Question 1: If   x \in \{-3,-2,-1,0,1,2,3 \} , find the solution set of each of the following:

i)  x+2<1      ii)  2x-1 < 4      iii)  2/3 x<1      iv)  1-x>0

v)  3-5x<-1      vi)  2-3x>1      vii)  -6 \geq 2x-4      viii)  3x-5 \geq -12

ix)  14-2x<6

Answer:

i)

 

 

x+2<1

\Rightarrow x \leq -1

Therefore Solution set = \{-3,-2\}

ii)

 

 

 

2x-1 < 4

\Rightarrow 2x < 5

\Rightarrow  x < \frac{5}{2}

Therefore Solution set = \{-3,-2,-1,0,1,2 \}

iii)

 

 

 2/3 x<1

\Rightarrow x<3/2

Therefore Solution set = \{-3,-2,-1,0,1 \}

iv)

 

 

1-x>0

\Rightarrow x<1

Therefore Solution set = \{-3,-2,-1,0 \}

v)

 

 

 

3-5x<-1

\Rightarrow 5x>4

\Rightarrow x>4/5

Therefore Solution set = \{ 1,2,3 \}

vi)

 

 

 

2-3x>1

\Rightarrow 3x<1

\Rightarrow x<1/3

Therefore Solution set = \{-3,-2,-1,0 \} 

vii)

 

 

 

-6 \geq 2x-4

\Rightarrow 2x \leq 2

\Rightarrow x \leq 1

Therefore Solution set = \{-3,-2,-1,0,1 \}

viii)

 

 

 

3x-5 \geq -12

\Rightarrow 3x \geq -7

\Rightarrow x \geq (-7)/3

Therefore Solution set = \{-2,-1, 0, 1, 2, 3 \}

ix)

 

 

 

14-2x<6

\Rightarrow 2x>8

\Rightarrow x>4

Therefore Solution set = \phi

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Question 2: If x \in N ; find the solution set of each of the following in equation: N=\{1, 2, 3, ... \}

i)  3x-8<0      ii)  7x+3 \leq 17      iii)  5-x>1 

iv)  1-3x>-4       v)  \frac{3}{2}-\frac{x}{2}>-1      vi)  \frac{-1}{4} \leq \frac{1}{2} - \frac{2}{3}

Answer:

i)

 

 

 

3x-8<0 

\Rightarrow 3x<8 

\Rightarrow x<\frac{8}{3} 

Solution Set = \{1,2  \}

ii)

 

 

 

7x+3 \leq 17 

\Rightarrow 7x \leq 14 

\Rightarrow x \leq 2 

Solution Set = \{1,2 \}

iii)

 

 

5-x>1 

\Rightarrow x<4 

Solution Set = \{1,2,3 \}

iv)

 

 

 

1-3x>-4 

\Rightarrow 3x<5 

\Rightarrow x<\frac{5}{3} 

Solution Set = \{1 \}

v)

 

 

 

\frac{3}{2}-\frac{x}{2}>-1 

\Rightarrow \frac{x}{2}<\frac{5}{2}  

\Rightarrow  x<5 

Solution Set = \{1,2,3,4 \}

vi)

 

 

 

 

\frac{-1}{4} \leq \frac{1}{2} - \frac{2}{3}

\Rightarrow \frac{-3}{4} \leq \frac{-x}{3}  

\Rightarrow \frac{x}{3}  \leq \frac{3}{4}  

\Rightarrow  x \leq \frac{9}{4}

Solution Set = \{1,2 \}

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Question 3: If x \in Z , find the solution set of the following in equations:  Z = \{...-3, -2, -1, 0, 1, 2, 3, ...\}

i)  9x-7 \leq 25+3x       ii)  -17<9x-8      iii)  -4(x+5)>10

iv)  4-3x<13+x      v)  5-4x<10-x      i)   10-2(1+4x)<20

Answer:

i)

 

 

 

 

 

9x-7 \leq 25+3x 

\Rightarrow  6x \leq 32

 \Rightarrow  x \leq \frac{32}{6}

Solution Set = \{...-3, -2, -1, 0, 1, 2, 3, 4, 5...\}

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ii)

 

 

 

 

 

-17<9x-8

 \Rightarrow  9x>-9

 \Rightarrow  x>-1

Solution Set = \{0, 1, 2, 3...\}

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iii)

