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Created by T. Madas
Created by T. Madas
QUADRATIC
INEQUALITIES EXAM QUESTIONS
Created by T. Madas
Created by T. Madas
Question 1 (**)
( ) 22 9 4f x x x= − + , x ∈� .
a) Sketch the graph of ( )f x .
The sketch must include the coordinates of any points where the graph of
( )f x meets the coordinate axes.
b) Solve the inequality
( ) 0f x > .
MP1-C , 1 42
x x< ∪ >
Created by T. Madas
Created by T. Madas
Question 2 (**+)
Solve each of the following inequalities.
a) ( )4 4 2 30x− < .
b) ( )23 4 2 0x x x+ − + > .
C1K , 74
x > , 2 33
x x< ∪ >
Question 3 (**+)
a) Solve the linear inequality
( )4 2 3 47 5x x x+ + > − .
b) Solve the quadratic inequality
( )( )5 2 1 0x x− + ≤ .
c) Hence determine the range of values of x that satisfy both the inequalities of
part (a) and part (b).
C1L , 52
x > , 1 52
x x≤ − ∪ ≥ , 5x ≥
Created by T. Madas
Created by T. Madas
Question 4 (**+)
The figure above shows the graphs of ( )2
1y x= + and 4 9y x= + .
a) Find the coordinates of the points of intersection between the two graphs.
b) Hence solve the inequality
( )2
1 4 9x x+ ≥ + ,
fully justifying the answer.
MP1-E , ( ) ( )2,1 , 4,25− , 2 4x x≤ − ≥∪
( )2
1y x= +
O
4 9x +
y
x
Created by T. Madas
Created by T. Madas
Question 5 (**+)
Solve each of the following inequalities.
a) 2
3 13
xx
+< − .
b) ( )26 20x x x+ + > .
MP1-L , 58
x < , 5 42 3
x x< − ∪ >
Created by T. Madas
Created by T. Madas
Question 6 (***)
A rectangle is such so that its length is 6 cm greater than its width.
Given the area of the rectangle is at least 40 2cm , determine the range of the possible
values of the length of the rectangle.
C1M , length 10>
Question 7 (***)
The rectangle shown above measures ( )2x − cm by ( )6x − cm .
Given the area of the rectangle is at most 60 2cm , and its perimeter at least 14 cm ,
determine the range of the possible values of x .
C1Q , 7.5 12x< <
( )2x −
( )6x −
Created by T. Madas
Created by T. Madas
Question 8 (***)
a) Solve the linear inequality
( )8 3 4 3 2x x+ > − + .
b) Solve the quadratic inequality
( )( ) ( )10 4 5 1 3x x x− − ≥ − − .
c) Hence determine the range of values of x that satisfy both the inequalities of
part (a) and part (b).
C1C , 18x < , 3 16x x≤ ∪ ≥ , 3 16 18x x≤ ∪ ≤ <
Created by T. Madas
Created by T. Madas
Question 9 (***)
a) Solve the linear inequality
( )6 2 2 10x− + < .
b) Solve the quadratic inequality
( )2
1 4 9x x+ ≥ + .
c) Hence determine the range of values of x that satisfy both the inequalities of
part (a) and part (b).
C1J , 4x > − , 2 4x x≤ − ∪ ≥ , 4 2 4x x− < ≤ − ∪ ≥
Created by T. Madas
Created by T. Madas
Question 10 (***+)
The curve C has equation
( )2
4 2y x= − + .
The line L has equation
13 2y x= −
a) Sketch on the same diagram the graph of C and the graph of L .
The sketch must include the coordinates of any points where these graphs
meet the coordinate axes.
b) Solve the equation
( )2
4 2 13 2x x− + = − .
c) Hence find the range of values of x for which
( )2
4 2 13 2x x− + < − .
C1E , 1,5x = , 1 5x< <
Created by T. Madas
Created by T. Madas
Question 11 (***+)
Find the set of values of x , that satisfy the following inequality.
4 13
1
x
x
+>
−.
MP1-B , 4 1x x< − >∪
Created by T. Madas
Created by T. Madas
Question 12 (***+)
A rectangle ABCD measures ( )3 2x + cm by ( )2 4x + cm .
A second rectangle PQRD is removed from the rectangle ABCD , as shown in the
figure above. The perimeter of the composite shape ABCPRQ is greater than 27 cm
but less than 52 cm .
a) Find the range of the possible values of x .
The area of the rectangle PQRD is 4x2
cm .
b) Given further that the area of the composite shape ABCPRQ is less than
982
cm , determine an amended range of the possible values of x .
C1D , 1.5 4x< < , 1.5 3x< <
D
B C( )3 2 cmx +
Q
R
P
A
( )2 4 cmx +
Created by T. Madas
Created by T. Madas
Question 13 (***+)
( ) 22 4f x x x= − − , x ∈� .
a) Express ( )f x in the form ( )2
x a b+ + , where a and b are constants.
b) Find in exact form the solutions of the equation ( ) 0f x = .
c) Hence solve the inequality
( ) ( )( )2 3 4 6 2 0x x x− − + − > .
MP1-G , ( )2
1 5x − − , 1 5x = ± , 1 5 1 5x− < < +
Created by T. Madas
Created by T. Madas
Question 14 (***+)
Determine the range of values of x that satisfy both the inequalities given below.
( ) ( )6 2 7 3 8 3 7x x− − ≥ − +
( )( ) ( )2 3 4 6x x x x− + < + .
