Linear Inequalities in One Unknown

1
Linear Inequalities in One Unknown
3
Case Study
3.1 Inequalities and Their Graphical
Representations
3.2 Basic Properties of Inequalities
3.3 Linear Inequalities in One Unknown
Chapter Summary
2
Case Study
What happened?
Let x L be the volume of juice in each of the
original cups.
In the above situation,
We can represent the above situation
mathematically x ? x ? 1 ? 3
Then we can solve for x using the same way we
solve linear equations.
2x ? 4
\ x ? 2
3
3.1 Inequalities and Their Graphical
Representations
A. Inequality Signs
For 2 different numbers, say 5 and 8, we cannot
use the sign ? to relate them.
In order to describe their relationship, we
should use other signs.
Consider the following inequality signs and their
meanings.
Sign Meaning Example
? Unequal 1 ? 2, 7 ? 10
? Greater than 2 ? 1, 9 ? 4
? Less than 2 ? 3, 5 ? 11
? Greater than or equal to, not less than, at least 7 ? 4, 10 ? 10
? Less than or equal to, not greater than, at most 12 ? 20, 5 ? 5
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3.1 Inequalities and Their Graphical
Representations
A. Inequality Signs
We can also use these signs to represent some
real-life situations.
For example
1. Ken is older than 10, that is, Kens age ? 10.
2. Maria is not taller than 150 cm, that is,
Marias height 150 cm.
3. The capacity of a box is at least 200 cm3,
that is, the boxs capacity ³ 200 cm3.
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3.1 Inequalities and Their Graphical
Representations
A. Inequality Signs
Suppose Maria is 160 cm tall and May is x cm
tall.
Then we know that May is either taller than,
equal to or shorter than Maria, that is,
(i) x ? 160 or (ii) x ? 160 or (iii) x ?
160.
In general, for any 2 numbers x and y, they must
satisfy either one of the following statements
1. x ? y 2. x ? y 3. x ? y
This is called the law of trichotomy.
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3.1 Inequalities and Their Graphical
Representations
B. Inequalities
An inequality is a statement describing the
relationship between 2 algebraic expressions
using an inequality sign.
In the law of trichotomy shown below, 1. x ?
y 2. x ? y 3. x ? y the first and third
statements are inequalities.
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3.1 Inequalities and Their Graphical
Representations
B. Inequalities
Example 3.1T
Use an inequality to represent each of the
following statements. (a) 7 times a is less than
the result of b minus 10. (b) t minus the
product of 5 and s is not greater than the sum of
u and 9.
Solution
(a) Since 7 times a ? 7a and b minus 10 ? b ? 10,
the required inequality is 7a ? b ? 10.
(b) Since t minus the product of 5 and s ? t ?
5s and the sum of u and 9 ? u ? 9,
the required inequality is t ? 5s ? u ? 9.
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3.1 Inequalities and Their Graphical
Representations
C. Solution and Graphical Representation of
Inequalities
We have learnt how to solve a linear equation in
one unknown.
For example, 4 is the solution of the equation x
? 4 ? 0.
How about an inequality such as x ? 4? Can we
find the values that satisfy an inequality?
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3.1 Inequalities and Their Graphical
Representations
C. Solution and Graphical Representation of
Inequalities
Suppose Peter is 160 cm tall. He is shorter than
his elder brother Steve.
Using an inequality to represent Steves height
(S cm), we have S ? 160.
Some possible heights of Steve 161 cm, 170 cm,
180 cm
There are other possible answers but it is not
possible to list out all of them.
From this example, we notice that an inequality
has many solutions.
There are so many solutions that one cannot list
out all of them.
For convenience, we present the solutions
graphically on a number line.
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3.1 Inequalities and Their Graphical
Representations
C. Solution and Graphical Representation of
Inequalities
We have learnt how to mark a cross ? on a
number line to represent a certain number, such
as x ? 1 as shown below.
Now consider the inequality x ? 1. This can be
represented graphically in the following way.
The newly added arrow pointing to the right shows
all the solutions.
