Book 6 Chapter 24 Inequalities and Linear Programming

1
24
Inequalities and Linear Programming
Case Study
24.1 Compound Linear Inequalities in One Unknown
24.2 Quadratic Inequalities in One Unknown
24.3 Linear Inequalities in Two Unknowns
24.4 Linear Programming
24.5 Applications of Linear Programming
Chapter Summary
2
Case Study
In an art lesson, students are asked to make
lantern frames using sticks and plasticine. Each
group of students is given 100 sticks and 70
pieces of plasticine.
There is also a requirement that the difference
between the number of rectangular prisms and the
number of pentagonal pyramids should not be more
than 3.
3
Case Study
The following table shows some combinations of
the lantern frames that can be made.
Number of rectangular prisms Number of pentagonal pyramids Number of sticks used Number of pieces of plasticine used
3 6 96 60
5 3 90 58
5 4 100 64
It is not easy to list all possible combinations.
In fact, we can use linear programming to find
the answer.
4
24.1 Compound Linear Inequalities in One
Unknown
A. Solving Compound Inequalities with and
In junior forms, we learnt that the solutions of
a linear inequality in one unknown can be
represented on a number line.
For example (a) x gt 4 (b) x 6
Now, we are going to investigate the solution of
compound inequalities connected by the word
and.
When we solve these kinds of compound
inequalities, we have to find the values of x
that can satisfy all the given inequalities.
If we want to represent the solution of compound
inequalities on a number line, we can shade the
region represented by each inequality
Then the overlapping region is the solution x
gt 4 and x 6, that is, 4 lt x 6.
5
24.1 Compound Linear Inequalities in One
Unknown
A. Solving Compound Inequalities with and
Example 24.1T
Solve 2x 3 ³ 7x 2 and 3x 1 lt 4x 2, and
represent the solution on the number line.
Solution
Solving 2x 3 ³ 7x 2, we have
2x 7x ³ 2 3
5x ³ 5
x 1..(1)
Solving 3x 1 lt 4x 2, we have
3x 4x lt 2 1
x lt 1
x gt 1..(2)
Combining (1) and (2), we have
Therefore, the required solution is 1 lt x 1.
6
24.1 Compound Linear Inequalities in One
Unknown
A. Solving Compound Inequalities with and
Example 24.2T
Solve the compound inequality
and 2(x 1) 3 lt x 7, and represent the
solution on the number line.
Solution
Solving , we have
Solving 2(x 1) 3 lt x 7, we have
6 x 3 lt 2x
2x 2 3 lt x 7
x 2x lt 3 6
2x x lt 7 2 3
3x lt 3
3x lt 12
x gt 1..(1)
x gt 4..(2)
Therefore, the required solution is x gt 4.
7
24.1 Compound Linear Inequalities in One
Unknown
A. Solving Compound Inequalities with and
Example 24.3T
Fanny has 200 and she wants to buy two types of
gifts, which are pencil cases and mouse pads, for
her students. The total number of gifts is 20
and the number of pencil cases should be at least
three. If the prices of a pencil case and a mouse
pad are 14 and 8 respectively, find the
possible number of pencil cases that she can buy.
Solution
Let x be the number of pencil cases, then the
number of mouse pads is (20 ? x). We have
Combining the above results, we have
14x 8(20 ? x) 200 and x ³ 3
14x 160 ? 8x 200 and x ³ 3
? The possible number of pencil cases are 3, 4,
5, 6.
160 6x 200 and x ³ 3
6x 40 and x ³ 3
8
24.1 Compound Linear Inequalities in One
Unknown
B. Solving Compound Inequalities with or
Now, we are going to study another kind of
compound inequalities connected by the word or.
For example, 3x 4 gt 5 or 2x 5 lt 2.
Solving compound inequalities with or, we need
to solve each inequality separately and shade the
region represented by each inequality.
The solution is the whole shaded region on the
number line.
9
24.1 Compound Linear Inequalities in One
Unknown
B. Solving Compound Inequalities with or
Example 24.4T
Solve or , and
represent the solution on the number line.
Solution
Solving , we have
x 4 gt 6
x gt 2..(1)
Solving , we have
2x ? 24
x gt 12..(2)
Therefore, the required solution is x gt 2.
10
24.1 Compound Linear Inequalities in One
Unknown
B. Solving Compound Inequalities with or
Example 24.5T
Solve or
, and represent the solution
on the number line.
