TM
REVISION PRACTICE SHEET MATHS
Path to success KOTA (RAJASTHAN )
Solving quadratic and rational inequalities. (Method of intervals)
Type-1 : Quadratic inequality involving non-repeated linear factors.
(1) 3x2 – 7x + 6 < 0
(2) (x2 – x – 6)(x2 + 6x) 0 )
5 2 x3
(3) Solve f ' (x) g ' (x) where f (x) = 5 – 3x + x – , g (x) = 3x – 7.
2 3
Type-2 : Quadratic inequality involving Repeated linear factos.
(1) (x + 1)(x – 3)(x – 2)2 0.
(2) x(x + 6)(x + 2)2(x – 3) > 0
(3) (x – 1)2(x + 1)3(x – 4) < 0
x3 (2x 3)2 ( x 4)6
(4) Number of positive integral solution of 0
( x 3)3 (3x 8) 4
(A) only one (B) 2 (C) 3 (D) 4
f ( x)
Type-3 : Quadratic / algebraic inequality of the type of . (Rational inequality)
g( x )
2x 3 x 2 5x 12
(1) >0 (2) >3
3x 7 x 2 4x 5
x 2 5x 6 ( x 1)2 (x 1)3
(3) <0 (4) <0
x2 x 1 x 4 ( x 2)
x 1 x 5 2( x 4) 1
(5) (6)
x 1 x 1 ( x 1)(x 7) x 2
x 2 6x 7 x 2 4x 4
(7) <0 (8) >0
| x 4| 2x 2 x 1
1 /2
TM
REVISION PRACTICE SHEET MATHS
Path to success KOTA (RAJASTHAN )
Type-4 : Double inequality and biquadratic inequality.
3x 2 7 x 8
(1) 1< 2
x2 1
(2) (x2 + 3x + 1) (x2 + 3x – 3) 0
( 2 x 3)
(3) (x2 + 3x)(2x + 3) – 16 0.
( x 2 3x )
MISCELLANEOUS EQUATIONS INEQUATIONS AND
LOGARITHMIC INEQUALITIES :
A. LINEAR EQUATION / INEQUATIONS INVOLVING MODULUS :
1. |x–3|+2|x+1|=4
3
2. |x+2|–|x–1|<x
2
3. Find the least +ve integer satisfying | x + 1 | + | x – 4 | > 7.
2 x 1 3x 1
4. Greater integer satisfying – >1
3 2
B. QUADRATIC EQUATION / INEQUATION INVOLVINGMODULUS &
EXPONENTIAL :
5x 16
1. | x2 + 4x + 2 | =
3
2. ( | x – 1 | – 3) (| x + 2 | – 5) < 0
3. | x – 5 | > | x2 – 5x + 9 |
4. 2| x + 2 | – | 2x + 1 – 1 | = 2x + 1 + 1
2 /2
TM
REVISION PRACTICE SHEET MATHS
Path to success KOTA (RAJASTHAN )
x 2 5x 4
5. 1
x2 4
x 2 3x 1
6. <3
x2 x 1
C. LOGARITHMIC EQUATION :
4x 6
1. log 1 0
x
5
2. (a) log2x+3(x2) < log2x + 3(2x + 3)
(b) logx+3(x2 – x) < 1
2x 6
3. log 7 >0
2x 1
4. log3 | 3 – 4x | > 2
5. log0.2 (x2 – x – 2 ) > log0.2 (– x2 + 2x + 3)
3x 6
log 1 log 2
x 2 2
6. (0.3) 3 >1
x 2 x
7. log0.5 log 6 <0
x 4
| x 2 4x | 3
8. log3 2 0
x | x 5|
4x 3 1
9. log 22 >–
4 3x 2
10. 2 log32 x 3log3 x 8 2 log32 x 3 log3 x 6 3
3 /2
