Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2008 | May-Jun | (P2-9709/02) | Q#1
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Question
Solve the inequality .
Solution
SOLVING INEQUALITY: PIECEWISE
Let and
, then;
We have to consider two separate cases;
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We have the inequality;
It can be written as;
We have to consider two separate cases;
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Therefore the inequality will hold for ;
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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;
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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;
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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 ;
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 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 below (smaller) than
for all values of;
Hence;
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