Question 18:  P  is the solution set of  7x-2 > 4x+1   and Q  is the solutions et of  9x-45 \leq 5(x-5)  ; where  x \in R  . Represent  i)  P \cap Q    ii)  P-Q     iii)  P \cap Q^{'}  on different number lines.

Answer:

P: 7x-2 > 4x+1 

3x > 3   or  x > 1 

Q:  9x-45 \geq 5(x-5) 

9x-45 \geq 5x-25   or  4x \geq 20   or   x \geq 5 

P \cap Q: x \geq 5 

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P-Q: 1 < x  < 5 

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P \cap Q^{'}: 1 < x < 5 

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Question 19: If  P = \{x:7x-4 > 5x+2, x \in R \}  and  Q = \{x : x-19 \geq 1-3x, x \in R \}  ; find the range of the set P \cap Q and represent it on number line.

Answer:

P: 7x-4 > 5x+2   or  2x > 6   or  x > 3 

Q: x-19 \geq 1-3x   or  4x \geq 20   or  x \geq 5 

P \cap Q: x \geq 5 

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Question 20: Find the range of values of x, which satisfy: -\frac{1}{3} \leq \frac{x}{2} +1\frac{2}{3} < 5\frac{1}{6}  Graph in each of the following cases for the values of x for each of the following cases: i)  x \in W    ii) x \in Z    iii)  x \in R 

Answer:

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

-2 \leq 3x+10 < 31   or  -12 \leq 3x < 21   or  -4 \leq x < 7 

i)  x \in W   : 0 \leq x  < 7 

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ii) x \in Z :   -4 \leq x < 7 

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iii)  x \in R : -4 \leq x < 7 

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Question 21: Given A = \{x: -8 < 5x+2 \leq 17, x \in I \} B = \{x: -2 \leq 7+3x < 17, x \in R \}  where R    is real numbers and I    is integers. Represent A \ and \ B  on two different number lines. Also write down the elements of A \cap B .

Answer:

A: -8 < 5x+2 \leq 17   or  -10 <5x \leq 15   or  -2 < x \leq 3 

B: -2 \leq 7+3x < 17   or  -9 \leq 3x < 10   or  -3 \leq < \frac{10}{3} 

A \cap B: -2 \leq x \leq 3 

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Question 22: 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 23: 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 24: 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 25: Find the set of values of x , satisfying: 7x+3 \geq 3x-5  and \frac{x}{4}-5 \leq \frac{5}{4}-x , where x \in N .

Answer:

7x+3 \geq 3x-5 

4x \geq -8   or  x \geq -2 

\frac{x}{4}-5 \leq \frac{5}{4}-x   or  \frac{5}{4} x \leq \frac{25}{4}   or  x \leq 5 

Therefore x \in \{1, 2, 3, 4, 5\} 

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Question 26: Solve  i)   \frac{x}{2} +5  \leq \frac{x}{3}+6 , where x is a positive odd integer. ii) \frac{2x+3}{3} \geq \frac{3x-1}{4} , where  x is a positive even integer.

Answer:

i)   \frac{x}{2} +5  \leq \frac{x}{3}+6 

\frac{1}{6} x \leq 1   or  x \leq 6 

x \in \{1, 3, 5 \} 

ii) \frac{2x+3}{3} \geq \frac{3x-1}{4} 

8x+12 \geq 9x-3   or  15 \geq x 

x \in \{2, 4, 6, 8, 10, 12, 14 \} 

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Question 27: Solve the inequation: -2\frac{1}{2}+2x \leq \frac{4x}{5} \leq \frac{4}{3}+2x, x \in W . Graph the solution on a number line.

Answer:

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

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

-\frac{5}{4}+2x \leq \frac{4x}{5} 

-25+20x \leq 8x   or  12x \leq 25   or  x \leq \frac{25}{12} 

\frac{4x}{5} \leq \frac{4}{3}+2x 

-\frac{4}{3} \leq 2x-\frac{4x}{5}   or   -\frac{4}{3} \leq \frac{6}{5}x   or   -\frac{20}{18} \leq x 

Therefore x \in {0, 1, 2} 

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Question 28: Find three consecutive largest positive integers such that the sum of one third of the first, one fourth of the second and one fifth of the third is 20.

Answer:

Let the three numbers be x, (x+1) and (x+2) 

Therefore

\frac{1}{3}x+\frac{1}{4}(x+1)+\frac{1}{5}(x+2) \leq 20 

\frac{20+15+12}{60}x+\frac{13}{20} \leq 20 

47x+39 \leq 1200 

x \leq 24.70 

Hence x = 24 . Therefore the numbers are 24, 25, and 26 .

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Question 29: 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 30: Solve the inequation: 3z-5 \leq z+3 < 5z-9, z \in R . Graph the solution set on a number line.

Answer:

3z-5 \leq z+3 < 5z-9

3z-5 \leq z+3   or  2z \leq 8   or   z \leq 4 

z+3 < 5z-9   or 12 < 4z   or  3 < z 

Hence \{ z: 3 < z  \leq 4, z \in R \} 

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Question 31: 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 32: 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 33: 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 34: 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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