Question 1: Solve the following inequation and graph the solution on a number line  2x- 5 \leq 5x+4 < 11 , where x \in I .     [2011]

Answer:

2x-5 \leq 5x+4 < 11 

2x-5 \leq 5x+4  or -9 \leq 3x   or -3 \leq x 

5x+4 < 11   or 5x < 7   or  x < \frac{7}{5} 

-3 \leq x <\frac{7}{5} 

Therefore x \in \{-3, -2, -1, 0, 1 \}  

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Question 2: Given that x \in I , solve the inequation and graph it on a number line: 3 \geq \frac{x-4}{2}+\frac{x}{3} \geq 2.     [2004]

Answer:

3 \geq \frac{x-4}{2}+\frac{x}{3} \geq 2

18 \geq 3(x-4)+2x \geq 12 

30 \geq 5x \geq 24 

6 \geq x \geq 4.8 

Therefore x \in \{5, 6 \} 

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Question 3: Given A = \{x: 11x-5 >  7x + 3, x \in R \} B = \{x: 18x-9  \geq 15+12x , x \in R \} . Find the range of the set A \cap B and represent it on a number line.      [2005]

Answer:

A: 11x-5 > 7x+3 

4x >8   or  x >2 

B: 18x-9 \geq 15+12x 

6x \geq 24   or  x \geq 4 

A \cap B = \{ x:  x \geq 4, x \in R \} 

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Question 4: Solve the given inequation and graph it on a number line: 2y-3 < y+1 \leq 4y+7, y \in R .     [2008]

Answer:

2y-3 < y+1 \leq 4y+7

2y-3 < y+1   or  y < 4 

y+1 \leq 4y+7   or  -6 \leq 3y   or  -2 \leq y 

Hence \{ x: -2 \leq y < 4, x \in R \} 

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Question 5: Solve the given inequation and graph it on a number line: -3 < -\frac{1}{2}-\frac{2x}{3} \leq \frac{5}{6}, x \in R .     [2010]

Answer:

-3 < -\frac{1}{2}-\frac{2x}{3} \leq \frac{5}{6}

-3 < -\frac{1}{2}-\frac{2x}{3} 

-18 < -3 -4x 

4x < 15   or  x < \frac{15}{4} 

-\frac{1}{2}-\frac{2x}{3} \leq \frac{5}{6}   or  -3-4x \leq 5 

-8 \leq 4x   or  -2 \leq x 

Therefore \{ x: -2 \leq x < \frac{15}{4}, x \in R \} 

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Question 6: Solve the given inequation and graph it on a number line: 4x-19 < \frac{3x}{5}-2 \leq -\frac{2}{5}+x, x \in R .     [2012]

Answer:

4x-19 < \frac{3x}{5}-2 \leq -\frac{2}{5}+x  

4x-19 < \frac{3x}{5}-2   or  20x-95 < 3x-10   or   17x < 85   or   x < 5 

\frac{3x}{5}-2 \leq -\frac{2}{5}+x   or   3x-10 \leq -2 +5x   or  -8 \leq 2x   or  -4 \leq x 

Therefore \{x :  -4 \leq x < 5, x \in R \} 

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Question 7: Solve the given inequation and graph it on a number line: -\frac{x}{3} \leq \frac{x}{2}-1\frac{1}{3} <\frac{1}{6}.x \in R .     [2013]

Answer:

-\frac{x}{3} \leq \frac{x}{2}-1\frac{1}{3} < \frac{1}{6}

-\frac{x}{3} \leq \frac{x}{2}-1\frac{1}{3} < \frac{1}{6} 

-2x \leq 3x-8 < 1   or -2x \leq 3x-8   or  8 \leq 5x 

\frac{8}{5} \leq x   or  3x-8 < 1   or  3x < 9   or   x < 3 

Therefore \{ x : \frac{8}{5} \leq x < 3,  x \in R \} 

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Question 8: Find the value of x which satisfies the inequation: -2\frac{5}{6} < \frac{1}{2} - \frac{2x}{3} \leq 2, x \in W .     [2014]

Answer:

-2\frac{5}{6} < \frac{1}{2} - \frac{2x}{3} \leq 2

-\frac{17}{6} < \frac{1}{2} -\frac{2x}{3} \leq 2 

-17 < 3-4x \leq 12   or -17 < 3-4x   or 4x < 20   or x < 5 

3-4x \leq 12   or -9 \leq 4x   or -2.25 \leq x 

Therefore \{x : -2.25 \leq x  < 5, x \in W \} 

x \in \{0, 1, 2, 3, 4\} 

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Question 9: 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 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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