Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2002  OctNov  (P29709/02)  Q#1
Question
Solve the inequality .
Solution
SOLVING INEQUALITY: PIECEWISE
Let, . We can write it as;
We have to consider both moduli separately and it leads to following cases;



When 







If then above four intervals translate to following with their corresponding
inequality;
When 
When 
When 



If then above four intervals translate to following with their corresponding inequality;
When 
When 
When 



We have the inequality;
In standard form it can be written as;
We have to consider both moduli separately and it leads to following cases;







Since then above four intervals translate to;



We can see that given inequality takes following forms for these intervals.


For interval 
For interval 
For interval











This is inconsistent with



Hence, the only solutions for the given inequality are;
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.
Now 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 x axis.
Therefore, next 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 .
Since both x and y intercepts of the line are on the same point we need coordinates of at least one more point to draw the line. For this we substitute in given equation of the line;
Hence, coordinates of another point on 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.
When we sketch the two graphs on the same axes and we get following.
We are looking for the solution of .
It is evident from the graphs that is below (smaller) than for all values of;
Comments