Inequalities
We have developed the real number line in the previous chapter. All the properties of real
numbers which deal with equalities are well understood by now.
Let us consider the real number line again.
→
……..−3 −2 −1 0 1 2 3………
There are several ways to explain what we meant by saying one real number is greater than
another. One way is merely checking the position of the numbers on the number line. Because
3 appear further to the right than 2, we can say that 3 is greater than 2. By putting it another
way, we say that 2 is less than 3.
A more rigorous definition about inequalities can be obtained by breaking the real number line
in to 3 portions. This we will do in the following manner.
Define
𝑅 + = {𝑥 | 𝑥 is located to the right hand side of 0}
𝑅 − = {𝑥 | 𝑥 is located to the left hand side of 0}
{0} = The single point 0
Now, we are in the position to define the order relations which are more commonly known as
the inequalities. When we write 𝑎 > 𝑏, we mean that 𝑎 is greater than 𝑏. If we write 𝑎 ≥ 𝑏,
that would mean 𝑎 is greater than or equal to 𝑏. Similarly, 𝑎 < 𝑏 means 𝑎 is less than 𝑏 and
𝑎 ≤ 𝑏 would mean 𝑎 is less than or equal to 𝑏.
The student should notice that
𝑅 = 𝑅 + ∪ {0} ∪ 𝑅 −
Definition 1
For any two real numbers 𝑎, 𝑏;
i) 𝑎 > 𝑏 ⟺ 𝑎 – 𝑏 ∈ 𝑅 +
ii) 𝑎 ≥ 𝑏 ⟺ 𝑎 – 𝑏 ∈ 𝑅 + ∪ {0}
iii) 𝑎 < 𝑏 ⟺ 𝑎 – 𝑏 ∈ 𝑅 −
iv) 𝑎 ≤ 𝑏 ⟺ 𝑎 – 𝑏 ∈ 𝑅 − ∪ {0}
𝑅 = 𝑅 + ∪ {0} ∪ 𝑅 − and this is a disjoint union. Hence, for any two reals 𝑎 and 𝑏, one and
only one of the following possibilities will occur.
i) 𝑎 – 𝑏 ∈ 𝑅 +
ii) 𝑎 – 𝑏 = 0
iii) 𝑎 – 𝑏 ∈ 𝑅 −
Combining this with definition 1, we can state the following principle.
Principle of Ordering
For any two real numbers 𝑎 and 𝑏, exactly one of the following statements will be true.
i) 𝑎 > 𝑏
ii) 𝑎 = 𝑏
iii) 𝑎 < 𝑏
1
, Now, we shall list 4 basic properties of the inequalities. Any other property pertaining to the
inequalities can be proven using those four. Here, 𝑎, 𝑏 and 𝑐 are real numbers.
Properties:
1. If 𝑎 < 𝑏, then 𝑎 ± 𝑐 < 𝑏 ± 𝑐
2. If 𝑎 < 𝑏, and 𝑐 > 0, then 𝑎𝑐 < 𝑏𝑐
3. If 𝑎 < 𝑏, and 𝑐 < 0, then 𝑎𝑐 > 𝑏𝑐
4. If 𝑎 < 𝑏 and 𝑏 < 𝑐, then 𝑎 < 𝑐
Proof 1: Suppose 𝑎 < 𝑏
(𝑎 ± 𝑐) – (𝑏 ± 𝑐) = 𝑎 – 𝑏 ∈ 𝑅 − 𝑎𝑠 𝑎 < 𝑏
∴ 𝑎 ± 𝑐 < 𝑏 ± 𝑐
The proof of the other properties is left as an exercise for the student.
Theorem 1 Let 𝑥 ∈ 𝑅
1
i) 𝑥 > 0 ⟹ 𝑥 > 0
1
ii) 𝑥 < 0 ⟹ 𝑥 < 0
Proof i) Let 𝑥 > 0
1
If = 0, then
𝑥
1
𝑥 ∙ 𝑥 = 𝑥 ∙ 0 = 0 ⟹ 1 = 0 - contradiction
1
If < 0, then
𝑥
1 property 3 1 1
0 < 𝑥& < 0⇒ 0∙ >𝑥∙ 𝑥⟹
𝑥 𝑥
0 > 1 - contradiction
Now, according to the principle of ordering, the only possibility left is
1
> 0.
𝑥
The proof of ii) is very similar.
Absolute Values
If we ask a student to tell us the difference between 5 and −5, he would tell us that they both
have the same size but having the opposite signs. It is this ‘size’ we consider as the absolute
value of a real number. The formal definition is given below.
Definition 2 For any 𝑥 ∈ 𝑅, the absolute value of 𝑥 which is denoted by |x| is defined as
𝑥 if 𝑥≥0
|𝑥| = {
−𝑥 if 𝑥<0
Theorem 2 Let 𝑎 > 0. Then
i) |𝑥| < 𝑎 ⟺ −𝑎 < 𝑥 < 𝑎
2
We have developed the real number line in the previous chapter. All the properties of real
numbers which deal with equalities are well understood by now.
