PURE MATHEMATICS - Algebra

 

Inequalities

 

 

first principles

notation

the formula

 

 

 

Symbols

   x greater than y

   x less than y    

                         x greater than or equal to y

                   x less than or equal to y

 

 

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<

 

 

As before, the inequality sign is reversed.

 

 

example #1

solve for x                                     

 

            multiplying each side by -2

   

 

            note how the inequality sign changes:    <    to    >

 

              

 

 

example #2

solve for x                            

 

            dividing each side by -5

 

     

 

     

 

back to top

 

 

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:

 

 

 

back to top

 

 

Inequalities with TWO variables - Solution is by arranging the equation into the form:

 

 

Then, above the line of the equation,

 

and below the line,

 

 

Consider the graph of   

 

 

 

note - the first term A must be made positive by multiplying the whole equation by -1

 

 

hence                              

 

 

inequalities#1

 

 

Look at the points(red) and the value of 2x - y for each.

 

(The blue line is the graph of the equation rearranged as y = 2x - 2 )

 

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

 

 

back to top

 

 

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

solve for x    

 

              

   

                                      

                          

                                 

 

 

                    

                                

                         

 

                    

 

 

 

 

back to top

 

 

creative commons license

All downloads are covered by a Creative Commons License.
These are free to download and to share with others provided credit is shown.
Files cannot be altered in any way.
Under no circumstances is content to be used for commercial gain.

 

 

 

 

©copyright a-levelmathstutor.com 2020 - All Rights Reserved