inequalities - madasmaths
TRANSCRIPT
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Created by T. Madas
Created by T. Madas
INEQUALITIES FURTHER PRACTICE
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Created by T. Madas
Created by T. Madas
RATIONAL
INEQUALITIES
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Created by T. Madas
Created by T. Madas
Question 1 (**)
Solve the following inequality.
31
4x≥
−.
4 7x< ≤
Question 2 (**)
Solve the following rational inequality.
4 31
2
x
x
−<
−.
FP3-O , 1 2x x< ∪ >
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Created by T. Madas
Created by T. Madas
Question 3 (**)
Find the set of values of x , that satisfy the following inequality.
2
5
4
xx
x<
+.
FP3-J , 1 0x− < < , 1x >
Question 4 (**)
Find the set of values of x , that satisfy the following inequality.
2 158
x
x
+> .
0 3 5x x< < ∪ >
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Created by T. Madas
Created by T. Madas
Question 5 (**)
Solve the following rational inequality.
( )( )5
12 4x x
<+ −
.
2 1 3 4x x x< − ∪ − < < ∪ >
Question 6 (**)
Solve the following inequality.
2
1x
x<
+
SYN-M , 2 1 1x x− < < − >∪
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Created by T. Madas
Created by T. Madas
Question 7 (**+)
Find the set of values of x , that satisfy the following inequality.
2 5
2
xx
x
+≤
−.
1 2 5x x≤ − ∪ < ≤
Question 8 (**+)
Find the set of values of x , that satisfy the following inequality.
3 4
3
x
x x
−>
+.
3 2 0 6x x− < < − < <∪
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Created by T. Madas
Created by T. Madas
Question 9 (**+)
Solve the following inequality.
( )( )( )( )2 1 3
23 2
x x
x x
− +<
− −.
FP3-L , 1 2 3x x< < <∪
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Created by T. Madas
Created by T. Madas
Question 10 (***)
Determine the range of values of x that satisfy the following inequality.
3
2
x x
x x
+≥
−.
FP3-P , 32 0 22
x x x≤ − ∪ < ≤ ∪ >
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Created by T. Madas
Created by T. Madas
Question 11 (***)
Determine the solution interval of the following inequality.
( )7 5
3
x
x x x
−≤
−.
FP3-V , 0 2 3 8x x< ≤ < ≤∪
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Created by T. Madas
Created by T. Madas
Question 12 (***)
Find, in terms of the positive constant k , the solution set of the following inequality.
4
x k k
x k x
+>
+.
FP3-M , 4 2 0 2x k k x x k< − − < < >∪ ∪
Question 13 (***+)
Solve the following rational inequality.
( )( )
2
02 1
x
x x≥
− +.
1 0 2x x x< − = >∪ ∪
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Created by T. Madas
Created by T. Madas
Question 14 (***+)
Solve the following rational inequality.
( ) ( )
( ) ( )
2
2
1 20
1 2
x x
x x
+ −<
− +.
FP3-R , 2 2, 1x x− < < ≠
Question 15 (***+)
Solve the following rational inequality.
2
2
2
xx
x x>
− +.
0 0 1x x< ∪ < <
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Created by T. Madas
Created by T. Madas
Question 16 (****+)
Solve the following inequality.
2 11x x
x− + > .
SP-I , 1 0 1 1x x x< − < < >∪ ∪
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Created by T. Madas
Created by T. Madas
MODULUS
INEQUALTIES
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Created by T. Madas
Created by T. Madas
Question 1 (**)
Find the set of values of x that satisfy the inequality
1 6 1x x− > −
FP3-Q , 27
x <
Question 2 (***)
Find the set of values of x that satisfy the inequality
5 11
2 3
x
x
−≥
−.
FP3-N , 34 ,7 2
x x≥ ≠
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Created by T. Madas
Created by T. Madas
Question 3 (***)
Find the set of values of x that satisfy the inequality
2 6 8 6 2x x x− + < − .
2 2 4 2x− < < −
Question 4 (**+)
Find the set of values of x that satisfy the inequality
2 2 2x x− < .
1 3 1 3x− + < < +
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Created by T. Madas
Created by T. Madas
Question 5 (****)
( )3 1
2
xf x
x
−≡
+, x ∈� , 2x ≠ − .
a) Sketch the graph of ( )f x .
The sketch must include the coordinates of any points where the graph meets the
coordinate axes and the equations of any asymptotes.
b) Hence, or otherwise, solve the inequality
3 12
2
x
x
−<
+.
SYNF-A , 3 55
x− < <
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Created by T. Madas
Created by T. Madas
Question 6 (****+)
Find the set of values of x , that satisfy the following inequality.
( )( )2
1 41
4
x x
x
− +<
+.
FP3-K , 3 8 or 02 3
x x< − < <
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Created by T. Madas
Created by T. Madas
Question 7 (*****)
Determine the range of values of x that satisfy the inequality
3
2
x x
x x
+≥
−.
SP-N , 32 0 0 62
x x x− ≤ < ∪ < ≤ ∪ ≥
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Created by T. Madas
Created by T. Madas
Question 8 (*****)
Find the set of values of x that satisfy the inequality
2 48 4
5
xx
x
−< −
+.
SP-U , 226 25
x x< − ∪ − < <
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Created by T. Madas
Created by T. Madas
Question 9 (*****)
Solve the following inequality.
( )( )5 5 9x x− − > , x ∈�
SPX-E , 4 2, 8x x− < < ∪ >
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Created by T. Madas
Created by T. Madas
Question 10 (*****)
Solve the following inequality in the largest real domain.
2
3 2
2 80
6 5 12
x x
x x x
− −≤
− +.
SPX-B , 4 4, 0x x− ≤ ≤ ≠
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Created by T. Madas
Created by T. Madas
Question 11 (*****)
Solve the following modulus inequality.
3 1 4 11x x+ − − ≤ , x ∈� .
SPX-J , C3T , 9 3x− ≤ ≤
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Created by T. Madas
Created by T. Madas
Question 12 (*****)
Find the set of values of x that satisfy the inequality
44
2
xx
x≥ −
+.
SP-U , 4 2 2 3 17 2x x x− ≤ < − ∪ − < ≤ − ∪ ≥
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Created by T. Madas
Created by T. Madas
VARIOUS
INEQUALITIES
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Created by T. Madas
Created by T. Madas
Question 1 (**)
Solve the inequality
2 1 3 2
7 3
x x− +< .
1715
x > −
Question 2 (**+)
An advertising sign has a rectangular design so the its length is x metres and its width is
( )6 x− metres.
Given that the area of the advertising sign must be at least 5 square metres, determine
the range of possible values of x .
1 5x≤ ≤
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Created by T. Madas
Created by T. Madas
Question 3 (**+)
8 12 7T x y= − + .
It is further given that 712 8
x− < < and 1 26 3
y− < < .
Determine the range of possible values of T .
MP1-H , 5 16T− < <
Question 4 (***)
Show clearly, without approximating and without using any calculating aid that
2 5 7+ > .
proof
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Created by T. Madas
Created by T. Madas
Question 5 (***+)
Given that 0k > show clearly that
12
k
k
+≥ .
MP1-L , proof
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Created by T. Madas
Created by T. Madas
Question 6 (****+)
Show clearly, without approximating and without using any calculating aid that
a) 6 2 6 3 2+ > + .
b) 3 3 2> .
c) 2 1 3 2− > − .
proof
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Created by T. Madas
Created by T. Madas
Question 7 (****+)
Show clearly that for all real numbers α , β and γ
2 2 2α β γ αβ βγ γα+ + ≥ + + .
SP-U , proof