Solution
Solving , we have
Solving , we have
Therefore, the required solution is x ? 17 or x
? 11.
11
24.2 Quadratic Inequalities in One
Unknown
A. Solving Quadratic Inequalities in One
Unknown by the Graphical Method
Consider the graph of a quadratic function y ?
ax2 bx c, where a gt 0.
If the graph meets the x-axis at x ? p and x ? q,
the x-axis can be divided into three intervals
Then we can solve the inequality by considering
the value of y in each interval.
For example (i) ax2 bx c gt 0, where a gt 0
The solution is x lt p or x gt q. (ii) ax2 bx c
lt 0, where a gt 0 The solution is p lt x lt
q. (iii) ax2 bx c ? 0, where a gt 0 The
solution is x ? p or x ? q. (iv) ax2 bx c ?
0, where a gt 0 The solution is p ? x ? q.
12
24.2 Quadratic Inequalities in One
Unknown
A. Solving Quadratic Inequalities in One
Unknown by the Graphical Method
Example 24.6T
Solve the inequality 6x x2 ³ 0 graphically and
represent the solution on the number line.
Solution
6x x2 ³ 0
x2 6x ? 0
x(x 6) ? 0
? The solution of the inequality is 0 x 6.
13
24.2 Quadratic Inequalities in One
Unknown
B. Solving by the Algebraic Method
In addition to using the graphical method, a
quadratic inequality can also be solved by the
algebraic method.
Before solving, it is necessary to factorize the
quadratic equation to get the root(s) or the
boundary point(s).
(1) Sign Testing Method The basic multiplication
sign rules state that

• If ab gt 0, then either a gt 0 and b gt 0 or a lt 0
  and b lt 0.
• That is, the product of two numbers that have the
  same sign is positive.

2. If ab lt 0, then either a gt 0 and b lt 0 or a lt
0 and b gt 0. That is, the product of two numbers
that have the same sign is negative.
14
24.2 Quadratic Inequalities in One
Unknown
B. Solving by the Algebraic Method
Example 24.7T
Solve the following quadratic inequalities
algebraically. (a) x2 6x 8 lt 0 (b) x(3x 8)
³ 4
Solution
(a) x2 6x 8 lt 0
(x 4)(x 2) lt 0
x 4 lt 0 and x 2 gt 0 or x 4
gt 0 and x 2 lt 0
x lt 4 and x gt 2 or x
gt 4 and x lt 2
There is no solution for x gt 4 and x lt 2.
? The solution is 2 lt x lt 4.
15
24.2 Quadratic Inequalities in One
Unknown
B. Solving by the Algebraic Method
Example 24.7T
Solve the following quadratic inequalities
algebraically. (a) x2 6x 8 lt 0 (b) x(3x 8)
³ 4
Solution
(b) x(3x 8) ³ 4
3x2 8x 4 ³ 0
(x 2)(3x 2) ³ 0
x 2 ³ 0 and 3x 2 ³ 0 or x 2
0 and 3x 2 0
x ³ 2 and x ³ or
x 2 and x
? The solution is x or x ³ 2.
16
24.2 Quadratic Inequalities in One
Unknown
B. Solving by the Algebraic Method
(2) Test Value Method
To solve a quadratic inequality by the test value
method,
? first find the boundary numbers (the roots)
and divide the number line into intervals by the
boundary numbers,
? then take a value of x from each interval and
check whether it satisfies the inequality or not.
Remarks We can choose any values of x in an
interval. Remember that we are only interested in
the sign (positive or negative) that we get after
substituting the value x into the quadratic
expression.
17
24.2 Quadratic Inequalities in One
Unknown
B. Solving by the Algebraic Method
Example 24.8T
Solve the inequality
by the test value method and represent the
solution on the number line.
Solution
When x ? ?2, (?2 ? 4)1(?2) 3 ? 6 gt 0.
When x ? 0, (0 ? 4)1(0) 3 ? ?12 lt 0.
When x ? 5, (5 ? 4)1(5) 3 ? 13 gt 0.
? The solution is or x ³ 4.
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24.2 Quadratic Inequalities in One
Unknown
B. Solving by the Algebraic Method
Example 24.9T
If the equation 2x2 k(x 1) 4(1 x) ? 0 has
two distinct real roots, find the range of values
of k.
