Symbols
The rules of inequalities (sometimes called 'inequations')
These are the same as for equations i.e that whatever you do to one side of the equation(add/subtract, multiply/divide by quantities) you must do to the other.
However, their are two exceptions to these rules.
When you multiply each side by a negative quantity
'<' becomes '>', or '>' becomes '<' .
That is, the inequality sign is reversed.
Similarly, when you divide each side by a negative quantity
< becomes >, or > becomes< .
That is, the inequality sign is reversed.
Examples
Inequalities with one variable
Example #1  Find all the integral values of x where,
The values of x lie equal to and less than 6 but greater than 5, but not equal to it.
The integral(whole numbers + or  or zero) values of x are therefore:
6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4
Example #2  What is the range of values of x where,
Since the square root of 144 is +12 or 12(remember two negatives multiplied make a positive), x can have values between 12 and 12.
In other words the value of x is less than or equal to 12 and more than or equal to 12. This is written:
Inequalities with two variables  Solution is by arranging the equation into the form
Ax + By = C
Then, above the line of the equation, Ax + By < C
and below the line, Ax + By > C
Consider the graph of 2x + y = 2
note  the first term A must be made positive by multiplying the whole equation by 1
The equation 2x + y = 2 becomes 2x  y =2
look at the points(red) and the value of 2x  y for each. The table below summarises the result.
point(x,y) 
2x  y 
value 
more than 2 ? 
above/below 
(1,1) 
2(1)(1) 
1 
no  less 
above 
(1,4) 
2(1)(4) 
2 
no  less 
above 
(2,3) 
2(2)(3) 
1 
no  less 
above 
(3,3) 
2(3)(3) 
3 
yes  more 
below 
(2,1) 
2(2)(1) 
3 
yes  more 
below 
(4,2) 
2(4)(2) 
6 
yes  more 
below 
The Modulus
The modulus is the numerical value of a number, irrespective of the sign it carries.
hence l x l < 3 means 3 < x < 3
Example

