free physics video tutorials for all

 

Coursework Notes - Algebra

 

Inequalities

 

symbols

 

the rules of inequalities

inequalities with one variable

 

inequalities with two variables

 

 

Symbols

 

inequalities #1

 

 

The rules of inequalities

 

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.

 

 

 

Examples

inequ#3transparentinequality#2

 

 

back to top

 

 

 

Inequalities with one variable

 

 

Example #1 - Find all the integral values of x where,

 

ineq#4

 

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,

 

inequalities#5

 

Since the square root of 144 is +12 or -12(remember two negatives multiplied make a positive), x can be equal to 12 or higher , or x can be equal to -12 or less.

 

inequalities#6

 

 

back to top

 

 

 

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 .

 

 

inequalities#1

 

 

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

 

The table below summarises the result.

 

 

point(x,y)

2x - y

value

more than 2 ?

above/below curve

(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

 

 

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 gcsemathstutor.com 2015 - All Rights Reserved