Solution
2x2 k(x 1) 4(1 x) ? 0
2x2 kx k 4 4x ? 0
2x2 (k 4)x (4 k) ? 0
The discriminant of the equation
? (k 4)2 4(2)(4 k)
? k2 8k 16 32 8k
? k2 16
Since the equation has two distinct real roots,
we have D gt 0,
k2 16 gt 0
(k 4)(k 4) gt 0
\ k lt 4 or k gt 4
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24.3 Linear Inequalities in Two Unknowns
A. Linear Inequalities in Two Unknowns
Linear inequalities in two unknowns are
inequalities which can be represented in the form
Ax By C gt 0 or Ax By C ³ 0, where A, B
and C are real numbers, with A and B not both
equal to zero.
Consider the graph of the straight line x y ?
1.
The straight line divides the coordinate plane
into two parts, one above the line and the other
below the line. Each region is called a
half-plane.
The region above the line is called the upper
half-plane, and that below the line is called the
lower half-plane.
The straight line itself is called the boundary
line of each half-plane.
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24.3 Linear Inequalities in Two Unknowns
A. Linear Inequalities in Two Unknowns
A dashed line is used to indicate that all points
on the line do not satisfy the inequality x y gt
1. It means that all the points on the line are
not included in the solution of the inequality x
y gt 1.
If we want to represent the solution of the
inequality x y ³ 1, we have to draw a solid
line. In this case, the points on the line x y
? 1 are also included in the solution of x y ³
1.
Every point in the upper half-plane satisfies the
inequality x y gt 1. Conversely, every point in
the lower half-plane satisfies the inequality x
y lt 1.
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24.3 Linear Inequalities in Two Unknowns
A. Linear Inequalities in Two Unknowns
We can summarize the steps of finding the
solution of a linear inequality in two unknowns
as follows
1. Draw the line for the corresponding equality
first. Use a solid line if the straight line is a
part of the solution and a dashed line if the
line is not a part of the solution.
2. Choose a point not on the line and substitute
its coordinates into the inequality. Check
whether the point satisfies the inequality.
3. The solution of the inequality is (a) the
half-plane containing the test point if the
inequality is satisfied by the coordinates of
the point or (b) the half-plane not containing
the test point if the inequality is not
satisfied by the coordinates of the point.
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24.3 Linear Inequalities in Two Unknowns
A. Linear Inequalities in Two Unknowns
Example 24.10T
Solve 2x y lt 10 graphically.
Solution
Step 1 Plot the graph of 2x y ? 10 with a
dashed line.
x 0 1 2 3 4
y 10 8 6 4 2
Step 2 Choose (0, 0) as a test point and
substitute its coordinates into 2x y (the
L.H.S. of the inequality)
2(0) 0 ? 0 lt R.H.S. which satisfies the
given inequality.
Step 3 Since the test point satisfies the
inequality, the graph of 2x y lt 10 is the
half-plane containing (0, 0).
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24.3 Linear Inequalities in Two Unknowns
B. System of Linear Inequalities in Two
Unknowns
We have just studied the method to represent the
solution of a linear inequality in two unknowns
graphically.
We can use the same technique to solve two or
more linear inequalities in two unknowns
graphically.
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24.3 Linear Inequalities in Two Unknowns
B. System of Linear Inequalities in Two
Unknowns
For example, to solve the system of inequalities

Step 1 Consider the graph of x y ? 8, and
choose (0, 0) as the test point.
? 0 0 ? 0 lt 8 ? The lower half-plane
represents the solution of x y 8.
Step 2 Consider the graph of x y ? 6 and (0,
0).
? 0 0 ? 0 lt 6 ? The upper half-plane
represents the solution of x y lt 6.
Combining the two graphs of the inequalities, the
overlapping region is the solution of the system
of inequalities.
Remarks Alternatively, we can use arrows to
indicate the solution of each inequality.
25
24.3 Linear Inequalities in Two Unknowns
B. System of Linear Inequalities in Two
Unknowns
Example 24.11T
(a) Plot the straight lines y x ? 0 and x y
2 ? 0 on the same coordinate plane. (b) Hence
solve the system of inequalities
.
Solution
(a) For y ? x ? 0, we have
x ?3 ?2 0 2 3
y ?3 ?2 0 2 3
For x y 2 ? 0, we have
x ?3 ?2 0 2 3
y 1 0 ?2 ?4 ?5
(b) The shaded region represents the solution
of the system of inequalities.
