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)  $14-2x<6$

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

 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)  $10-2(1+4x)<20$

 i) $9x-7 \leq 25+3x$ $\Rightarrow 6x \leq 32$ $\Rightarrow x \leq \frac{32}{6}$ Solution Set $= \{...-3, -2, -1, 0, 1, 2, 3, 4, 5...\}$ ii) $-17<9x-8$ $\Rightarrow 9x>-9$ $\Rightarrow x>-1$ Solution Set $= \{0, 1, 2, 3...\}$ iii) $-4(x+5)>10$ $\Rightarrow -4x>30$ $\Rightarrow x<\frac{(-15)}{2}$ Solution Set $= \{...-10, -9, -8 \}$ iv) $4-3x<13+x$ $\Rightarrow 4x>-9$ $\Rightarrow x>\frac{(-9)}{4}$ Solution Set $= \{-2, -1, 0, 1, 2, 3, ...\}$ v) $5-4x<10-x$ $\Rightarrow -5<3x$ $\Rightarrow x >\frac{(-5)}{3}$ Solution Set $= \{-1, 0, 1, 2, ...\}$ 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)  $\frac{1}{3}$ $(4x-1)+3\leq 4+$ $\frac{2}{5}$ $(6x+2)+$ $\frac{4}{5}$

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 $= \{-1,0,1\}$

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

$\Rightarrow{} 5<2x<10$

$\Rightarrow{}$ $\frac{5}{2}$ $

Solution Set $= \{3,4\}$

iv)  $-3<$ $\frac{x}{2}$ $- 1<1, x \in Z$

$\Rightarrow{} -2<$ $\frac{x}{2}$ $<2$

$\Rightarrow{} -4

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

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

$\Rightarrow{} -5<$ $\frac{2x}{5}$ $<-4$

$\Rightarrow{} 25<2x<-20$

$\Rightarrow{}$ $\frac{-25}{2}$ $

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

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

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 $= \{x \epsilon{} R: x\geq{}1\}$

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 $= \{x \in R: x\geq{}-2\}$