It is the process of obtaining all the possible solution of an inequation.
* SOLUTION SET:
The set of all possible solutions of an inequation is known as its solution set.
=> SOLVING LINEAR INEQUATION IN ONE VARIABLE:
• RULE-1: Same number may be added to (or subtracted from) both sides of an inequation without changing the sign of inequality.
•RULE-2: Both sides of an inequation can be multiplied(or divided) by the same positive real number without changing the sign of inequality. However, the sign of inequality is reversed when both sides of an inequation are multiplied or divided by a negative number.
• RULE-3: Any term of an inequation may be taken to the other side with its sign changed without affecting the sign of inequality.
A linear inequation in one variable is of the form
ax+b<0 or, ax+b≤0 or, ac+b> 0 or, ax+ b ≥ 0.
We follow the following STEP to solve a linear inequation in one variable.
• STEP-1: Obtain the linear inequation.
• STEP -2 : Collect all terms involving the variable on one side of the inequation and the constant terms on the other side.
• STEP -3: Simplified both sides of inequality in their simplest form to reduce the inequation in the form
ax <b, or ax≤b, or ax>b, or ax≥b
• STEP -4: Solve the inequation obtained in step - 3 by dividing both sides of the inequation by the coefficient of the variable.
• STEP -5: Write the solution set obtained in step -4 in the form of an interval on the real line.
EXERCISE - 1
😀 ------------------ 😀
* Solve the following inequation:
1) 2x - 4≤ 0. (-∞, 2]
2) - 3x +12< 0. (4,∞)
3) 4x - 12 ≥ 0. [3,∞)
4) 12x < 50, when
a) x ∈ R. (-∞,25/6)
b) x ∈ Z. {....,-3,-2,-1,0,1,2,3,4}
c) x ∈ N. {1,2,3,4}
5) - 4x > 30, when
a) x ∈ R. (-∞,-15/2)
b) x ∈ Z. {....,-9,-8}
c) x ∈ N. null set
6) 7x +9> 30. (3,∞)
7) 4x -2 < 8, when
a) x ∈ R. (-∞,5/2)
b) x ∈ Z. {....-2,-1,0,1,2}
c) x ∈ N. {1,2}
8) 5x -3< 3x+1 when
a) x is a real number. (-∞, 2)
b) x is integer number. {..,-4,-3, -2, -1, 0,1}
c) x is a natural number. {1}
8) 3x - 7 > x+1. (4,∞)
9) x+ 5 > 4x -10. (-∞, 5)
10) 3x+ 9 ≥ - x+19. (5/2,∞)
11) 3x+17≤ 2(1-x). (-∞, 3]
12) 2(2x+3)-10≤6(x-2). [4,∞)
13) 2(3-x)≥ x/5 + 4. (-∞, 10/11]
14) -(x-3)+4< 5- 2x. (-∞,-2)
15) (2x-3)/4 +9≥ 3+ 4x/3. (-∞,63/10]
16) (3x-2)/5 ≤ (4x-3)/2. [11/14,∞)
17) (5x-2)/3 -(7x-3)/5 > x/4. (4,∞)
18) 1/2(3x/5 +4) ≥ (x- 6)/3. (-∞,120]
19) 3(x- 2)/5 ≥ 5(2 - x)/3. [2,∞)
20) x/5< (3x-2)/4 - (5x-3)/5. (-∞,2/9)
21) 2(x-1)/5 ≤ 3(2+x)/7. [-44,∞)
22) 5x/2 + 3x/4 ≥ 39/4. [3,∞)
23) (x-1)/3 +4< (x -5)/5 - 2. (-∞,50)
24) (2x+3)/4 - 3 < (x-4))3 - 2. (-∞, - 13/2)
25) (5-2x)/3 < x/6 - 2. (8,∞)
26) (4+2x)/3 ≥x/2 - 3. (-26,∞)
27) (2x+3)/5 - 2 < 3(x-3)/5. (-1,∞)
28) (x-2)≤(5x+8)/3. (-7,∞)
29) 1/(x -2)< 0. (-∞,2)
30) (x+1)/(x +2)≥ 1. (-∞,-2)
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Type ::2
EQUATION OF THE FORM
* (ax+b)/(cx+d)> k, OR
* (ax+b)/(cx+d)≥ k, OR
* (ax+b)/(cx+d)< k, OR
* (ax+b)/(cx+d) ≤ k.