26
24.3 Linear Inequalities in Two Unknowns
B. System of Linear Inequalities in Two
Unknowns
Example 24.12T
(a) Solve the system of inequalities
graphically. (b) How many integral
pairs of (x, y) satisfy the above system of
inequalities?
Solution
The shaded region represents the solution of the
system of inequalities.
(a) For 2x y ? 12,
x 3 4 5 6
y 6 4 2 0
For x ? y ? 10,
x 4 6 8 10
y ?6 ?4 ?2 0
(b) 44 pairs
27
24.4 Linear Programming
A. Optimum Values of a Linear Function in
Two Variables
A linear function in two variables x and y is a
function in the form C ? ax by, where a and b
are constants.
For each pair of x and y, there is a
corresponding value of the function.
Usually, we want to find the optimum values
(maximum or minimum) of a linear function C,
where x and y are under some restrictions.
Consider a linear function C ? x 2y.
x 2y 6
For different values of C, the graphs of the
linear function C ? x 2y are a group of
parallel lines and the value of C increases as
the line shifts to the right.
x 2y 4
x 2y 2
28
24.4 Linear Programming
A. Optimum Values of a Linear Function in
Two Variables
The above results demonstrate a fundamental
concept of a new branch of mathematics, which is
called linear programming.
Linear programming is the method of finding the
optimum values of a linear function under some
restrictions, which are called constraints.
The linear functions we want to optimize are
called the objective functions.
29
24.4 Linear Programming
B. Methods of Finding the Optimum Values of
a Linear Function
There are two methods to find the optimum values
of a linear function under given constraints.
Method I Method of Sliding Line
? First draw the line x y ? 0 which is
parallel to the graph of the objective function C
? x y.

Then move the line in a parallel direction to
obtain the optimum values of C.
? The value of C increases as the line shifts to
the right, so the maximum value of C will be
obtained at the extreme right vertex of the
solution region, while the minimum value of C
will be obtained at the extreme left vertex of
the solution region.
? From the graph, C attains its maximum and
minimum values at R(8, 1) and Q(0, 1)
respectively.
Maximum value of C ? 8 (1) ? 7
Minimum value of C ? 0 (1) ? 1
30
24.4 Linear Programming
B. Methods of Finding the Optimum Values of
a Linear Function
Method II Method of Testing Vertices
When solving problems of linear programming, the
objective function always attains its maximum and
minimum values at the vertices of the shaded
region.
We can find the optimum values of C by
substituting the coordinates of all the vertices
into the objective function.
At P(2, 3), C ? 2 3 ? 5
At Q(0, 1), C ? 0 (1) ? 1
At R(8, 1), C ? 8 (1) ? 7

• Maximum value of C ? 7
• minimum value of C ? 1

31
24.4 Linear Programming
B. Methods of Finding the Optimum Values of
a Linear Function
Example 24.13T
Find the maximum and minimum values of the linear
function C ? x 2y, subject to the constraints
Solution
Solve the system of inequalities graphically.
By the method of testing vertices,
At A(0, 15), C ? 0 ? 2(15) ? ?30
At B(0, ?1), C ? 0 ? 2(?1) ? 2
At C(12, 3), C ? 12 ? 2(3) ? 6
? Maximum value
minimum value
32
24.4 Linear Programming
B. Methods of Finding the Optimum Values of
a Linear Function
Example 24.14T
Find the maximum and minimum values of the linear
function C ? x 2y, where x and y are integers,
subject to the constraints x y ³ 2, 3y 2x
10 and y ³ 4x 19.
Solution
Solve the system of inequalities graphically.
Using the method of sliding line, P attains its
maximum value at (4, ?2) and minimum value at (4,
6) and (6, 7).
? Maximum value of C ? 4 2(2)
Minimum value of C ? 6 2(7)
33
24.5 Applications of Linear Programming
There are many applications of linear programming
in daily life.
People can use it to allocate resources (such as
capital, materials, labour force, land, etc.) in
order to achieve the best performance.
To solve a real-life problem of linear
programming, we can follow the steps below.
Step 1 Identify the variables x and y. Step
2 Set up the system of inequalities for the
constraints. Step 3 Solve the system of
inequalities graphically and shade the region of
feasible solution. Step 4 Write down the
objective function C. Step 5 Find the optimal
solution of the objective function by the methods
we learnt in the previous section.
34
24.5 Applications of Linear Programming
Example 24.15T
The table below shows the ingredients per unit of
product A and product B. Suppose there are 60
units of chemical R and 25 units of chemical S.
