Question 1 : Represent the following inequalities on real number lines:

i) 2x-1 < 5

 2x < 6 or  x < 3

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ii) 3x+1 \geq -5 

 3x \geq -5 or  3x \geq -6 or  x \geq -2

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iii)  2 (2x-3) \leq 6 

 2x-3 \leq 3 or  2x \leq 6 or  x \leq 3

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iv)  -4 < x < 4 

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v)  -2 \leq x < 5 

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vi)  8 \geq x > -3 

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vii)  -5 < x \leq -1 

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Question 2: For each graph given alongside, write an inequation taking x as the variable:

i) x \leq -1

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ii) x \geq 2

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iii) -4 \leq x < 3

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iv) 5 \geq x > -1

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Question 3: For the following inequations, graph the solution set on the real number line:

i) -4 \leq 3x-1 < 8   

 -3 \leq 3x < 9   or  -1 \leq x < 3  

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ii) x-1 < 3-x \leq 5  

 x-1 < 3-x     or  2x < 4     or  x < 2  

 3-x \leq 5     or    -2 \leq x  

Hence -2 \leq x < 2  

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Question 4: Represent the solution of each of the following inequalities on a real number line:

i) 4x-1 > x + 11  

 3x > 12   or  x > 4  

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ii) 7-x \leq 2-6x  

 5x \leq -5   or  x \leq -1  

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iii) x+3 \leq 2x+9  

 -6 \leq x  

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iv) 2-3x > 7-5x  

 2x > 5   or  x > \frac{5}{2}  

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v) 1+x \geq 5x - 11  

 12 \geq 4x   or  3 \geq x  

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vi) \frac{2x+5}{3} > 3x-3  

 2x+5 > 9x-9   or  14 > 7x   or  2 > x  

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Question 5: x \in \{real \ numbers \} \ and \ -1 < 3-2x \leq 7 , evaluate x and represent it on a number line.

Answer:

-1 < 3-2x \leq 7

-1 < 3-2x   or 2x < 4    or  x < 2

3-2x \leq 7   or -3+2x \geq -7    or 2x \geq -4   or x \geq -2

2 \leq x < 2

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Question 6:  List the elements of the solution set of the inequation -3 <x-2 \leq 9-2x, x \in N .

Answer:

 -3 <x-2 \leq 9-2x

 -3 < x-2   or  -1 < x

 x-2 \leq 9-2x   or   3x \leq 11    or     x \leq \frac{11}{3}

Hence x \in \{1, 2, 3 \}

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Question 7: Find the range of values of x which satisfies  -2\frac{2}{3} \leq x+\frac{1}{3} < 3\frac{1}{3}, \ x \in R

Answer:

2\frac{2}{3} \leq x+\frac{1}{3} < 3\frac{1}{3}

-\frac{8}{3} \leq x+\frac{1}{3} < \frac{10}{3}

-8 \leq 3x+1 < 10

-8 \leq 3x+1  or   -9 \leq 3x  or   -3 \leq x

3x+1 < 10  or   3x < 9  or   x < 3

Therefore -3 \leq x < 3

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Question 8: Find the range of values of x which satisfies -2 \leq \frac{1}{2} -\frac{2x}{3} \leq 1\frac{5}{6}, \ x \in N . Graph the solution on a number line.

Answer:

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

 -2 \leq \frac{1}{2}-\frac{2x}{3} \leq \frac{11}{6}

 -12 \leq 3-4x \leq 11  or  -12 \leq 3-4x4x \leq 15  or    x \leq \frac{15}{4}

 3-4x \leq 11  or   -8 \leq 4x  or    -2 \leq x

Therefore

 -2 \leq x \leq \frac{15}{4}

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Question 9: Given x \in  \{ real \ number \} , find the range of the values of x  for which  -5 \leq 2x-3 < x+2   and represent it on number line.