STEP -1: Obtain the inequation.
STEP-2:Transpose all terms on LHS
STEP-3: Simplify LHS of the inequation obtained in STEP-2 to obtain an inequation of the form
(px+q)/(rx+s)> 0 OR
(px+q)/(rx+s)≥0 OR
(px+q)/(rx+s)< 0, OR
(px+q)/(rx+s) ≤ 0
STEP-4: Make coefficient x positive in numerator and denominator if they are not.
STEP-5: Equate numerator and denominator separately to zero and obtain the values of x. These values of x are generally called critical points.
STEP-6: Plot the critical points obtained in STEP-5 on real line. These points will divide the real line in three regions.
STEP-7: In the right most region the expression on LHS of the inequation obtained in STEP-4 will be positive and in other regions it will be alternatively negative and positive. So, mark positive signs in the right most region and then mark alternatively negative and positive signs in other regions.
STEP-8: Select appropriate region on the basis of the sign of the inequation obtained in STEP-4 , Write these region in the form of interval to obtain the desired solution sets of the given inequation.
EXERCISE-2
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Solve the inequation of followings:
1) (2x+4)/(x-1) ≥ 5. (1,3]
2) (x+3)/(x-2) ≤2. (-∞,2)U(7,∞)
3) (2x-3)/(3x-7) ≤2. (-∞,3/2) U (7/3,∞)
4) 3/(x-2) < 1. (-∞,2)U(5,∞)
5) 1/(x-1) ≤2. (-∞,2)U(3/2,∞)
6) (4x+3)/(2x-5) ≤2. (-∞,5/2) U (33/8,∞)
7) (5x-6)/(x+6) <1. (-6,3)
8) (5x+8)/(4-x) <2. (-∞,0) U (4,∞)
9) (x-1)/(x+3) >2. (-7,-3)
10) (7x-5)/(8x+3)>4. (-17/25, -3/8)
11) x/(x-5) > 1/2. (-∞,-5)U(5,∞)
12) (x-3)/(x+4) >0, x ∈R. {x∈ R: x < -4}U {x ∈ R: x> 3}
13) (x+5)/(x-2) >0, x ∈R. {x∈ R: x ≤ -5}U {x ∈ R: x> 2}
14) (2x+5)/(x+3) >1, x ∈R. {x∈ R: x < -3} U {x∈ R: x > -2}
15) (x+7)/(x+4) >1, x ∈R. {x∈ R: x > - 4}
16) (x+4)/(x+6) >1, x ∈R. {x∈ R: x < -6}
17) 3/(x -2) >2, x ∈R. {x∈ R: 2< x < 7/2}
18) (x-3)/(x+1) < 0, x ∈R. {x∈ R: -1 < x < 3}
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TYPE -3:
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* If a is a positive real number, then
1) |x| < a <=> -a<x< a i.e. x∈ (-a, a).
2) |x|≤a <=> -a≤x≤a i.e. x x∈ [-a, a].
3) | x|> a <=> x<-a or x> a.
4) | x|≥ a <=> x≤-a or x≥ a.
* Let r be a positive real number and a be a fixed real number. Then,
1) |x - a| < r <=> a-r <x< a+r i.e. x∈ (a - r, a+ r).
2) |x -a|≤r <=> a-r ≤x≤a+r i.e.x∈ [a - r, a +r].
3) | x - a|> r <=> x<a -r or x> a +r
4) | x -a|≥ r <=> x≤-a- r or x≥ a+r.
