AS level

Pure Maths C1
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Inequalities
This is all about how to deal with equations where < or > replace =		A Revision Aid


There aren't really many differences.
The process with Quadratic inequalities is as follows:
1)Treat it like a normal equation replacing the inequality by "="

2) Make it equal 0 (if it doesn't already)

3) Factorise and determine the values of x

4)This gives you the points where the curve crosses the x axis of the graph
The solution to finding the range is either ..
4a) the values on the x axis OUTSIDE these two points
or
4b) the values BETWEEN these two points.

5) Test any three random values
5a) one that is less than the two points
5b) one that is greater than the two points
5c) and one inside (i,e, between) the two points
this will tell you whether it is 4a) or 4b)
(I realise that this is confusing - so maybe best if you just look at the example below)
The way in which we deal with Quadratic inequalities is interesting.
Here is an example :-

Find the range of values satisfying :
2x2 + 7x + 3 < 0
There are a couple of things to note here ..#
1) The co-efficient of x2 is +2 (a positive number)
This means the curve will be "U" shaped (negative co-efficient would be "n" shaped)
2) The inequality is < (less than)
It might have been > (greater than) , <= (less than or equal) or => (greater than or equal)
so here we go ....
1) 2x2 + 7x + 3 = 0
2)  it already = 0
3) (2x +1) (x + 3) so x = -3 or - 0.5
4) so the range is either
        a) less than -3 and greater than -0.5
                or
       b) between -3 and -0.5

Using -10 as a random value below the range -3 to -0.5 we get
2x2 + 7x + 3 < 0 = 200 -70 + 3 = <0
which is clearly not true
Using +10 as a random value above the range -3 to -0.5 we get
2x2 + 7x + 3 < 0 = 200 +70 + 3 = <0
which is also not true
Using -1 as a random value inside the range -3 to -0.5 we get
2x2 + 7x + 3 < 0 = 2 -7 + 3 = <0
which is true

So we now know that the correct answer is -3< x<-0.5


notes
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