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Inequalities and Modulus

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Inequalities
For any two real numbers a and b, one of three possibilities exists: either a is less than b (a < b), a is equal to b (a = b), or a is greater than b (a > b). “a > b” or “b > a” is an inequality. Inequalities can also involve equations, and these equations can be solved to arrive at a solution i.e. range of values that satisfies the inequality.

Inequality signs
> means greater than
< means less than
≥ means greater than or equal to
≤ means less than or equal to

Properties of inequalities

(1) Trichotomy property:
For any two real numbers a and b, only one of the following can be true.
a <b
a = b
a > b

(2) Transitive properties of inequality:
If a < b and b < c, then a < c.
If a > b and b > c, then a > c.

(3) Properties of addition and subtraction:
An equal quantity may be added to or subtracted from both sides of an inequality without changing the inequality.
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

(4) Properties of multiplication and division:
i. An equal positive quantity may multiply or divide both sides of an inequality without changing the inequality.
If a < b and c > 0, then ac < bc and a/c < b/c
If a > b and c > 0, then ac > bc and a/c > b/c

ii. If both sides of an inequality are multiplied or divided by a negative quantity, then the inequality is reversed.
If a < b and c < 0, then ac > bc and a/c > b/c
If a > b and c < 0, then ac < bc and a/c < b/c

Modulus or absolute value
Absolute value of a real number is its numerical value without regard to its sign – this is the common definition that one might have used. When we are talking about absolute value inequalities, this definition is useless for our purposes. Absolute value of a real number, from an analytical geometry perspective, is that number’s distance from zero along the number line.

For any real number ‘a’ the absolute value or modulus of ‘a’ is denoted by | a | and is defined as

Properties of Modulus

1.  x = 0 ⇔ | x | = 0
2. | x | ≥ 0 and -| x | ≤ 0
3. | x + y | ≤  | x | + | y |
4. || x | - | y || ≤  | x - y |
5. - | x | ≤  x  ≤  | x |
6. | x . y | = | x | . | y |
7. | x/y | = | x | / | y | ; ( y ≠ 0 )
8.  | x |2 = x2


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