EXERCISE -3
------------------
Solve the inequation of followings:
1) | x | < 5, x ∈R. {x∈ R: -5< x < 5}
2) |x |≥ 5, x ∈R. {x∈ R: x < -5} U {x∈ R: x ≥ 5}
3) |3x -2|≤ 1/2. {x∈ R: x < -5}
4) | 4x - 5 | ≤ 1/3, x∈[1/2,5/7}
5) | x - 2|≥5. (-∞,-3)U[7,∞)
6) |5 - 2x | ≤ 3, x ∈R. {x∈ R: 1≤x< 4}
7) | 3x - 7| > 4, x ∈R. {x∈ R: x < 1} U {x∈ R: x > 11/3}
8) |2(3-x)/5| < 9/5, x ∈R. {x∈ R: -3/2 < x < 15/2}
9) |x +1/3| > 8/3. (-∞,-3)U(7/3,∞)
10) |4 - x| + 1<3. (2,6)
11)|(3x-4)/2|≤ 5/12. (19/18,29/18)
12) |x -1|≤ 5, |x|≥ 2. (-∞,-2]U[2,∞)
13) 1≤| x-2|≤ 3. [-1,1]U[3,5]
14) |x- 1|≤5, | x |≥2. (-∞,-2]U[2,∞)
15) |x-2|/(x-2) > 0. (2,∞)
16) (|x| -1)/(|x| -2) ≥0, x∈R, x≠ ±2. [-1,1]U(-∞,-2]U(2,∞)
17) -1/(|x| -2) ≥ 1, where x∈R, x≠ ±2. [-2,-1]U[1,2)
18) |2/(x-4)|> 1, x ≠4. (2,4)U(4,6)
19) (|x+3| + x)/(x+2) > 1. (-5,-2) U(-1,∞)
20) 1/(|x| -3)< 1/2. (-∞,-5) U(-3, 3) U(5,∞)
21) (|x +2| - x)/x < 2. (-∞,2]U(1,∞)
22) |x -1| + |x-2|≥ 4. (-∞,-1/2]U [7/2,∞)
23) |x-1|+ |x-2| + |x -3|≥6. (-∞,0] U [4,∞).
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EXERCISE-4
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1) 3x² - 7x + 4≤0
A) x> 0 B) x<0 C) all x D) no solution E) none
2) 3x² - 7x - 6 <0
A) -0.66< x < 3 B) x<-0.66 or x> 3 C) 3< x < 7 D) -2< x< 2 E) none
3) 3x² - 7x + 6 <0
A) 0.66< x < 3 B) -0.66<x<3 C) -1< x < 3 D) -0.66< x< 0.66 E) none
4) x² - 3x +5>0
A) x> 0 B) x< 0 C) both A and B D) - ∞< x< ∞ E) none
5) x² - 14x - 15 <0
A) x < -1 B) 15< x C) both A and B D) -1< x< 15 E) none
6) 2 - x - x² ≥ 0
A) - 2≤x ≤ 1 B) -2< x<2 C) x< -2 D) x> 1 E) none
7) |x² - 4x| < 5
A) -1≤ x ≤5 B) 1≤ x ≤5 C) -1< x ≤ 1 D) - 1< x< 5 E) none
8) |x² +x | <0
A) x< 0 B) x> 0 C) all values of x D) x> 5 E) none
9) |x² - 5x| < 6
A) - 1< x < 2 B) 3< x< 6 C) both A and B D) - 1< x< 6 E) none
10) |x² - 2x| < x
A) 1< x < 3 B) -1< x< 3 C) 0< x <4 D) x > 3 E) none
11) |x² - 2x - 3| < 3x - 3
A) 1< x < 3 B) -2< x< 5 C) x> 5 D) 2< x< 5 E) none
12) |x² - 3x| + x - 2 < 0
A) (1 - √3) < x < (2+ √2) B) 0< x< 5 C) (1-√3),2-√2) D) 1< x< 4 E) none
13) x²- 7x +12< |x - 4|.