Let x units and y units be the amounts of
product A and product B that are produced
respectively. (a) Set up the system of
constraints in terms of x and y. (b) Draw and
shade the region of feasible solution that
represents the constraints in (a). (c) If the
profit on selling product A is 100 and the
profit on selling product B is 120, find the
maximum profit.
Chemical R Chemical S
Product A 2 units 1 unit
Product B 3 units 1 unit
Solution
(a) The constraints are
35
24.5 Applications of Linear Programming
Example 24.15T
Let x units and y units be the amounts of product
A and product B that are produced respectively.
(a) Set up the system of constraints in terms of
x and y. (b) Draw and shade the region of
feasible solution that represents the
constraints in (a). (c) If the profit on selling
product A is 100 and the profit on selling
product B is 120, find the maximum profit.
Solution
(b) Refer to the figure.
(c) The objective function is P ? 100x 120y.
Draw a straight line 100x 120y ? 0.
By the method of sliding line, we observe P
attains its maximum value at (15, 10).
? The maximum profit
? 100(15) 120(10)
36
24.5 Applications of Linear Programming
Example 24.16T
In a candy shop, there are two packets of
candies. Packet A contains 5 pieces of milk
candies and 20 pieces of fruit candies. Packet B
contains 25 pieces of milk candies and 25 pieces
of fruit candies. Peter wants to buy at least 140
pieces of milk candies and 375 pieces of fruit
candies. If the prices of packets A and B are 15
and 20 respectively, find the minimum cost that
Peter needs to pay.
Solution
Let x be the number of packets A and y be the
number of packets B.
The constraints are
The objective function is C ? 15x 20y.
37
24.5 Applications of Linear Programming
Example 24.16T
In a candy shop, there are two packets of
candies. Packet A contains 5 pieces of milk
candies and 20 pieces of fruit candies. Packet B
contains 25 pieces of milk candies and 25 pieces
of fruit candies. Peter wants to buy at least 140
pieces of milk candies and 375 pieces of fruit
candies. If the prices of packets A and B are 15
and 20 respectively, find the minimum cost that
Peter needs to pay.
Solution
In the figure, we draw a line 15x 20y ? 0 and
move it to the right in a parallel direction.
As x, y are non-negative integers, C attains its
minimum value at (15, 3).
? The minimum cost
? 15(15) 20(3)
38
24.5 Applications of Linear Programming
Example 24.17T
David buys x toy cars and y toy robots from a
factory. The cost of a car and a robot is 35 and
15 respectively and Davids budget is 900. He
uses a plastic bag to carry the toys. The
capacity of the bag is 50 items. The weight of a
toy car and a toy robot are 300 g and 550 g
respectively and David can carry a load of 20 kg.
If the profit from selling a car and a robot is
15 and 10 respectively, find the maximum profit
he can make and the number of toy cars and robots
he should buy.
Solution
The constraints are
The objective function is P ? 15x 10y.
39
24.5 Applications of Linear Programming
Example 24.17T
David buys x toy cars and y toy robots from a
factory. The cost of a car and a robot is 35 and
15 respectively and Davids budget is 900. He
uses a plastic bag to carry the toys. The
capacity of the bag is 50 items. The weight of a
toy car and a toy robot are 300 g and 550 g
respectively and David can carry a load of 20 kg.
If the profit from selling a car and a robot is
15 and 10 respectively, find the maximum profit
he can make and the number of toy cars and robots
he should buy.
Solution
Draw a line 15x 10y ? 0 and move it to the
right in a parallel direction.
We observe that P attains its maximum value at
(13, 29).
? The maximum profit
Number of toy cars ? 13
number of robots ? 29.
40
Chapter Summary
24.1 Compound Linear Inequalities in One Unknown
Compound inequalities that contain the word and
are the set of values which satisfy both
inequalities.
Compound inequalities that contain the word or
are the set of values which satisfy one or both
of the inequalities.
41
Chapter Summary
24.2 Quadratic Inequalities in One Unknown
A quadratic inequality can be solved by the
1. graphical method
2. algebraic method (a) Sign testing
method (b) Test value method
42
Chapter Summary
24.3 Linear Inequalities in Two Unknowns
1. Linear inequalities in two unknowns can be
represented on the coordinate plane.
2. The line y ? mx c divides the plane into
two parts, the upper half-plane and the lower
half-plane. (a) All the points in the upper
half-plane satisfy the inequality y gt mx c.
