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

Question 1: State True or False:

\displaystyle \text{i)  }  x < -y \Rightarrow -x> y  \text{ : True }

\displaystyle \text{ii)  }  -5x \geq 15 \Rightarrow x \geq -3  \text{ : False }

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

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

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

\displaystyle \text{i) If  } a < b  \text{ , then } a-c <b-c  \text{ : True }

\displaystyle \text{ii) If  } a > b  \text{ , then } a+c > b+c  \text{ : True }

\displaystyle \text{iii) If  } a < b  \text{ , then } ac > bc  \text{ : False }

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

\displaystyle \text{v) If  } a-c > b-d ; then \displaystyle a+d > b+c  \text{ : True }

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

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

\displaystyle \text{i)  }  5x+3 \leq 2x+18     \displaystyle \text{ii)  }  3x-2 < 19-4x  

Answer

\displaystyle \text{i)  }  5x+3 \leq 2x+18

 \displaystyle \Rightarrow 3x \leq 15  

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

\displaystyle \text{ii)  }  3x-2 < 19-4x  

\displaystyle 3x-2 < 19-4x  

\displaystyle \Rightarrow 7x <21  

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

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

\displaystyle \text{i)  }  x+7 \leq 11           \displaystyle \text{ii)  }  3x-1 > 8           \displaystyle \text{iii)  }  x- \frac{3}{2} < \frac{3}{2} - x           \displaystyle \text{iv)  }  18 \leq 3x-2  

Answer

\displaystyle \text{i)  }  x+7 \leq 11  

\displaystyle x+7 \leq 11  

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

\displaystyle \text{ii)  }  3x-1 > 8  

\displaystyle 3x-1 > 8  

\displaystyle \Rightarrow 3x > 9  

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

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

\displaystyle x- \frac{3}{2} < \frac{3}{2} - x  

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

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

\displaystyle \text{iv)  }  18 \leq 3x-2  

\displaystyle 18 \leq 3x-2  

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

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

Answer

\displaystyle 3-2x \geq x-12  

\displaystyle \Rightarrow 3x \leq 15  

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

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

Answer

\displaystyle 25-4x \leq 16  

\displaystyle \Rightarrow 4x \geq 9  

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

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

\displaystyle \text{i)  }  -4x \geq -16           \displaystyle \text{ii)  }  8-3x \leq 20       \displaystyle \text{iii)  }  5+ \frac{x}{4} > \frac{x}{5} +9           \displaystyle \text{iv)  }  \frac{x+3}{8} < \frac{x-3}{5}  

Answer

\displaystyle \text{i)  }  -4x \geq -16  

\displaystyle -4x \geq -16  

\displaystyle \Rightarrow 4x \leq 16  

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

\displaystyle \text{ii)  }  8-3x \leq 20  

\displaystyle 8-3x \leq 20  

\displaystyle \Rightarrow 3x \geq -12  

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

\displaystyle \text{iii)  }  5+ \frac{x}{4} > \frac{x}{5} +9  

\displaystyle 5+ \frac{x}{4} > \frac{x}{5} +9  

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

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

\displaystyle \text{iv)  }  \frac{x+3}{8} < \frac{x-3}{5}  

\displaystyle \frac{x+3}{8} < \frac{x-3}{5}  

\displaystyle \Rightarrow 5x+15< 8x-24  

\displaystyle \Rightarrow 39 < 3x  

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

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

Answer

\displaystyle 5-2x<5 \frac{1}{2} - \frac{5}{3} x  

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

\displaystyle \Rightarrow 30-12x<33-10x  

\displaystyle \Rightarrow -3<2x  

\displaystyle \Rightarrow x> - \frac{3}{2}  

Therefore \displaystyle x = -1  

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

Answer

\displaystyle 2(x-1) \leq (9-x)  

\displaystyle \Rightarrow 2x-2 \leq 9-x  

\displaystyle \Rightarrow 3x \leq 11  

Therefore \displaystyle x =3  

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

Answer

\displaystyle 12+1 \frac{5}{6} x \leq 5+3x  

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

\displaystyle \Rightarrow 7 \leq \frac{7}{6} x  

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

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

Answer

\displaystyle -5 \leq 2x-3 < x+2  

Equation 1:\displaystyle -5 \leq 2x-3  

\displaystyle \Rightarrow -2 \leq 2x  

\displaystyle \Rightarrow -1 \leq x  

Equation 2: \displaystyle 2x-3 < x+2  

\displaystyle \Rightarrow x < 5  

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

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

Answer

\displaystyle -1 \leq 3+4x < 23  

Equation 1: \displaystyle -1 \leq 3+4x  

\displaystyle \Rightarrow -4 \leq 4x  

\displaystyle \Rightarrow -1 \leq x  

Equation 2: \displaystyle 3x+4 < 23  

\displaystyle \Rightarrow 4x < 20  

\displaystyle \Rightarrow x < 5  

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