A) x < 2 B) x> 4 C) 2< x <4 D) 2≤ x≤ 4 E) 1< x< 4
14) x²- | 5x - 3| - x < 2
A) x >3+ 2√2 B) x< 3+2√2 C) x> - 5 D) - 5 < x< 3+2√2 E) none
15) |x - 6| > x²- 5x +9
A) 1 ≤ x < 3 B) 1< x< 3 C) 2< x< 5 D) -3< x< 1 E) none
16) |x - 6|< x²- 5 x +9
A) x < 1 B) x > 3 C) 1< x< 3 D) both A and B E) x≥ 1 & x ≥ 3
17) |x - 2| ≤ 2x²- 9x +9
A) (x > (4- √2)/2 B) x< (5+ 3√2)/2 C) both A and B D) x≥(4- √2)/2 E) none
18) 3x²- |x - 3| > 9 x - 2
A) x < (4- √19)/3 B) x>(4+√19)/3 C) both A and B D) 2< x< 2 E) none
19) x²- |5x + 8| > 0
A) x < (5- √57)/2 B) x<(5+√57)/2 C) x> (5+ √57)/2 D) x> (5-√57)/2 E) both A and B
20) 3 |x - 3| + x² - 7> 0
A) x> -1 B) x< -1 C) x > 2 D) both B and C E) none
21) |x - 6| > |x²- 5x +9 |
A) x < 1 B) x> 3 C) 1< x < 3 D) Both A and B E) none
22) |x - 3| - 3 (|x +2| -5)< 0
A) -7<x < -2 B) x<-7 and x> 4 C) x< -2 and x> 3 D) any of these E) none of these
23) |x² - x - 8| > 2x
A) x < 2√2 B) x< 3+ 3√5 C) x> 2+ 2√3 D) both A and C E) none
24) (x-1) √(x² - x - 2) ≥ 0
A) x≤ 2 B) x ≥ 2 C) x≤ - 2 D) x≥ 0 E) none
25) (x² -1) √(x² - x - 2) ≥ 0
A) x≤ -1 B) x ≥ -1 C) x≥2 D) A and C E) none
26) √{(x - 2)/(1- 2x)} ≥ -2
A) 0.5> x B) x > 2 C) both A and B D) 0.5 < x ≤ 2 E) none
27) √{(3x - 2)/(2- x)} > 1
A) 0 < x< 2 B) 0.75< x < 4C) 0.75< x < 2 D) 0< x < 4 E) none
28) √(3x - 10) > (6- x)
A) 4< x≤ 6 B) x< 4 or x > 6 C) x< 4 D) x > 8 E) 4< x < 8
29) √{(x² - 2x - 3)< 1
A) -1- √5< x < -3 B) 1≤ x <(√5-1) C) x > 1 D) both A and B E) none
30) √{ 1 - (x +2)/x²}< 2/3
A) -6/5 < x ≤ -1 or 2≤ x <3 B) -6/5 ≤ x < -1) C) 2≤ x < 3 D) -6/5 ≤ x < 3 E) none
31) 2 - √(x - 2)< x
A) x> 1 B) x≥ 1, x ≠ 2 C) x < 1 D) 1< x < 5 E) none
32) √{(x +18)< 2- x.
A) x≤ -18 B) x< -2 C) x > -2 D) -18≤ x < -2 E) none
33) x> √(24+ 5x(
A) x< 3 B) 3<x≤ 4.8 C) x ≥ 24/5 D) x> 8 E) none
34) √{(9x - 20)< x.
A) 4< x < 5 B) 20/9≤ x< 4 C) x > 5 D) both B and C E) none
35) √{(x +7 )< x.
A) x> 2 B) x > √30/2 C) x> (1+ √29)/2 D) x> 1+ √29/2 E) none
36) √{(2x - 1)< x- 2
A) x< 5 B) x> 5 C) x > 5 or x < -5 D) 5< x < 15 E) none
37) √{(x +78)< x + 6
A) x 3 B) x> 3 or x< 2 C) x > 3 D) 3< x < 10 E) none
38) √{(5 - 2x)< 6x - 1
A) 0.5< x B) x< 2.5 C) 0.5< x < 2.5 D) x > 2.5 E) none
39) √{(x +61)< x + 5
A) x < 3 B) x> 3 or x <1 C) x > 3 D) 3< x < 15 E) none
40) x < √(3 - x)
A) x> 1 B) x< 1 C) -2 <x < 1 D) -1 < x E) none
41) x + 3 < √(x + 33)
A) x> 3 B) x< 3 C) -4 <x < 3 D) -33 < x < 3 E) none
42) √(2x +14)> x +3
A) x< -7 B) -7≤ x< 1 C) x> 1 D) -7 < x< 1 E) none
43) x - 3 < √(x - 2)
A) 2≤x<(7+5√2)/2 B) 2≤ x C) x < (7+√5)/3 D) x ≤ 2 E) none
44) x + 2 < √(x + 14)
A) -14≤ x < 2 B) x> -14 C) x < 2 D) -1 < x< 2 E) none
45) x - 1 < √(7 - x)
A) x> 3 B) x< 3 C) -53 <x < 3 D) -103 < x < 3 E) none
46) √(3 - x) > x
A) x< 4 B) x > 5 C) x≤ 4 or x ≥ 5 D) 4 < x < 5 E) none
47) √(11 - 5x)> x - 1
A) x> 3, x < 5 B) -3 < x< 2 C) -25 <x < 2 D) x < 2 E) none
48) √(3 - x) > x
A) -2≤ x < 2 B) -2≤ x C) x < 2 D) x= -2 or x > 2 E) none
** Find the largest integral x that satisfies the following::::
49) (x -2)/(x² -9) < 0
A) x=-4 B) x =-2 C) x= 3 D) none
50) 1/(x +1 ) - 2/(x²- x +1) < (1- 2x)/(x³ +1)
A) x= 1 B) x = 2 C) x= -1 D) none
51) (x +4)/(x² -9) - 2/(x +3) < 4x/(3x - x²).
