The document discusses inequalities and modulus. It covers the basics of inequalities including the sense of inequality, trichotomy property, and properties such as transitivity, reversal, addition, subtraction, multiplication, division, inverses, and powers. It then provides examples of applying these properties to compare values and solve various inequality problems involving quadratic and rational inequalities.

1. Inequalities and Modulus – Session 1
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2. Inequalities - Basics
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Sense of the Inequality
The < and > signs define what is known as the sense of the inequality (indicated by the direction
of the sign).
Two inequalities are said to have
(a) the same sense if the signs of inequality point in the same direction; and
(b) the opposite sense if the signs of inequality point in the opposite direction.
Trichotomy Property
For any two real numbers a and b , exactly one of the following is true:
a < b , a = b , a > b
The expression a<b is read as a is less than b
The expression a>b is read as a is greater than b.

3. Inequalities - Properties
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If a < b and b < c , then a < c .
If a > b and b > c , then a > c .
Note: These properties also apply to "less than or equal to" and "greater than or equal to":
If a ≤ b and b ≤ c , then a ≤ c .
If a ≥ b and b ≥ c , then a ≥ c .
Transitive Property
Reversal Property
We can swap a and b over, if we make sure the symbol still "points at" the smaller value.
If a > b then b < a
If a < b then b > a

4. Inequalities - Properties
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Properties of Addition and Subtraction
Addition Properties of Inequality:
If a < b , then a + c < b + c
If a > b , then a + c > b + c
Subtraction Properties of Inequality:
If a < b , then a - c < b - c
If a > b , then a - c > b - c
These properties also apply to ≤ and ≥ :
If a ≤ b , then a + c ≤ b + c
If a ≥ b , then a + c ≥ b + c
If a ≤ b , then a - c ≤ b - c
If a ≥ b , then a - c ≥ b - c
Adding and Subtracting Inequalities
If a < b and c < d, then a + c < b + d
If a > b and c > d, then a + c > b + d

5. Inequalities - Properties
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Properties of Multiplication and Division
When we multiply both a and b by a positive number, the inequality stays
the same.
But when we multiply both a and b by a negative number, the
inequality swaps over!
Here are the rules:
•If a < b, and c is positive, then ac < bc
•If a < b, and c is negative, then ac > bc (inequality swaps over!)

6. Inequalities - Properties
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Inverses
Additive Inverse
As we just saw, putting minuses
in front of a and b changes the
direction of the inequality. This
is called the "Additive Inverse":
If a < b then -a > -b
If a > b then -a < -b
Multiplicative Inverse
Taking the reciprocal (1/value) of both a
and b
When a and b are both positive or both
negative
can change the direction of the
inequality.
If a < b then 1/a > 1/b
If a > b then 1/a < 1/b
When a and b are of opposite signs, the
inequality remains as it is.

7. Inequalities - Properties
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Powers
For two positive numbers a and b,
If a > b, then a2 > b2
In general, for any positive value of
n, an > bn
For two negative numbers a and
b,
If a > b, then a2 < b2
In general, for any even positive
value of n, an > bn
More on powers
For any positive value of x ≥ 1
2 ≤ 1 +
1
𝑥
𝑥
< 2.8

8. Problems
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Which of the following is greater?
11^12 or 12^11
Generalisation
If a < b < e then , ab < ba
If e < a < b then ab > ba

9. Inequalities – for comparison of values
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When the signs of two numbers a and b are known
When both are positive
If a/b > 1 , then a > b
If a/b < 1 , then a < b
If a/b = 1 , then a = b
When both are negative
If a/b > 1 , then a < b
If a/b < 1 , then a > b
If a/b = 1 , then a = b
When the signs of two numbers a and b are NOT known
If a - b > 0 , then a > b
If a - b < 0 , then a < b
If a - b = 0 , then a = b

10. Relation between AM, GM and HM
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Basic AM-GM Inequality
For positive real numbers a, b
𝑎 + 𝑏
2
= 𝑎 ∗ 𝑏
Proof:
Squaring, this becomes (a + b) 2 ≥ 4ab, which is equivalent to (a − b) 2
≥ 0. Equality holds if and only if a = b.
In general,
𝐴𝑀 ≥ 𝐺𝑀 ≥ 𝐻𝑀

11. Some applications of AM ≥ GM
1. For any positive real number x,
𝑥 +
1
𝑥
≥ 2
2. For any negative real number x,
𝑥 +
1
𝑥
≤ −2
3. When a+b is constant, ab is maximum when a = b
4. When ab is constant, a+b is minimum when a = b
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12. Problems involving AM ≥ GM
If 𝑥 + 𝑦 + 𝑧 = 19 , what is the maximum value of
(𝑥 – 2) (𝑦 – 3) (𝑧 – 2)?
1. 64
2. 16
3. 125
4. None of these
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13. Problems involving AM ≥ GM
If a, b and c are distinct positive integers, then
(𝑎 + 𝑏) ( 𝑏 + 𝑐) ( 𝑐 + 𝑎) is
1. Greater than or equal to 8
2. Greater than 8
3. Lesser than or equal to 8
4. Lesser than 8
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14. Problems involving AM ≥ GM
If p, q and r are positive real numbers, then the least value of
𝑝 + 𝑞 + 𝑟
1
𝑝
+
1
𝑞
+
1
𝑟
is
1. 1
2. 9
3. 27
4. 64
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15. Solving quadratic Inequalities
Solve 𝑥2
− 7𝑥 + 10 > 0
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16. Solving rational Inequalities
Solve for x
2𝑥 + 7
𝑥 − 4
≥ 3
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Answer: 4 < t ≤ 19 - in interval notation: (4; 19]