This document provides an overview of topics related to algebra, including quadratic equations, inequalities, absolute value, and modulus. It discusses how to form quadratic equations given different conditions on the roots. Methods for finding roots such as splitting the middle term and using the quadratic formula are presented. Properties and techniques for solving different types of inequalities and absolute value equations are outlined. Maximum and minimum values of functions involving modulus are also addressed.

1. GRE
Concept Session: Algebra
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2. Topics Quadratic Equations
Inequalities
Absolute Value
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3. • Quadratic Function
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4. Quadratic function and its Graph
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5. • Forming a Quadratic Equation
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6. Quadratic expression and quadratic equation
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7. Forming a quadratic equation
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1. When roots are given
2. When sum of the roots and the product of the roots are given
3. When the roots are related to the roots of another quadratic equation
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8. Forming a quadratic equation
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1. When roots are given
Form a quadratic equation whose roots are 1 and 2
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9. Forming a quadratic equation
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2. When sum of the roots and the product of the roots are given
Form a quadratic equation such that the sum of the roots is 4 and
the product of the roots is 3
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10. Forming a quadratic equation
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3. When the roots are related to the roots of another quadratic equation
Changed roots Changed Q.Eq.
1 α + p and β + p a (x - p)2 + b (x - p) + c =0
2 α - p and β - p a (x + p)2 + b (x + p) + c =0
3 αp and βp a (x / p)2 + b (x / p) + c =0
4 α/p and β/p a (x p)2 + b (x p) + c =0
5 1/ α and 1/ β a (1/x )2 + b (1/x ) + c =0
6 -α and -β
a (-x )2 + b (-x ) + c =0
a x 2 - b x + c =0
7 α2 and β2 ax + b root x + c =0
8 αn and βn ax2/n + bx1/n + c=0
Form a quadratic equation whose roots are two more than the roots of the equation x2
-3x +2 = 0
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11. • Roots of a Quadratic Equation
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12. What is a root of a quadratic equation?
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13. Finding roots of a quadratic equation
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1. Splitting the middle term
2. Quadratic formula
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14. Finding roots of a quadratic equation
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 Find the roots of the quadratic equation x2 + 5x + 6 = 0
 Find the roots of the quadratic equation 6x2 - 5x - 6 = 0
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15. Finding roots of a quadratic equation
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 Find the roots of the quadratic equation x2 + 6x + 10 = 0
 Quadratic Formula
 Roots =
−𝑏 ± 𝑏2−4𝑎𝑐
2𝑎
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16. Solving Quadratic Inequalities
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Solve for x in x2 – 5x +6 >0
Solve for x in x2 – 5x +6 <0
Solve for x in x2 – 5x +6 ≥0
Solve for x in x2 – 5x +6 ≤0
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Solve for x in
𝑥2 – 5𝑥 + 6
(𝑥 + 2)(4𝑥 − 1)
> 0
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Solve for x in x2 – 5x +6 >0
Solve for x in x2 – 5x +6 <0
Solve for x in x2 – 5x +6 ≥0
Solve for x in x2 – 5x +6 ≤0
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Solve for x in
𝑥2 – 5𝑥 + 6
(𝑥 + 2)(4𝑥 − 1)
> 0
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21. • Inequalities
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22. 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.

23. 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

24. 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

25. 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!)

26. 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.

27. 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

28. 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
Where e = 2.71

29. 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

30. 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

31. 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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32. Solving quadratic Inequalities
Solve 𝑥2
− 7𝑥 + 10 > 0
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33. Solving rational Inequalities
Solve for x
2𝑥 + 7
𝑥 − 4
≥ 3
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Answer: 4 < t ≤ 19 - in interval notation: (4; 19]

34. Absolute Value ( Modulus)
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35. If x is the coordinate of a point on a real number line, then
the distance of a from the origin is represented by 𝑥 .
x is called Absolute value of x or modulus of x.
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Modulus and Distance

36. Definition of Absolute value
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Modulus and Distance

37. Properties of Modulus
1. 𝑥 ≥ 0
2. − 𝑥 ≤ 0
3. 𝑥 + 𝑦 ≤ 𝑥 + 𝑦
4. 𝑥 − 𝑦 ≤ 𝑥 − 𝑦
5. 𝑥 ∗ 𝑦 = 𝑥 ∗ 𝑦
6.
𝑥
𝑦
=
𝑥
𝑦
( as long as y≠0)
7. 𝑥 2
= 𝑥2
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Modulus

38. 𝑥 = 3
𝑥 < 3
𝑥 > 3
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Modulus and Distance

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Modulus and Distance
What if you have to measure a distance from a point other
than 0?
Distance of a coordinate x from a is given by 𝑥 − 𝑎
Then what does 𝑥 + 𝑎 represent?
Distance of x from -a

40. 𝑥 − 1 = 3
𝑥 − 1 < 3
𝑥 − 1 > 3
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Modulus and Distance

41. Find the value of x if 3𝑥 + 2 = 5
Answer : 1 or −
7
3
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Modulus

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Modulus
General Properties of 𝑥

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Modulus
General Properties of 𝑎𝑥 + 𝑏

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Modulus - Problems
Solve for x if 2𝑥 − 1 < 3
Answer : (-1, 2)
Solve for x if 7 − 3𝑥 > 2
Answer : −∞,
5
3
∪ (3, ∞)

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Modulus
How about solving for x for the expression below?
𝑥 − 4 = 3𝑥 − 8
Answers: 𝑥 = 2 𝑜𝑟 3

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Modulus - Problems
Solve for x for the expression below?
3𝑥 − 4 = 𝑥 + 5

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Modulus – Maximum and Minimum Values
What is the maximum value of 𝑓 𝑥 = 15 − 9 + 𝑥 ?

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Modulus – Maximum and Minimum Values
Maximum and Minimum values when more than one
modulus functions are given.
What is the minimum value of the function
𝑥 − 2 + 𝑥 − 9 + 𝑥 + 4
Answer : 13

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Modulus – Maximum and Minimum Values
How many integer solutions are possible for the inequality
𝑥 − 6 + 𝑥 − 8 + 𝑥 + 4 < 11
1. 1
2. 2
3. 0
4. Infinitely many
Answer : 0

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Modulus – Maximum and Minimum Values
How many integer solutions are possible for the inequality
𝑥 − 6 ∗ 𝑥 − 7 ≤ 15
1. 6
2. 5
3. 8
4. More than 8
Answer : 8