A) x= 1 B) x = 2 C) x= -1 D) none
52) (4x +19)/(x + 5) < (4x-17)/(x - 3).
A) x= 1 B) x = 2 C) x= -1 D) none
53) (x +1)(x -3)²(x -5)(x -4)²(x -2) < 0
A) x= 1 B) x = 2 C) x= -1 D) none
** Solve the following inequalities:
54) (x -1)(3- x)(x -2)² > 0
A) 1<x < 3 B) 1< x < 3 but x ≠ 2 C) 0< x < 2 D) -1< x < 3 E) none
55) (6x -5)/(4x+1) < 0
A) -1/4<x < 1 B) -1/2 < x < 1 C) -1< x < 1 D) -1/4 < x < 5/6 E) none
56) (2x -3)/(3x- 7) > 0
A) x < 3/2 B) 3/2< x < 7/3 C) x> 7/3 D) -1< x < 3 E) none
57) 3/(x -2) < 1
A) 2<x < 5 B) x < 2 C) x> 5 D) x< 2 or x > 5 E) none
58) 1/(x -1) ≤ 2
A) x < 1 B) x ≥ 1.5 C) -5< x < 1 D) both A and B E) none
59) (4x +3)/(2x-5) < 6
A) x < 2.5 B) x < 33/8 C) x≥ 2.5 D) x < 2.5 or x > 33/8 E) none
60) (5x -6)/(x + 6)< 1
A)-6 <x < 6 B) -6< x < 0 C) -6< x < 4 D) -1< x < 3 E) none
61) (5x +8)/(4- x)< 2
A) x<0 or x> 4 B) 0< x < 4 C) 0≤ x < 4 D) x > 4E) none
62) (x -1)/(x + 3) > 2
A) x<-7 B) x< -3 C) -7< x < -3 D) x > 4 E) none
63) (7x -5)/(8x + 3) > 4
A) -17/25 < x<-3/8 B) x> -17/25 C) 0< x < 3/8 D) -17/25 < x< 0 E) none
64) x/(x -5) > 1/2
A) -5 < x< 5 B) -5< x < 0 C) -5 ≤ x ≤ 5 D) x < -5 or x> 5 E) none
65) x ≤ 6/(x -5).
A) x< -1 B) x> 5 C) -x < 6 D) x ≤ -1 or 5< x ≤ 6 E) none
66) (30x -9)(x -2)≥ 25(x +2)
A) x< - 1.4 or x > 2 B) x< -1.4 or 2< x ≤ 2.6 C) x ≤ 1.4 or 2 < x ≤ 2.6 D) x < -5 or x> 5 E) none
67) 4/(x +2) > 3- x
A) -2 < x< -1 or x > 2 B) -2< x < 2 C) -2 < x < -1 D) 0< x < 3 E) none
68) x - 17 ≥ 60/x.
A) -x< -3 B) x < 20 C) -3 ≤ x < 0 D) -3< x ≤ 0 or x≥ 20 E) none
69) √x² < x +1
A) x > 0.5 B) x> 0 C) all x D) x>-0. 5 E) none
** Find the smallest integral x satisfying the inequalities:::
70) (x -5)/(x²+ 5x -14) > 0.