The hollow circle ? means that 1 is NOT
included as a solution.
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3.1 Inequalities and Their Graphical
Representations
C. Solution and Graphical Representation of
Inequalities
Similarly, the following shows the possible
solutions of the inequality x ? 2.
The newly added arrow pointing to the left shows
all the solutions.
The solid circle ? means that 2 is included
as a solution.
In general, we can draw simple graphs to
represent solutions of inequalities.
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3.1 Inequalities and Their Graphical
Representations
C. Solution and Graphical Representation of
Inequalities
Example 3.2T
Represent the solutions of the following
inequalities graphically. (a) x ³ 4 (b) x ?
2.5
Solution
(a) x ³ 4
(b) x ? 2.5
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3.1 Inequalities and Their Graphical
Representations
C. Solution and Graphical Representation of
Inequalities
Example 3.3T
Write down the inequalities in x represented by
the following graphs. (a) (b)
Solution
(a) x ? ?18.3
(b) x ? 1
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3.2 Basic Properties of Inequalities
A. Transitive Property
Matthew is older than Christy and Dora is younger
than Christy.
Let a, b, and c be the ages of Matthew, Christy
and Dora respectively.
Then we have a ? b and c ? b.
That is, a ? b and b ? c.
We can also relate a and c directly using an
inequality sign.
Since we observe that Matthew is older than Dora,
we have a ? c.
This illustrates a property of inequalities.
Consider any numbers a, b and c.
If a ? b and b ? c, then a ? c.
This is called the transitive property.
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3.2 Basic Properties of Inequalities
B. Additive Property
The following table shows the present ages of the
father and the mother and their ages after 20
years.
Family member Father Mother
Present age 48 46
Age after 20 years
66
68
At present Fathers age ? Mothers age
We observe that Fathers age ? 20 ? Mothers age
? 20
This illustrates another property of inequalities.
Consider any numbers a, b and c.
If a ? b, then a ? c ? b ? c.
This is called the additive property.
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3.2 Basic Properties of Inequalities
B. Additive Property
If we minus a constant on both sides of an
inequality, the inequality sign remains
unchanged.
For example, 15 ? 12 then 15
3 ? 12 3
that is, 12 ? 9
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3.2 Basic Properties of Inequalities
C. Multiplicative Property
Henry and Elaine have 4 and 3 candies
respectively.
Suppose the numbers of candies they have are
doubled at the same time.
Original number 4 ? 3 New number 8 ? 6
That is, 4 ? 2 ? 3 ? 2
We can observe that when multiplying a positive
number on both sides of an inequality, the
inequality sign remains unchanged.
What if a negative number is multiplied on both
sides instead?
Original number 4 ? 3 New number ?8 ? ?6
That is, 4 ? (?2) ? 3 ? (?2)
We can observe that when multiplying a negative
number on both sides of an inequality, the
inequality sign is reversed.
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3.2 Basic Properties of Inequalities
C. Multiplicative Property
This illustrates another property of
inequalities.
Consider any numbers a, b and c.
If a ? b and c ? 0, then ac ? bc. If a ? b and c
? 0, then ac ? bc.
This is called the multiplicative property.
The following are also true.
1. If a ? b and c ? 0, then ac ? bc. 2. If a ? b
and c ? 0, then ac ? bc.
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3.2 Basic Properties of Inequalities
C. Multiplicative Property
For the multiplicative property of an inequality,
if we multiply both sides by 0.5, it is the same
as dividing both sides by 2.
In general, the multiplicative property is
applicable when multiplying or dividing a
non-zero number on both sides of an inequality.
Consider any number c.
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3.2 Basic Properties of Inequalities
D. Reciprocal Property
Suppose there are x boys and y girls. They order
2 pizzas. The boys share one pizza and the girls
share the other one.
Each girl can get of a pizza.
Each boy can get of a pizza.
If there are more boys than girls (i.e., x ? y),
each girl can get a larger piece of pizza.
Hence we can write
This illustrates another property of
inequalities.
Consider any numbers a and b.
