LINEAR INEQUALITY

Inequality had been part of life when two things differs in content, values privilege and quantity when it should be on the same ground State. The phenomenon called INEQUALITY had been set in place

Inequalities means ordering. The order or position of 2 in relation to 3 or 0, in the real number system, can be described as follows: 2 is less than 3 and 2 is greater than 0.

The symbol '<' is used to mean 'strictly less than' , and the symbol '>' means 'strictly greater than'. The symbol '≥' means 'greater than or equal to' while the symbol '≤' means 'less than or equal to'.

PROPERTIES OF INEQUALITIES

If a, b and c are real numbers then the following relation hold:

1. a < b⇔b - a > 0 , where the symbol '⇔' means the expression on the right is true if the previous statement holds.

2. One and only one of a < b, a = b , a > b is always true

3. If a > 0 and b > 0 , then a + b >0

4. If a >0 and b > 0 , then ab > 0

5. If a < b , there exists a 'c' such that a < c < b

6. If 'a' is given , then we can find some b and c such that b < a < c.

Example 1: If a, b and c ∈ R and a < b , where R is the set of real numbers, prove that

(i) a + c < b + c

(ii) a - c < b - c

(iii) ac < bc if c > 0

(iv) ac > bc if c < 0

Solution

(i) a < b ⇔ b - a > 0 by property 1 

But ( b + c ) - ( a + c ) = b - a for any c

( b + c ) - ( a + c ) > 0

b + c > a + c

An inequality is preserved if a fixed quantity is added to each side of it

(ii) Since a < b ⇒ b - a > 0, the symbol '⇒' is read as 'implies that;

     Then ( b - c ) - ( a - c ) = b - a

    ( b - c ) - ( a - c ) > 0

    b - c > a - c

An inequality is preserved if a fixed quantity is subtracted from each side of it.

(iii) Since a < b ==> b - a > 0

      Then for any c > 0, ( b - a )c > 0

      bc - ac > 0

     bc > ac

Multiplying each side of an inequality by a fixed positive quantity doesn't alter the inequality.

(iv) If a < b and c < 0 then

      b - a > 0 and ( -c ) > 0

     ( b - a )( -c ) > 0

     -bc + ac > 0

     ac > bc

Multiplying each side of an inequality by a fixed negative quantity reverses the inequality.

Example 2: Find the range of values of x for which each of the following will hold:

(i) ¾x - 2 < 4

(ii) ⅔( x - 1 ) > ½( 1 - 2x ) + ½

Solution

(i) ¾x - 2 < 4

     Multiplying both sides by 4, gives 3x - 8 < 16

     ⇔ 3x - 8 + 8 < 16 + 8

      ⇔ 3x < 24

           x < 8

(ii) ⅔( x - 1 ) > ½( 1 - 2x ) + ½

      Multiplying both sides by 6

     4( x -1 ) > 3( 1 - 2x ) + 3

    4x - 4 > 3 - 6x + 3

    10x > 10

     x > 1

Example 3: Solve the inequality 4x - 3 ≥ 2x + 3

Solution

4x - 3 ≥ 2x + 3

⇔ 4x - 2x - 3 + 3 ≥ 2x - 2x + 3 + 3

 ⇔ 2x ≥ 6 or x ≥ 3