A) x= -6 B) x = -3 C) x = -7 D) x = -5 E) none
71) x² - 5|x| + 6 < 0
A) -3 < x< -2 B) 2< x < 3 C) both A and B D) -3< x < 3 E) -3< x< 3 F) none
72) x² - |x| -2 ≥ 0
A) -2 < x< 2 B) x≤ -2 or x≥ 2 C) x< -2 or x> 1 D) -2< x < 1 E) none
** Solve the inequalities:
73) (x -1)(3- x)(x -2)²> 0
A) 1< x < 2 B) -1< x < 3 C) -3<x < -1 D) 1< x < 3, x≠ 2 E) 1< x < 3
74) 0.5/(x - x² -1) < 0
A) x> 0 B) x ≤ 0 C) x ≥ 0 D) x < 0 E) for all real x
75) (x² - 5x +6)/(x²+x +1)< 0
A) x < 2 B) x > 3 C) 2 <x < 3 D) x< 2 or x > 3, x≠ 2 E) none
76) (x² +2x -3)/(x²+1)< 0
A) x < -3 B) -7<x <- 3 C) -3 <x < 2 D) -7< x< 1 E) none
77) (x -1)(x +2)²/(-1 - x)< 0
A) x < -2 B) x <-1 or x> 1 C) x<-2 and x ≠ 2 D) x< -2 or x > 1, x≠ 2 E) none
78) (x² +4x +4)/(2x²-x -1)> 0
A) x < -2 B) x > 1 C) x≠ 2 D) all of the above E) none
79) x⁴ - 5x² + 4 < 0
A) -2< x < 1 B) -2< x < 2 C) -2 <x < -1 or 1 < x < 2 D) 1< x < 2 E) none
80) x⁴ - 2x² - 63 ≤ 0
A) x≤ -3 or x≥ 3 B) -3≤ x ≤0 C) 0≤x ≤ 3 D) -3≤ x ≤3 E) none
81) (5x -1)/(x²+3)< 1
A) x< 4 B) 1< x < 4 C) x< 1 or x > 4 D) 1< x < 3 E) none
82) (x -2)/(x²+ 1) < -1/2
A) -3< x< 3 B) x < -3 C) -3< x< 1< 6 D) -3 < x < 2 E) 1< x < 6
83) (x +1)/(x - 1)² < 1
A) x > 3 or x is negative B) x> 3 C) x > 3 or -23< x < 0 D) x is negative and x > 2 E) x< 3
84) (x² - 7x +12)/(2x²+4x +5)> 0
A) x< 3 or x> 4 B) 3< x < 4 C) 4< x< 24 D) 0< x < 3 E) 0< x < 4
85) (x²+ 6x -7)/(x²+1)≤ 2
A) x is negative B) x≥ 0 C) x > 0 or x< 0 D) always E) never
86) (x⁴+x²+1)/(x²-4x - 5)< 0
A) x<-1 or x> 5 B) -1< x < 5 C) x > 5 D) -5< x < -1 E) -5< x < 6
87) (1+ 3x²)/(2x²- 21x +40)< 0
A) 0< x<8 B) 2.5 < x < 8 C) -8< x < 8 D) 3 < x < 8 E) none
88) (1 - x²)/(x²- 5x +6)< 0
A) x< 2 B) x> 3 C) both A and B D) 2 < x < 3 E) none
89) (x⁴+x²+1)/(x²-4x - 5)> 0
A) -1< x< 5 B) x <-1 or x> 5 C) x≤ -1 or x> 5 D) - 1 < x < 1 E) 1 < x < 5
90) (1- 2x - 3x²)/(3x - x² - 5)> 0
A) x<-1 or x> 1/3 B) x < -1 or x= 1/3 C) -1 < x < 13 D) x <1/3 E) none
91) (x² - 5x+7)/(-2x²+ 3x +2)> 0.
A) x>0. 5 B) x> 0.5 C) -0.5< x < 5 D) - 0.5< x < 2 E) 0.5< x < 2
92) (2x²- 3x- 459)/(x²+ 1)> 0
A) x> -20 B) x < 0 C) x <-20 D) -20< x < 20 E) 0 < x < 20
93) (x²- 1)/(x²+ x +1)< 1
A) x> -2 B) x> 2 C) -2< x < 2 D) x <2 E) none
94)
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