This is called the reciprocal property.
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3.2 Basic Properties of Inequalities
D. Reciprocal Property
Example 3.4T
Determine whether each of the following
statements is true. (a) If a ? 2, then (b) If
p ? q ? 0, then
Solution
(a) a ? 2
(b) ? p ? q ? 0
?
Hence the statement is true.
Hence the statement is false.
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3.3 Linear Inequalities in One Unknown
A. Solving Linear Inequalities in One Unknown
If an inequality contains only one unknown and
the index of the unknown is one, then the
inequality is called a linear inequality in one
unknown.
x ? 3, 4y ? 5 ? 10 and are
some examples.
When solving a linear equation in one unknown, we
try to simplify the equation until the unknown is
left alone on the left hand side of the equal
sign.
Linear inequalities in one unknown can be solved
in a similar way.
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3.3 Linear Inequalities in One Unknown
A. Solving Linear Inequalities in One Unknown
Consider the equation 4x ? 7 ? 33 and the
inequality 4x ? 7 ? 33.
\ 10 is the unique solution of 4x ? 7 ?
33.
\ All numbers greater than 10 are the
solutions of 4x ? 7 ? 33.
Graphical representation of the solutions
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3.3 Linear Inequalities in One Unknown
A. Solving Linear Inequalities in One Unknown
Example 3.5T
Solve ?7 and represent the solutions
graphically.
Solution
Graphical representation of the solutions
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3.3 Linear Inequalities in One Unknown
A. Solving Linear Inequalities in One Unknown
Example 3.6T
Solve 4(3 ? x) 7(2x ? 1) ? 1 and represent the
solutions graphically.
Solution
Graphical representation of the solutions
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3.3 Linear Inequalities in One Unknown
A. Solving Linear Inequalities in One Unknown
Example 3.7T
Solve and represent the
solutions graphically.
Solution
Graphical representation of the solutions
27
3.3 Linear Inequalities in One Unknown
B. Applications of Linear Inequalities in One
Unknown
In our daily lives, we often come across the
terms of exceed, maximum, at least, less
than etc.
For example, the Hong Kong Observatory records
the maximum UV Index every day.
Problems relating to these ideas involve
inequalities.
In such cases, we can set up inequalities to
solve the problems.
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3.3 Linear Inequalities in One Unknown
B. Applications of Linear Inequalities in One
Unknown
Example 3.8T
The sum of 2 consecutive odd integers is not less
than 18. Find the smallest possible value of the
larger odd integer.
Solution
Let x and x ? 2 be the 2 consecutive odd integers.
Since the smallest odd integers not less than 8
is 9, the smallest possible value of x is 9.
\ The smallest possible value of the larger odd
integer is 11.
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3.3 Linear Inequalities in One Unknown
B. Applications of Linear Inequalities in One
Unknown
Example 3.9T
The length of each orange rod is 3 m and the
length of each purple rod is 5 m. Jason has 20
orange rods and some purple rods. If the total
length of rods that Jason has is longer than 200
m, at least how many purple rods does Jason have?
Solution
Let x be the number of purple rods.
\ Jason has at least 29 purple rods.
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3.3 Linear Inequalities in One Unknown
B. Applications of Linear Inequalities in One
Unknown
Example 3.10T
The weight of each medium and large-sized gas
bottle is 7 kg and 10 kg respectively. There are
100 gas bottles in a warehouse. If the total
weight of gas bottles is at least 800 kg, at most
how many medium-sized gas bottles are there in
the warehouse?
Solution
Let x be the number of medium-sized gas bottles.
Then the number of large-sized gas bottles is 100
x.
Since x should be an integer, the maximum value
of x is 66.
\ There are at most 66 bottles of medium-sized
gas.
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Chapter Summary
3.1 Inequalities and Their Graphical
Representations
1. ?, ?, ?, ? and ? are inequality
signs.
2. We can represent solutions of an inequality
graphically on a number line.
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Chapter Summary
3.2 Basic Properties of Inequalities
Consider any numbers a, b and c.
