ∴ The general term of the sequence is –2n. Index 4.3 Sequences 1A_Ch4(18) Back to Question (b) First term: 5=5×1 Second term: 10 = 5 × 2 Third term: 15 = 5 × 3 Fourth term: 20 = 5 × 4 … … … The nth term: 5n = 5 × n ∴ The general term of the sequence is 5n. Index 4.3 Sequences 1A_Ch4(19) The general term of a sequence is 2n – 1. Find (a) the first 5 terms, (b) the 15th term of the sequence. General term an = 2n – 1 (a) Substituting 1, 2, 3, 4 and 5 respectively for n in an, we obtain a1 = 2(1) – 1 = 1 a3 = 2(3) – 1 = 5 a2 = 2(2) – 1 = 3 a4 = 2(4) – 1 = 7 a5 = 2(5) – 1 = 9 ∴ The first 5 terms of the sequence are 1, 3, 5, 7 and 9. Index 4.3 Sequences 1A_Ch4(20) Back to Question (b) Substituting n = 15 in an, we obtain a15 = 2(15) – 1 = 29 ∴ The 15th term of the sequence is 29. Fulfill Exercise Objective Find the terms of sequences from their general terms. Index 4.3 Sequences 1A_Ch4(21) Consider the sequence 3, 6, 9, 12, 15, .... (a) Write down the next 3 terms of the sequence. (b) (i) Use an algebraic expression to represent the general term an of the sequence. Soln (ii) Use the result of (b)(i) to find the 20th term of the sequence. Soln (a) 【We can obtain the subsequent term of the sequence by adding 3 to the previous term.】 The next 3 terms after the term 15 are 18, 21, 24. Index 4.3 Sequences 1A_Ch4(22) Back to Question (b) (i) a1 = 3(1) = 3 a2 = 3(2) = 6 a3 = 3(3) = 9 a4 = 3(4) = 12 a5 = 3(5) = 15 … … an = 3(n) = 3n ∴ The required algebraic expression is 3n. Index 4.3 Sequences 1A_Ch4(23) Back to Question (b) (ii) Substituting n = 20 in an, we obtain a20 = 3(20) = 60 ∴ The 20th term of the sequence is 60. Fulfill Exercise Objective Find the general terms and some specified terms of sequences. Key Concept 4.3.1 Index 4.4 Introduction to Functions 1A_Ch4(24) Example Functions ‧ A function is like a number-producing machine. Whenever we input a number, the machine will output a corresponding number. E.g. y = 2x can represent a function where y is a function of x. For every value of x, there is one (and only one) corresponding value of y. Index 4.4 Introduction to Functions 1A_Ch4(25) It is known that y is a function of x, and y = 8 – x. Find the value of y when x is (b) –2. (a) 3, By the method of substitution, (a) when x = 3, y =8–x (b) when x = –2, y =8–x =8–3 = 8 – (–2) = 5 = 10 Index 4.4 Introduction to Functions 1A_Ch4(26) It is known that y is a function of x, and y = 2x + 3. Find the value of y when x is (a) –1, (b) 0, (c) 1. By the method of substitution, (a) when x = –1, y = 2x + 3 = 2(–1) + 3 = 1 Index 4.4 Introduction to Functions 1A_Ch4(27) Back to Question By the method of substitution, (b) when x = 0, y = 2x + 3 = 2(0) + 3 = 3 (c) when x = 1, y = 2x + 3 Fulfill Exercise Objective Find the values of functions. = 2(1) + 3 = 5 Key Concept 4.4.1 Index