Transcript Document

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
x3
x5
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
hb
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