 

 

 

 

 

-4(x+5)>10

 \Rightarrow  -4x>30

 \Rightarrow  x<\frac{(-15)}{2}

Solution Set = \{...-10, -9, -8 \}

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iv)

 

 

 

 

 

4-3x<13+x

\Rightarrow  4x>-9

 \Rightarrow x>\frac{(-9)}{4}

Solution Set = \{-2, -1, 0, 1, 2, 3, ...\}

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v)

 

 

 

 

 

5-4x<10-x

 \Rightarrow  -5<3x

 \Rightarrow  x >\frac{(-5)}{3}

Solution Set = \{-1, 0, 1, 2, ...\}

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vi)

 

 

 

 

 

 

 10-2(1+4x)<20

  \Rightarrow  10-2-20<8x

  \Rightarrow  8x>-12

  \Rightarrow  x>-3/2

Solution Set = \{-1, 0, 1, 2, ...\}

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Question 4: Find the Solution Set of each of the following in equations and represent the solution on a real line.

i)  1-4x\geq{}-1, x \in N      ii)  -3\leq{}4x+1<9, x \in N

iii)    0<2x-5<5, x \in W \ \ \ \ \ W= \{0,1,2,3\ldots{}.\}      iv)  -3< \frac{x}{2} - 1<1, x \in Z

v)     -4< \frac{2x}{5} +1<-3, x\in Z      vi)     3+ \frac{x}{4} < \frac{2x}{3} +5, x\in R

vii)     \frac{(3x+1)}{4} \leq \frac{(5x-2)}{3} , x\in R      viii)  \frac{1}{3} (4x-1)+3\leq 4+ \frac{2}{5} (6x+2)+ \frac{4}{5}

Answer:

i)  1-4x\geq{}-1, x \in N

  \Rightarrow{} 4x\leq{}2

  \Rightarrow{} x\leq{} \frac{1}{2}

Solution Set = \varnothing{}

ii)  -3\leq{}4x+1<9, x \in N

  \Rightarrow{} -4\leq{}4x<8

  \Rightarrow{} -1\leq{}x<2

Solution Set = \{-1,0,1\}

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iii)    0<2x-5<5, x \in W \ \ \ \ \ W= \{0,1,2,3\ldots{}.\}

  \Rightarrow{} 5<2x<10

  \Rightarrow{} \frac{5}{2} <x<5

Solution Set = \{3,4\}

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iv)  -3< \frac{x}{2} - 1<1, x \in Z

  \Rightarrow{} -2< \frac{x}{2} <2

  \Rightarrow{} -4<x<4

Solution Set = \{-3,-2,-1,0,1,2,3\}

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v)    -4< \frac{2x}{5} +1<-3, x\in Z

  \Rightarrow{} -5< \frac{2x}{5} <-4

  \Rightarrow{} 25<2x<-20

  \Rightarrow{} \frac{-25}{2} <x<-10

Solution Set = \{-12,-11\}

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vi)    3+ \frac{x}{4} < \frac{2x}{3} +5, x\in R

  \Rightarrow{} \frac{x}{4} < \frac{2x}{3} +2

  \Rightarrow{} \frac{2x}{3} - \frac{x}{4} >-2

  \Rightarrow{} 5x>-24

  \Rightarrow{} x> \frac{(-24)}{5}

Solution Set = \{x\in R :x> \frac{(-24)}{5} \}

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vii)    \frac{(3x+1)}{4} \leq{} \frac{(5x-2)}{3} , x\in R

  \Rightarrow{} 9x+3\leq{}20x-8

  \Rightarrow{} 11x\geq{}11

  \Rightarrow{} x\geq{}1

Solution Set = \{x \epsilon{} R: x\geq{}1\}

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viii)    \frac{1}{3} (4x-1)+3\leq{}4+ \frac{2}{5} (6x+2)+ \frac{4}{5}

  \Rightarrow{} \frac{4}{3} x- \frac{1}{3} +3\leq{}4+ \frac{12}{5} x+ \frac{4}{5}

  \Rightarrow{} \frac{(-32)}{15} \leq{} \frac{16}{15} x

  \Rightarrow{} x\geq{}-2

Solution Set = \{x \in R: x\geq{}-2\}

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