Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | May-Jun | (P2-9709/02) | Q#1

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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;

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 solution for the given inequality is;

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  & 1.

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;

Therefore;

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

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 sketch by first drawing and then reflecting in x-axis 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 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 .

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 ;

desmos-graph (6).png

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

desmos-graph (4).png

Next 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.

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  sketch by first drawing and then reflecting in x-axis that part of the line  which is below x-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 ;

desmos-graph (6).png

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

desmos-graph (4).png

When we sketch the two lines on the same graph we get following.

desmos-graph (35).png

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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