Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/23) | Q#3

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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;

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 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  & .

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;

Hence;

SOLVING INEQUALITY: GRAPHICALLY

We aregiven 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.

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, 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 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

Next we have to solve;

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 is ca 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
.

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 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;

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