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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,

·        a + c > b + c

·        a - c > b - c

·        ac > bc

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

 

2. Inequalities may be transposed:

·        if a - c > b then a > b + c

·        If a > b and b > c, then a > c

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

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

·        If a > b then an > bn ; 1/an < 1/bn

·        All squares are greater than zero

·        a + 1/a 2 if a 1

·        ab (a + b) /2 if a > b (geometric mean is less than arithmetic mean).

·        2 (1 + 1/n)n 3.

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

 

The function mod x (written as ½x½).

½x½ means that only the positive value of x is taken.

If x = -5, ½x½ will be 5.

 

Illustration :

What are the values of x is ½2x + 3½ < 5.

Solution

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 (ab)

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

 

2. If ai > 0, i = 1, 2, 3,… n, then

(a1 + a2 + a3 +……. + an)/n (a1, a2, a3….. an)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 ai 0, i = 1, 2, … n) and a1 + a2 +… an = constant, then the product a1a2….an is greatest when a1 = a2 = a3 =…….= an.

 

5. For a, b 0

(am + bm)/2 [ (a + b)/2]m; m â‰⁄ 0 OR m 1

(am + bm)/2 ≤ [ (a + b)/2]m; 0 < m < 1

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

 





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