Note: Refer to the following if you need clarifications. Reference on Numbers

Question 1:  State True or False:

i) x < -y \Rightarrow -x> y : True

ii) -5x \geq 15 \Rightarrow x \geq -3  : False

iii) 2x \leq -7 \Rightarrow \frac{2x}{-4} \geq \frac{-7}{-4}  : True

iv) 7 > 5 \Rightarrow \frac{1}{7} < \frac{1}{5}  : True

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Question 2: State True or False: Given that a, b, c \ and \  d , are real numbers and c \neq 0

i) If a < b , then a-c <b-c  : True

ii) If a > b , then a+c > b+c  : True

iii) If a < b , then ac > bc  :  False

iv) If a > b , then \frac{a}{c} < \frac{b}{c}  :  False

v) If a-c > b-d ; then a+d > b+c  : True

vi) If a < b, \ and \ c>0 , then a-c > b-c  :  False

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Question 3: If x \in N , find the solution set of the inequations:

i) 5x+3 \leq 2x+18 

Answer

 5x+3 \leq 2x+18 

 \Rightarrow 3x \leq 15 

 \Rightarrow x \leq 5 \ or \  x \in \{1, 2, 3, 4, 5 \} 

ii) 3x-2 < 19-4x 

Answer:

3x-2 < 19-4x 

\Rightarrow 7x <21  

\Rightarrow x < 3 \ or \  x \in \{1, 2 \}  

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Question 4: If the replacement set is a set of whole numbers, solve:

i) x+7 \leq 11 

Answer:

x+7 \leq 11 

\Rightarrow x \leq 4 \ or \  x \in \{0, 1 2, 3, 4 \}  

ii) 3x-1 > 8 

Answer:

3x-1 > 8 

\Rightarrow 3x > 9

\Rightarrow x > 3 \ or \  x \in \{4, 5, 6, ... \}

iii) x- \frac{3}{2} < \frac{3}{2} - x 

Answer:

x- \frac{3}{2} < \frac{3}{2} - x 

\Rightarrow 2x < \frac{3}{2} + \frac{3}{2} 

\Rightarrow 2x < 3 \ or \  x \in \{0, 1 \} 

iv) 18 \leq 3x-2 

Answer:

18 \leq 3x-2 

\Rightarrow 3x \geq 20 \ or \  x \in \{7, 8, 9, ... \}  

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Question 5: Solve the inequation: 3-2x \geq x-12     given that x \in N     [1987]

Answer:

3-2x \geq x-12  

\Rightarrow 3x \leq 15 

\Rightarrow x \leq 5 \ or \  x \in \{1, 2, 3, 4, 5 \} 

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Question 6: If 25-4x \leq 16    , find:   i) the smallest value of x    , when x     is a real number   ii) smallest value of x     when x     is an integer

Answer:

25-4x \leq 16  

\Rightarrow 4x \geq 9   

Therefore if x     is a real number the x=2.25    and if x     is an integer then x = 3   

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Question 7: If the replacement set is a set of real numbers, solve

i) -4x \geq -16 

Answer:

-4x \geq -16 

\Rightarrow 4x \leq 16 

\Rightarrow x \leq 4 or  \{x: x\in R \ and \  x \leq 4 \}

ii) 8-3x \leq 20 

Answer:

8-3x \leq 20 

\Rightarrow 3x \geq -12

\Rightarrow x \geq -4 \ or\  \{x: x\in R \ and \  x \geq 4 \}  

iii) 5+ \frac{x}{4} > \frac{x}{5} +9 

Answer:

5+ \frac{x}{4} > \frac{x}{5} +9 

\Rightarrow  \frac{x}{4} - \frac{x}{5} >4 

\Rightarrow  x > 80 \ or\  \{x: x\in R \ and \  x \geq 80  \}

iv) \frac{x+3}{8} < \frac{x-3}{5} 

Answer:

\frac{x+3}{8} < \frac{x-3}{5} 

\Rightarrow 5x+15< 8x-24  

\Rightarrow 39 < 3x  

\Rightarrow x>13  \ or\  \{x: x\in R \ and \  x \geq 13  \}  

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Question 8: Find the smallest value of x    for which 5-2x < 5 \frac{1}{2} - \frac{5}{3} x   , where x \in I   .

Answer:

5-2x<5 \frac{1}{2} - \frac{5}{3} x 

\Rightarrow 5-2x < \frac{11}{2} - \frac{5}{3} x 

\Rightarrow 30-12x<33-10x 

\Rightarrow -3<2x 

\Rightarrow x> - \frac{3}{2} 

Therefore x = -1 

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Question 9: Find the largest value of x    for which 2(x-1) \leq (9-x)    and x \in W   .

Answer:

2(x-1) \leq (9-x) 

\Rightarrow 2x-2 \leq 9-x  

\Rightarrow  3x \leq 11  

Therefore x =3  

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Question 10: Solve the inequation: 12+1 \frac{5}{6} x \leq 5+3x    and x \in R  .    [1999]

Answer:

12+1 \frac{5}{6} x \leq 5+3x 

 \Rightarrow 12+ \frac{11}{6} x \leq 5+3x 

 \Rightarrow 7 \leq \frac{7}{6} x 

 \Rightarrow x \geq 6  \ or\  \{x: x\in R \ and \  x \geq 6 \} 

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Question 11: Given x \in I   , find the solution set for -5 \leq 2x-3 < x+2 

Answer:

-5 \leq 2x-3 < x+2 

Equation 1:-5 \leq 2x-3 

\Rightarrow -2 \leq 2x 

\Rightarrow -1 \leq x 

Equation 2: 2x-3 < x+2 

\Rightarrow x < 5 

Therefore \{ x : x \in I \ and \  -1 \leq x < 5 \} \ or \  x \in \{-1, 0, 1, 2, 3, 4 \}

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Question 12: x \in W   , find the solution set for -1 \leq 3+4x < 23 

Answer:

-1 \leq 3+4x < 23 

Equation 1: -1 \leq 3+4x  

\Rightarrow -4 \leq 4x 

\Rightarrow -1 \leq x 

Equation 2: 3x+4 < 23  

\Rightarrow 4x < 20  

\Rightarrow x < 5  

Therefore \{ x: x \in W and -1 \leq x < 5 \}  \ or \ x \in \{0, 1, 2, 3, 4 \}