C1H , 1 4x≤ <
Question 15 (***+)
Solve the following quadratic inequality.
22 4 0x x− − > .
C1X , 1 5 1 5x x< − ∪ > +
Created by T. Madas
Created by T. Madas
Question 16 (***+)
( ) 2 12 30f x x x= − + , x ∈� .
a) Find in exact surd form the solutions of the equation ( ) 0f x = .
b) Hence solve the inequality …
i. … ( ) 0f x < .
ii. … 2 12 30 0n n− + < , where n is an integer.
MP1-I , 6 6x = ± , 6 6 6 6x− < < + , 4,5,6,7,8n =
Created by T. Madas
Created by T. Madas
Question 17 (***+)
The figure above shows the plan of a rectangular enclosure to be built next to a
farmhouse. One of the farmhouse’s walls will form one of the sides of the enclosure
and 25 metres of fencing will form the other three sides.
The width of the enclosure is x metres, as shown in the figure.
The area of the enclosure must be at most 75 2m .
Given further that the width of the enclosure must be at least 3 metres but no more
than 9 metres, determine the range of the possible values of x .
C1U , 3 5 7.5 9x x< < ∪ < <
x x
Farmhouse Wall
Enclosure
Created by T. Madas
Created by T. Madas
Question 18 (***+)
Determine the range of values of x that satisfy both the inequalities given below.
( )5 8 4 1x x+ ≥ +
( ) ( )( )2
1 8 1 2 0x x x+ − + + < .
MP1-V , 154 17
x x− ≤ < − > −∪
Created by T. Madas
Created by T. Madas
Question 19 (****)
Find, as exact simplified surds, the solution interval that satisfies both the following
inequalities.
( )( )2 4 10 7x x x+ + > +
( )2 2 13 2
3
xx
−< + .
MP1-K , 2 2 3 2 3 2 3x x− < < − ∪ > +
Created by T. Madas
Created by T. Madas
Question 20 (****)
The cost for framing a picture is
• 2 pence per 2cm of glass.
• 5 pence per cm of wooden frame.
A rectangular picture is such so that its length is 4 cm greater than its width, x cm .
If a maximum of £10 is available for framing, determine the range of the possible
values of x .
C1I , 0 16x< <
Created by T. Madas
Created by T. Madas
Question 21 (****)
The figure above shows a square piece of lawn ABCD of side length x metres.
The lawn is surrounded by a path which is 2 metres wide, as shown in the figure.
The area of the lawn must be less than the area of the path.
a) Show clearly that
2 8 16 0x x− − < .
b) Hence determine the range of the possible values of x , in terms of surds
where appropriate.
SYN-H , 0 4 4 2x< < +
C
A B
D
x
x
2
2
Created by T. Madas
Created by T. Madas
Question 22 (****)
The figure above shows a right angled trapezium ABCD where 3 1AB x= + ,
3AD x= , 7 1DC x= + and 90DAB CDA= = °� � .
a) Express the perimeter of the trapezium in terms of x .
b) Show that the area of the trapezium is 215 3x x+ .
The perimeter of the trapezium has to be less than 92 and its area greater than 66 .
c) Determine the range of the possible values of x .
C1N , 18 2P x= + , 2 5x< <
C
A B
D
3 1x +
3x
7 1x +
Created by T. Madas
Created by T. Madas
Question 23 (****)
A rectangular piece of card has length x cm and an area of 1200 2cm .
A square of side length 5 cm is removed from each corner and the sides of the
remaining card are folded upwards to form an open box of height 5 cm .
The resulting box must have a volume greater than 2850 3cm .
a) Show clearly that
2 73 1200 0x x− + < .
b) Hence determine the range of the possible values of x .
SYN-E , 25 48x< <
Created by T. Madas
Created by T. Madas
Question 24 (****)
The figure above shows the graphs of 2 2 7y x x= − + and 2 7y x= + .
Express the shaded region in the figure as a set of inequalities.
MP1-F , { }22 7 2 7 : 0 4x y x x x a a+ ≤ ≤ − + ≥ ≤ ≤∩
O
2 2 7y x x= − +
y
x2 7y x= +
Created by T. Madas
Created by T. Madas
Question 25 (****)
Solve the inequality
2 22 3x y xy+ < ,
and hence indicate the solution in a suitable sketch.
SYN-S , ( ) ( )1 12 2
y x y x y x y x< > > <∩ ∪ ∩
Created by T. Madas
Created by T. Madas
Question 26 (****+)
Consider the following inequalities
( )5 13 4 2x x+ > +
( ) ( )( )2
2 2 3 0x k x x− − − + < ,
where k is a non zero constant.
The common solution interval of both these inequalities is
1754
x x m− < < − >∪ ,
where m is a non zero constant.
Determine, in any order, the value of k and the value of m .
MP1-Y , 2m = , 5k =
Created by T. Madas
Created by T. Madas
Question 27 (*****)
Consider the following inequality
( )( )2 2 1 1 3kx x x x+ + ≤ + − ,
where k is a real constant.
Find, in terms of k where appropriate, the solution intervals of the above inequality
for all possible values of k .
SPX-C ,
2 2 2 21
1 1
1 1
2 2 2 21 2
1 1
2 2
2 no solutions
k kk k k
k k
k x
k kk k k
k k
k x
k
− − − − − −< < >
− −= ≤ −
− − − − − −
< < < >− −
= = − >
∪
∪
