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

i) 2x-1 < 5

2x < 6 \text{ or }  x < 3

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ii) 3x+1 \geq -5 

3x \geq -5 \text{ or } 3x \geq -6 \text{ or } x \geq -2

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iii)  2 (2x-3) \leq 6 

2x-3 \leq 3 \text{ or } 2x \leq 6 \text{ 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  \text{ or } -1 \leq x < 3  

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ii) x-1 < 3-x \leq 5  

x-1 < 3-x  \text{ or } 2x < 4  \text{ or } x < 2  

3-x \leq 5  \text{ 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  \text{ or } x > 4  

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ii) 7-x \leq 2-6x  

5x \leq -5  \text{ 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  \text{ or } \displaystyle x > \frac{5}{2} 

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v) 1+x \geq 5x - 11  

12 \geq 4x  \text{ or } 3 \geq x  

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vi) \displaystyle \frac{2x+5}{3} > 3x-3  

2x+5 > 9x-9  \text{ or } 14 > 7x  \text{ 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 \text{ or } 2x < 4 \text{ or } x < 2

3-2x \leq 7 \text{ or } -3+2x \geq -7 \text{ or } 2x \geq -4 \text{ 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 \text{ or } -1 < x

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

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

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

Answer:

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

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

-8 \leq 3x+1 < 10

-8 \leq 3x+1 \text{ or } -9 \leq 3x \text{ or } -3 \leq x

3x+1 < 10 \text{ or } 3x < 9 \text{ or } x < 3

Therefore -3 \leq x < 3

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

Answer:

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

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

\displaystyle -12 \leq 3-4x \leq 11 \text{ or }  -15 \leq -4x  \text{ or }  -4x \leq 8 

4x \leq 15 \text{ or } 4x \geq -8 \text{ or } x \geq -2

Therefore

\displaystyle -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 \text{ or } -2 \leq 2x \text{ or } -1 \leq x

2x-3 < x+2 \text{ 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 \text{ or } 2x \leq 8 \text{ or } x \leq 4

5+3x \leq 4x+2 \text{ 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 \text{ or } x < 5

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

\displaystyle - \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 \text{ or } 2x < 16 \text{ or } x < 8

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

Therefore \displaystyle 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 \text{ or } 2x \leq 16 \text{ or } x \leq 8

3x+9 > 25 \text{ or } 3x > 16 \text{ or } \displaystyle 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)  \text{ or } 9 \geq 3x  \text{ or } 3 \geq x  

3-2x <-7  \text{ or } 2x > 10  \text{ 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 \text{ or } 3-2x \geq -7; x \in R  

3x-2 > 19  \text{ or } 3x > 21  \text{ or } x > 7  

3-2x \geq -7  \text{ or } 2x \leq 10  \text{ or } x \leq 5  

Therefore x \leq 5 \text{ or } x > 7  

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

5 > p-1 > 2  \text{ or } 6 > p > 3  

7 \leq 2p-1 \leq 17  \text{ or }  8 \leq 2p \leq 8  \text{ 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 \text{ and }   0 < x < 6

  3<x< 6 

ii)   x < 0 \text{ and }   -3 \leq x < 1

  -3 \leq x < 0 

iii)  -1 < x \leq 6  \text{ 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 \text{ 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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