Inequalities

This chapter is about inequality relationships, and how to solve inequalities. When you have completed it, you should
1. know the rules for working with inequality symbols
2. be able to solve linear inequalities
3. be able to solve quadratic inequalties


However, the examiners usually combine inequalities with other topic like quadratics in one question. So make sure you read the quadratics notes too !


These for expressions are equivalent.

a > b              a is greater than b
b < a              b is less than a




a ≥ b              a is greater than or equal than b
b ≤ a              b is not greater than a


The symbols < and  > are called strict inequalities , and the symbols ≤ and ≥are called weak inequalities.




Example :


3x + 10 > 10x - 11


Trick : I will teach you the faster way to do these type of questions. You don't have to visualize it geometrically. It is a waste of time. 


3x - 10x > -11 - 10
- 7x > -21
 To get rid of the (-) of x , you need to change the direction of the inequality.
so it becomes : 
7x < 21
 x < 3


Example :

Solve the inequality 1/3 (4x + 3) - 3(2x - 4) ≥ 20.


1/3(4x + 3) - 6x + 12 ≥ 20
4x/3 + 1 - 6x + 12 ≥  20
4x/3 - 6x ≥ 20 - 12 - 1
4x/3 - 6x ≥ 7

Multiply them by 3 to get rid of the fraction.

4x - 18x ≥ 21
-14x ≥ 21       Remember to change the direction .
x  ≤  - 3/2 #


Example :

Solve the inequalities (2x + 1)(x - 3) < 0 


First of all you write it as
 2x + 1 = 0
    x - 3 = 0
If you don't know how it'd become like this, read the notes about quadratics.
So x = -1/2
    x = 3


okay, now watch this carefully.


STEP 1 : Identify the sign of x^2. It is a positive x^2.
               So the graph will have a smiling face instead of a 'sad face'







STEP 2 : Identify the direction of the inequality.
               Since it is < 0 , so it will be inside the curve. If it is > 0 ,it would be outside the curve.

There you go !


The answer is    - 1/2  < x < 3 .






More exercises will be uploaded soon.

Posted in: inequality,quadratics