(b) All the points in the lower half-plane
satisfy the inequality y lt mx c.
3. The plane which satisfies the inequality is
called the solution region.
4. The common region of all the inequalities
represents the feasible solution of the system of
inequalities.
Notes The straight line y ? mx c is included
as part of the solution for inequalities with
signs ? or ?.
43
Chapter Summary
24.4 Linear Programming
Linear programming is a method for finding the
optimum values of a linear function under given
constraints.
44
Chapter Summary
24.5 Applications of Linear Programming
Steps in solving problems of linear programming
(i) Identify the decision variables x and y.
(ii) Set up the system of inequalities for the
constraints.
(iii) Set up the system of inequalities for the
constraints.
(iv) Solve the system of inequalities graphically
and shade the region of feasible solution.
(v) Write down the objective function C.
(vi) Find the optimum solution of the objective
function.
45
Follow-up 24.1
24.1 Compound Linear Inequalities in One
Unknown
A. Solving Compound Inequalities with and
Solve 2x 1 lt 9 and 2(x 3) lt 12, and represent
the solution on the number line.
Solution
Solving 2x 1 lt 9, we have
2x lt 10
x lt 5(1)
Solving 2(x 3) lt 12, we have
x 3 lt 6
x lt 9(2)
Combining (1) and (2), we have
Therefore, the required solution is x lt 5.
46
Follow-up 24.2
24.1 Compound Linear Inequalities in One
Unknown
A. Solving Compound Inequalities with and

• Solve the compound inequality 2(x 3) 5(x 1)
  11 and
• (b) Represent the solution in (a) on the number
  line.

Solution
Solving , we have
(a) Solving 2(x 3) 5(x 1) 11, we have
2x 6 5x 5 11
3x 1 11
3x 12
x 10 12
x ? 4..(1)
x 2..(2)
Combining (1) and (2), we have 4 x 2.
(b)
47
Follow-up 24.3
24.1 Compound Linear Inequalities in One
Unknown
A. Solving Compound Inequalities with and
There are some 50 notes and 100 notes in a
purse. The number of 50 notes is three times
that of the 100 notes. If the total value of the
notes is at most 1000 dollars and is greater than
500 dollars, find the possible numbers of 100
notes.
Solution
Let x be the number of 100 notes, then the
number of 50 notes is 3x. we have 500 lt 50(3x)
100x 1000
The inequality can be rewritten as 500 lt 150x
100x and 150x 100x 1000
500 lt 250x and 250x
1000
2 lt x and
x 4
Combining the above results, we have
Therefore, the possible numbers of 100 notes are
3 or 4.
48
Follow-up 24.4
24.1 Compound Linear Inequalities in One
Unknown
B. Solving Compound Inequalities with or
Solve 5(x 1) 2x 1 or 2x ³ x 12, and
represent the solution on the number line.
Solution
Solving 5(x 1) 2x 1, we have
5x 5 2x 1
5x 2x 1 5
3x 6
x 2(1)
Solving 2x ³ x 12, we have
2x x ³ 12
3x ³ 12
x 4..(2)
Therefore, the required solution is x 4.
49
Follow-up 24.5
24.1 Compound Linear Inequalities in One
Unknown
B. Solving Compound Inequalities with or
Solve or
, and represent the solution on the
number line.
Solution
Solving , we have
Solving we have
8x 9 lt 15
3(x 2) lt 2(x 1)
8x lt 24
3x 6 lt 2x 2
x lt 3..(1)
3x 2x lt 6 2
x lt 8..(2)
Therefore, the required solution is x lt 8.
50
Follow-up 24.6
24.2 Quadratic Inequalities in One
Unknown
A. Solving Quadratic Inequalities in One
Unknown by the Graphical Method
Solve the inequality 15 x2 2x graphically and
represent the solution on the number line.
Solution
15 x2 2x
x2 2x 15 ³ 0
(x 5)(x 3) ³ 0
\ The solution of the inequality is x 3 or x
³ 5.
51
Follow-up 24.7
24.2 Quadratic Inequalities in One
Unknown
B. Solving by the Algebraic Method
Solve the following quadratic inequalities
algebraically. (a) x2 19x 34 gt 0 (b) 2 3x2
³ 5x
Solution
(a) x2 19x 34 gt 0
(x 2)(x 17) gt 0
x 2 gt 0 and x 17 gt 0 or x 2 lt 0
and x 17 lt 0
x gt 2 and x gt 17 or x lt 2
and x lt 17
? The solution is x lt 17 or x gt 2.
