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Algebra : Inequalities
 

[ rules ][ one variable ][ two variables ][ modulus ]

 

 

 

 

Symbols

inequalities #1

 

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

inequ#3

 

inequality#2

 

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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 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#6

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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 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

 

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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

inequality problems involving the modulus problem#1

 

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