Let us consider the real number line again.
→
……..−3 −2 −1 0 1 2 3………
There are several ways to explain what we meant by saying one real number is greater than
another. One way is merely checking the position of the numbers on the number line. Because
3 appear further to the right than 2, we can say that 3 is greater than 2. By putting it another
way, we say that 2 is less than 3.
A more rigorous definition about inequalities can be obtained by breaking the real number line
in to 3 portions. This we will do in the following manner.
Define
𝑅 + = {𝑥 | 𝑥 is located to the right hand side of 0}
𝑅 − = {𝑥 | 𝑥 is located to the left hand side of 0}
{0} = The single point 0
Now, we are in the position to define the order relations which are more commonly known as
the inequalities. When we write 𝑎 > 𝑏, we mean that 𝑎 is greater than 𝑏. If we write 𝑎 ≥ 𝑏,
that would mean 𝑎 is greater than or equal to 𝑏. Similarly, 𝑎 < 𝑏 means 𝑎 is less than 𝑏 and
𝑎 ≤ 𝑏 would mean 𝑎 is less than or equal to 𝑏.
The student should notice that
𝑅 = 𝑅 + ∪ {0} ∪ 𝑅 −
Definition 1
For any two real numbers 𝑎, 𝑏;
i) 𝑎 > 𝑏 ⟺ 𝑎 – 𝑏 ∈ 𝑅 +
ii) 𝑎 ≥ 𝑏 ⟺ 𝑎 – 𝑏 ∈ 𝑅 + ∪ {0}
iii) 𝑎 < 𝑏 ⟺ 𝑎 – 𝑏 ∈ 𝑅 −
iv) 𝑎 ≤ 𝑏 ⟺ 𝑎 – 𝑏 ∈ 𝑅 − ∪ {0}
𝑅 = 𝑅 + ∪ {0} ∪ 𝑅 − and this is a disjoint union. Hence, for any two reals 𝑎 and 𝑏, one and
only one of the following possibilities will occur.
i) 𝑎 – 𝑏 ∈ 𝑅 +
ii) 𝑎 – 𝑏 = 0
iii) 𝑎 – 𝑏 ∈ 𝑅 −
Combining this with definition 1, we can state the following principle.
Principle of Ordering
For any two real numbers 𝑎 and 𝑏, exactly one of the following statements will be true.
i) 𝑎 > 𝑏
ii) 𝑎 = 𝑏
iii) 𝑎 < 𝑏
1
, Now, we shall list 4 basic properties of the inequalities. Any other property pertaining to the
inequalities can be proven using those four. Here, 𝑎, 𝑏 and 𝑐 are real numbers.
Properties:
1. If 𝑎 < 𝑏, then 𝑎 ± 𝑐 < 𝑏 ± 𝑐
2. If 𝑎 < 𝑏, and 𝑐 > 0, then 𝑎𝑐 < 𝑏𝑐
3. If 𝑎 < 𝑏, and 𝑐 < 0, then 𝑎𝑐 > 𝑏𝑐
4. If 𝑎 < 𝑏 and 𝑏 < 𝑐, then 𝑎 < 𝑐
Proof 1: Suppose 𝑎 < 𝑏
(𝑎 ± 𝑐) – (𝑏 ± 𝑐) = 𝑎 – 𝑏 ∈ 𝑅 − 𝑎𝑠 𝑎 < 𝑏
∴ 𝑎 ± 𝑐 < 𝑏 ± 𝑐
The proof of the other properties is left as an exercise for the student.
Theorem 1 Let 𝑥 ∈ 𝑅
1
i) 𝑥 > 0 ⟹ 𝑥 > 0
1
ii) 𝑥 < 0 ⟹ 𝑥 < 0
Proof i) Let 𝑥 > 0
1
If = 0, then
𝑥
1
𝑥 ∙ 𝑥 = 𝑥 ∙ 0 = 0 ⟹ 1 = 0 - contradiction
1
If < 0, then
𝑥
1 property 3 1 1
0 < 𝑥& < 0⇒ 0∙ >𝑥∙ 𝑥⟹
𝑥 𝑥
0 > 1 - contradiction
Now, according to the principle of ordering, the only possibility left is
1
> 0.
𝑥
The proof of ii) is very similar.
Absolute Values
If we ask a student to tell us the difference between 5 and −5, he would tell us that they both
have the same size but having the opposite signs. It is this ‘size’ we consider as the absolute
value of a real number. The formal definition is given below.
Definition 2 For any 𝑥 ∈ 𝑅, the absolute value of 𝑥 which is denoted by |x| is defined as
𝑥 if 𝑥≥0
|𝑥| = {
−𝑥 if 𝑥<0
Theorem 2 Let 𝑎 > 0. Then
i) |𝑥| < 𝑎 ⟺ −𝑎 < 𝑥 < 𝑎
2