52
Follow-up 24.7
24.2 Quadratic Inequalities in One
Unknown
B. Solving by the Algebraic Method
Solve the following quadratic inequalities
algebraically. (a) x2 19x 34 gt 0 (b) 2 3x2
³ 5x
Solution
(b) 2 3x2 ³ 5x
3x2 5x 2 0
(3x 1)(x 2) 0
3x 1 0 and x 2 ³ 0 or 3x 1 ³ 0
and x 2 0
x and x ³ 2 or x ³
and x 2
There is no solution for and x 2.
? The solution is .
53
Follow-up 24.8
24.2 Quadratic Inequalities in One
Unknown
B. Solving by the Algebraic Method
Solve the inequality x(3x 5) ³ 2 by the test
value method and represent the solution on the
number line.
Solution
x(3x 5) ³ 2
3x2 5x 2 ³ 0
(3x 1)(x 2) ³ 0
When x ? ?3, 3(?3) ? 1(?3 2) ? 10 gt 0.
When x ? 0, 3(0) ? 1(0 2) ? ?2 lt 0.
When x ? 1, 3(1) ? 1(1 2) ? 6 gt 0.
? The solution is x 2 or .
54
Follow-up 24.9
24.2 Quadratic Inequalities in One
Unknown
B. Solving by the Algebraic Method
Given an equation x2 2kx x 4 ? 0. If the
equation does not have any real root, find the
range of values of k.
Solution
x2 2kx x 4 ? 0
x2 (2k 1)x 4 ? 0
The discriminant of the equation
? (2k 1)2 4(1)(4)
? 4k2 4k 1 16
? 4k2 4k 15
Since the equation does not have any real root,
we have D lt 0,
4k2 4k 15 lt 0
(2k 3)(2k 5) lt 0
\ 1.5 lt k lt 2.5
55
Follow-up 24.10
24.3 Linear Inequalities in Two Unknowns
A. Linear Inequalities in Two Unknowns
Solve x y ³ 5 graphically.
Solution
Step 1 Plot the graph of x y ? 5 with a solid
line.
x 2 0 2 4 6
y 7 5 3 1 1
Step 2 Choose (0, 0) as a test point and
substitute its coordinates into x y (the L.H.S.
of the inequality)
0 0 ? 0 lt R.H.S. which does NOT satisfy
the given inequality.
Step 3 Since the test point does NOT satisfy the
inequality, the graph of x y ? 5 is the
half-plane NOT containing (0, 0).
56
24.3 Linear Inequalities in Two Unknowns
B. System of Linear Inequalities in Two
Unknowns
Follow-up 24.11
(a) Plot the straight lines x y ? 6 and 2x y
? 6 on the same coordinate plane. (b) Hence
solve the system of inequalities
.
Solution
(a) For x y ? 6, we have
x ?2 0 2 4 6
y 8 6 4 2 0
For 2x y ? 6, we have
x ?4 ?2 0 2 4
y ?2 2 6 10 14
(b) The shaded region represents the solution of
the system of inequalities.
57
Follow-up 24.12
24.3 Linear Inequalities in Two Unknowns
B. System of Linear Inequalities in Two
Unknowns
(a) Solve the system of inequalities
graphically. (b) Find the point(s) of
(x, y), where x and y are integers, which
satisfies/satisfy the above system of
inequalities.
Solution
The shaded region represents the solution of the
system of inequalities.
(a) For x ? y ? 6,
x 0 2 4 6
y ?6 ?4 ?2 0
For x ? 2y ? 8,
x 0 2 8 6
y ?4 ?3 ?2 ?1
(b) The possible point is (7, 0) only.
58
Follow-up 24.13
24.4 Linear Programming
B. Methods of Finding the Optimum Values of
a Linear Function
Find the maximum and minimum values of the linear
function C ? x y subject to the constraints
.
Solution
Solve the system of inequalities graphically.
By the method of testing vertices,
At P(0, 10), C ? 0 10 ? 10
At Q(0, ?8), C ? 0 (?8) ? ?8
At R(6, ?2), C ? 6 (?2) ? 4
? Maximum value
minimum value
59
Follow-up 24.14
24.4 Linear Programming
B. Methods of Finding the Optimum Values of
a Linear Function
(a) Find the minimum value of the linear function
P ? y 2x subject to the constraints
. (b) Repeat (a) if x and y are
integers.