1. Transitive Property If a ? b and b ? c, then
a ? c.
2. Additive Property If a ? b, then a ? c ? b ?
c.
3. Multiplicative Property (a) If a ? b and c ?
0, then ac ? bc. (b) If a ? b and c ? 0, then ac
? bc.
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Chapter Summary
3.3 Linear Inequalities in One Unknown
1. An inequality contains only 1 unknown and the
index of the unknown is 1, then this is a linear
inequality in one unknown.
2. Solving linear inequalities in one unknown is
similar to solving linear equations in one
unknown. However, the direction of the sign when
doing multiplication should be taken into
consideration.
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3.1 Inequalities and Their Graphical
Representations
B. Inequalities
Follow-up 3.1
Use an inequality to represent each of the
following statements. (a) h divided by k is at
least equal to f . (b) a plus b is not greater
than 4 minus c.
Solution
(a) The required inequality is .
(b) The required inequality is a ? b ? 4 ? c.
35
3.1 Inequalities and Their Graphical
Representations
C. Solution and Graphical Representation of
Inequalities
Follow-up 3.2
Represent the solutions of the following
inequalities graphically. (a) x ? 5 (b) x 7
(c) x ³ (d) x ? 2.3
Solution
(a) x ? 5
(b) x ? 7
(d) x ? 2.3
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3.1 Inequalities and Their Graphical
Representations
C. Solution and Graphical Representation of
Inequalities
Follow-up 3.3
Write down the inequalities in x represented by
the following graphs. (a) (b) (c) (d)
Solution
(a) x ? 3.4
(b) x ? ?100
(c) x ? ?12
(d) x ? 17.6
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3.2 Basic Properties of Inequalities
D. Reciprocal Property
Follow-up 3.4
Determine whether each of the following
statements is true. (a) If g ³ ?2, then
0. (b) If r ? s ? 0, then
.
Solution
(a) g ³ ?2
(b) ? r ? s ? 0
?
Hence the statement is true.
Hence the statement is false.
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3.3 Linear Inequalities in One Unknown
A. Solving Linear Inequalities in One Unknown
Follow-up 3.5
Solution
Graphical representation of the solutions
Graphical representation of the solutions
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3.3 Linear Inequalities in One Unknown
A. Solving Linear Inequalities in One Unknown
Follow-up 3.6
Solve the following inequalities and represent
the solutions graphically. (a) x 2(x 5) ³
5 (b) 2(3x 6) ? 15 3x
Solution
Graphical representation of the solutions
Graphical representation of the solutions
40
3.3 Linear Inequalities in One Unknown
A. Solving Linear Inequalities in One Unknown
Follow-up 3.7
Solution
41
3.3 Linear Inequalities in One Unknown
B. Applications of Linear Inequalities in One
Unknown
Follow-up 3.8
The sum of 2 consecutive odd integers is greater
than 50. Find the least possible value of the
smaller odd integer.
Solution
Let x and x ? 2 be the 2 consecutive odd integers.
Since the smallest odd integers greater than 24
is 25, the the least possible value of the
smaller odd integer is 25.
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3.3 Linear Inequalities in One Unknown
B. Applications of Linear Inequalities in One
Unknown
Follow-up 3.9
A schools basketball team made 22 two-point
shots in a match. If the total score of two-point
and three-point shots was greater than 60, at
least how many three-point shots did the team
make?
Solution
Let x be the number of three-point shots made.
Since x should be an integer, the minimum value
of x is 6.
\ At least 6 three-point shots were made.
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3.3 Linear Inequalities in One Unknown
B. Applications of Linear Inequalities in One
Unknown
Follow-up 3.10
Mary bought a total of 12 bottles of cola and
orange juice with more than 100. The cost of
each bottle of cola and orange juice was 8 and
9 respectively. At most how many bottles of cola
did Mary buy?
Solution
Let x be the number of bottles of cola.
Then the number of bottles of orange juice was
(12 x).
Since x should be an integer, the maximum value
of x is 7.
\ Mary bought at most 7 bottles of cola.