Coursework Notes - Algebra

Inequalities

 symbols inequalities with one variable

Symbols

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

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 be equal to 12 or higher , or x can be equal to -12 or less.

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 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

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