Solution
Solve the system of inequalities graphically.
(a) By the method of sliding lines, P is minimum
at (1.5, ?8).
? Minimum value ? ?8 ? 2(1.5)
(b) If x and y are integers, then P is minimum at
(1, ?7).
? Minimum value ? ?7 ? 2(1)
60
Follow-up 24.15
24.5 Applications of Linear Programming
A factory produces two types of television sets.
The standard set requires two hours for assembly
and one hour for testing and packaging. The
deluxe set requires three hours for assembly and
four hours for testing and packaging. There are a
total of 150 hours and 160 hours available for
assembly, testing and packaging process
respectively. The profit on each standard set is
150 and the profit on each deluxe set is 250.
(a) Let x be the number of standard sets and y
be the number of deluxe sets produced. Set up
the system of constraints in terms of x and y.
(b) Draw and shade the region of feasible
solution that represents the constraints in (a).
(c) Set up an objective function P and find the
maximum profit.
Solution
(a) The constraints are
61
Follow-up 24.15
24.5 Applications of Linear Programming
The profit on each standard set is 150 and that
on each deluxe set is 250. (a) Let x be the
number of standard sets and y be the number of
deluxe sets produced. Set up the system of
constraints in terms of x and y. (b) Draw and
shade the region of feasible solution that
represents the constraints in (a). (c) Set up
an objective function P and find the maximum
profit.
Solution
(b) Refer to the figure.
(c) The objective function is P ? 150x 250y.
Draw a straight line 150x 250y ? 0.
By the method of sliding line, we observe P
attains its maximum value at (24, 34).
? The maximum profit ? (150 ? 24 250
? 34)
62
Follow-up 24.16
24.5 Applications of Linear Programming
The owner of an amusement park wants to install
two types of new machines. Each small machine
takes up a floor area of 4 m2 and consumes 120
units of electricity per day. Each large machine
takes up a floor area of 6 m2 and consumes 480
units of electricity per day. It is expected that
the total area used should be at least 54 m2 and
the total electricity consumption should be at
least 4000 units per day. If the running cost of
each small machine and large machine are 1000
and 2500 per day respectively, find the minimum
cost of operating the machines each day.
Solution
Let x be the number of small machines and y be
the number of large machines.
The constraints are
The objective function is C ? 1000x 2500y.
63
Follow-up 24.16
24.5 Applications of Linear Programming
The owner of an amusement park wants to install
two types of new machines. Each small machine
takes up a floor area of 4 m2 and consumes 120
units of electricity per day. Each large machine
takes up a floor area of 6 m2 and consumes 480
units of electricity per day. It is expected that
the total area used should be at least 54 m2 and
the total electricity consumption should be at
least 4000 units per day. If the running cost of
each small machine and large machine are 1000
and 2500 per day respectively, find the minimum
cost of operating the machines each day.
Solution
In the figure, we draw a line 2x 5y ? 0 and
move it to the right in a parallel direction such
that C increases.
As x, y are non-negative integers, C attains its
minimum value at (2, 8).
? The minimum cost ? 1000(2) 2500(8)
64
Follow-up 24.17
24.5 Applications of Linear Programming
A company decides to employ x full-time workers
and y part-time workers. Each full-time worker
earns 300 per day and each part-time worker
earns 200 per day. The company cannot afford to
pay more than 15 000 a day. Also, the total
number of workers should not exceed 70 and the
number of part-time workers should not exceed the
number of full-time workers by more than 10.
(a) Set up the system of constraints in terms of
x and y. (b) Draw and shade the feasible region
that represents the constraints in (a). (c) If a
full-time worker generates a profit of 2000 per
day and a part-time worker generates a profit
of 1500 per day, find the maximum profit of the
company.
Solution
(a) The constraints are
65
Follow-up 24.17
24.5 Applications of Linear Programming
(a) Set up the system of constraints in terms of
x and y. (b) Draw and shade the feasible region
that represents the constraints in (a). (c) If a
full-time worker generates a profit of 2000 per
day and a part-time worker generates a profit
of 1500 per day, find the maximum profit of the
company.
Solution
(b) Refer to the figure.
(c) The objective function is P ? 2000x
1500y.
? P is maximum at (26, 36).
Maximum profit
? 2000(26) 1500(36)