Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2006  MayJun  (P29709/02)  Q#1
Question
Solve the inequality .
Solution
SOLVING INEQUALITY: PIECEWISE
Let and , then;
We have to consider two separate cases;
When 
When 










We have the inequality;
It can be written as;
We have to consider two separate cases;
When 
When 









Therefore the inequality will hold for ;


SOLVING INEQUALITY: ALGEBRAICALLY
Let, .
Since given equation/inequality is of the form or or , we can solve this inequality by taking square of both sides;
We are given inequality;
Therefore, we can solve it algebraically;
To find the set of values of x for which , we solve the following equation to find critical values of ;
Now we have two options;




Hence the critical points on the curve for the given condition are & .
Standard form of quadratic equation is;
The graph of quadratic equation is a parabola. If (‘a’ is positive) then parabola opens upwards and its vertex is the minimum point on the graph.
If (‘a’ is negative) then parabola opens downwards and its vertex is the maximum point on the graph.
We recognize that given curve , is a parabola opening upwards.
Therefore conditions for are;
SOLVING INEQUALITY: GRAPHICALLY
We are given inequality;
To solve the inequality graphically, we need to sketch both sides of inequality;
Let’s sketch both equations onebyone.
First we have to sketch;
Let, .
It can be written as;
We have to draw two separate graphs;
When ; 
When ; 


Therefore;

Therefore;

It is evident that and are reflection of each other in xaxis. So we can draw line of by first drawing and then reflecting in xaxis that part of the line which is below xaxis.
It can be written as;
We have to draw two separate graphs;
When ; 
When ; 
Therefore;

Therefore;

It is evident that and are reflection of each other in xaxis. So we can sketch by first drawing and then reflecting in xaxis that part of the line which is below xaxis.
Therefore, first we sketch the line .
To sketch a line we only need x and y intercepts of the line.
The point at which curve (or line) intercepts xaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
Therefore, we substitute in given equation of the line.
Hence, coordinates of xintercept of the line with are .
The point at which curve (or line) intercepts yaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
Therefore, we substitute in given equation of the line.
Hence, coordinates of yintercept of the line with are .
Hence, we get following graph for ;
We can reflect in xaxis that part of the line which is below xaxis to make it graph of , as shown below.
The graph of is a straight horizontal line.
When we sketch the two line on the same graph we get following.
We are looking for the solution of .
It is evident from the graphs that is above (greater) than for all values of;
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