Answer:

  -5 \leq 2x-3 < x+2

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

  2x-3 < x+2  or     x < 5

Therefore  -1 \leq x < 5

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Question 10: If  5x-3 \leq 5+3x \leq 4x+2  , express it as  a \leq x \leq b   and state the values of  a \ and \ b  .

Answer:

  5x-3 \leq 5+3x \leq 4x+2

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

  5+3x \leq 4x+2   or     3 \leq x

Therefore   3 \leq x \leq 4

Hence a = 3 \ and \ b = 4

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Question 11: Solve the following inequation and graph the solution on a number line:   2x-3 < x+2 \leq 3x+5; x \in R

Answer:

 2x-3 < x+2 \leq 3x+5

 2x-3 < x+2   or     x < 5

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

 -\frac{3}{2} \leq x < 5

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Question 12: Solve and graph the solution set of the following:

Answer:

i) 2x-9 < 7 \ and \  3x+9 \leq 25; x \in R 

2x-9 < 7    or    2x < 16    or    x < 8 

 3x+9 \leq 25    or     3x \leq 16    or    x \leq \frac{16}{3} = 5\frac{1}{3} 

Therefore x \leq 5\frac{1}{3} 

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ii) 2x-9 \leq 7 \ and \  3x+9 > 25; x \in I 

 2x-9 \leq 7   or    2x \leq 16   or    x \leq 8 

 3x+9 > 25   or  3x > 16   or    x >5\frac{1}{3} 

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iii) x+5 \geq 4(x-1) \ and \  3-2x <-7; x \in R 

 x+5 \geq 4(x-1)    or    9 \geq 3x    or   3 \geq x  

 3-2x <-7    or    2x > 10    or    x > 5  

Therefore solution set is Empty Set.

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Question 13:  Solve and graph the solution set of:

Answer:

i) 3x-2 > 19 \ or \ 3-2x \geq -7; x \in R  

 3x-2 > 19     or    3x > 21    or    x > 7  

 3-2x \geq -7    or   2x \leq 10    or   x \leq 5  

Therefore x \leq 5 or x > 7  

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ii)  5 > p-1 > 2 \ or \ 7 \leq 2p-1 \leq 17; p \in R  

 5 > p-1 > 2    or   6 > p > 3  

 7 \leq 2p-1 \leq 17    or   8 \leq 2p \leq 8    or   4 \leq p \leq 8  

Therefore 3 < p \leq 8  

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Question 14: Given    A = \{ x \in R:  -2 \leq x < 5 \}  \ and \ B = \{x \in R: -4 \leq x < 3 \} .  Represent   A \cap B   and   A \cap B^{'}   on two different number lines.

Answer:

 A = -2 \leq x < 5 

 B = -4 \leq x < 3 

 A \cap B = \{ x: -2 \leq x < 3, x \in R \}

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 B^{'} x \leq -4 or x \geq 3 

Therefore A \cap B^{'} = \{x: 3 \leq x < 5,   x \in R \}  

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Question 15: Use real number line to find the range of value of x for which:

Answer:

i)   x > 3 \ and \  0 < x < 6 

   3<x< 6 

ii)   x < 0 \ and \   -3 \leq x < 1 

   -3 \leq x < 0 

iii)  -1 < x \leq 6  \ and \   -2 \leq x \leq 3 

   -1 < x \leq 3 

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Question 16: Illustrate the set   \{x:-3 \leq x < 0 \ or \ x >2; x \in R \}  on a real number line.

Answer:

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Question 17: Given  A =\{x:-1<x\leq5, x \in R \}  and  B=\{ x:4 \leq x <3, x \in R\} . Represent on different number lines

Answer:

i)  A \cap B     ii)  A^{'} \cap B      iii)  A - B 

A \cap B : -1 < x < 3 

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A^{'}= x > 5 or x \leq -1 

A^{'} \cap B: -4 \leq x \leq -1 

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A-B : 3 \leq x \leq 5 

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