# Inequalities

Any Quantity, 'a' may have a relationship with any other quantity 'b'. For example, if a = 2 and b = 3, we say that a < b.

**Properties of Inequalities**:

**1. If a > b,**

Â· a + c > b + c

Â· a - c > b - c

Â· ac > bc

Â· a/c > b/c, where c is a positive number

**2. Inequalities may be transposed**:

Â· if a - c > b then a > b + c

Â· If a > b and b > c, then a > c

Â· If a > b and c > d then a + c > b + d

Â· If a > b, then - a < - b and - ac < - bd where c is positive

Â· If a > b then a^{n} > b^{n} ; 1/a^{n} < 1/b^{n}

Â· All squares are greater than zero

Â· a + 1/a â‰¥ 2 if a â‰¥ 1

Â· âˆšab â‰¤ (a + b) /2 if a > b (geometric mean is less than arithmetic mean).

Â· 2 â‰¤ (1 + 1/n)^{n} â‰¤ 3.

Most sums on inequalities can be done by substituting values, or eliminating choices,

**The function mod x (written as ****Â½****x****Â½****).**

Â½xÂ½ means that only the positive value of x is taken.

If x = -5, Â½xÂ½ will be 5.

**Illustration :**

What are the values of x is Â½2x + 3Â½ < 5.

We take two cases. First, positive value: 2x + 3 < 5, which gives us x < 1.

Second, the negative case: - (2x + 3) = - 2x - 3 < 5 which gives x > 4.

Hence the values of x will be x < 1 and x > 4.

# The quadratic function

If we have a quadratic function, say (x + 2)(x - 3) > 0.

To solve for the value of x, we consider two cases.

First, both factors are negative, then both factors are negative (only then will the inequality be maintained).

So, either (x + 2) > 0 and (x - 3) > 0 OR (x + 2) < 0 and (x - 3) < 0

That is, either x > -2 and x > 3 OR x < -2 and x < 3.

Taking common values, we get: x > 3 and x < -2.

** **

**Some theorems on inequalities:**

1. (a + b)/2 â‰¥ âˆš(ab)

The Arithmetic Mean of two positive quantities is greater than or equal to their Geometric Mean. Similarly âˆš(ab) â‰¥ [2ab /(2ab/(a + b)]. Hence the Geometric Means â‰¥ Harmonic Mean

2. If a_{i} > 0, i = 1, 2, 3,â€¦ n, then

(a_{1} + a_{2} + a_{3} +â€¦â€¦. + a_{n})/n â‰¥ (a_{1}, a_{2}, a_{3}â€¦.. a_{n})^{1/n}

that is, the geometric mean of n positive quantities cannot exceed their arithmetic mean.

3. If the sum of two positive quantities is constant, then their product *is greatest when they are equal*; and if their product is constant, then their sum is least when they are equal.

4. If a_{i} â‰¥ 0, i = 1, 2, â€¦ n) and a_{1} + a_{2} +â€¦ a_{n} = constant, then the product a_{1}a_{2}â€¦.a_{n} is greatest when a_{1} = a_{2} = a_{3} =â€¦â€¦.= a_{n}.

5. For a, b â‰¥ 0

(a^{m} + b^{m})/2 â‰¥ [ (a + b)/2]^{m}; m â‰⁄ 0 OR m â‰¥ 1

(a^{m} + b^{m})/2 â‰¤ [ (a + b)/2]^{m}; 0 < m < 1

That is, arithmetic mean of the m^{th} powers of n positive quantities is greater than the m^{th} power of their arithmetic mean in all cases except when 0 < m < 1.