# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2007 | Oct-Nov | (P2-9709/02) | Q#3

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Question

Solve the inequality .

Solution

SOLVING INEQUALITY: PIECEWISE

Let  and , then;

We consider two separate cases.

 When When

We have the inequality;

We have to consider two separate cases;

 When When

Therefore the inequality will hold for ;

Hence;

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 one-by-one.

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 x-axis. So we can  draw line of by first drawing and then reflecting in x-axis that part of the line  which is below x-axis.

We have to draw two separate graphs;

 When ; When ; Therefore; Therefore;

It is evident that  and are reflection of each other in y-axis. So we can sketch  by first drawing  and then reflecting in y-axis that part of the line which  is on left side of  y-axis.

To sketch a line we only need x and y intercepts of the line.

The point at which curve (or line) intercepts x-axis, 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 x-intercept of the line with are .

The point at which curve (or line) intercepts y-axis, 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 y-intercept of the line with   are Hence, we get following graph  for ;

We can reflect in x-axis that part of the line which is below x-axis to make it graph of , as  shown below.

The graph of is a straight vertical 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 below (smaller) than for all values of;

Hence;