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

 $i)$ $x+2<1$ $\Rightarrow x \leq -1$ $Therefore \ Solution \ set = \{-3,-2\}$ $ii)$ $2x-1 < 4$ $\Rightarrow 2x < 5$ $\Rightarrow x < \frac{5}{2}$ $Therefore \ Solution \ set = \{-3,-2,-1,0,1,2 \}$ $iii)$ $2/3 x<1$ $\Rightarrow x<3/2$ $Therefore \ Solution \ set = \{-3,-2,-1,0,1 \}$ $iv)$ $1-x>0$ $\Rightarrow x<1$ $Therefore \ Solution \ set = \{-3,-2,-1,0 \}$ $v)$ $3-5x<-1$ $\Rightarrow 5x>4$ $\Rightarrow x>4/5$ $Therefore \ Solution \ set =\{ 1,2,3 \}$ $vi)$ $2-3x>1$ $\Rightarrow 3x<1$ $\Rightarrow x<1/3$ $Therefore \ Solution \ set = \{-3,-2,-1,0 \}$ $vii)$ $-6 \geq 2x-4$ $\Rightarrow 2x \leq 2$ $\Rightarrow x \leq 1$ $Therefore \ Solution \ set = \{-3,-2,-1,0,1 \}$ $viii)$ $3x-5 \geq -12$ $\Rightarrow 3x \geq -7$ $\Rightarrow x \geq (-7)/3$ $Therefore \ Solution \ set =\{-2,-1, 0, 1, 2, 3 \}$ $xi)$ $14-2x<6$ $\Rightarrow 2x>8$ $\Rightarrow x>4$ $Therefore \ Solution \ set = \phi$

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

 $i)$ $3x-8<0$ $\Rightarrow 3x<8$ $\Rightarrow x<\frac{8}{3}$ $Solution \ Set = \{1,2 \}$ $ii)$ $7x+3 \leq 17$ $\Rightarrow 7x \leq 14$ $\Rightarrow x \leq 2$ $Solution \ Set = \{1,2 \}$ $iii)$ $5-x>1$ $\Rightarrow x<4$ $Solution \ Set = \{1,2,3 \}$ $iv)$ $1-3x>-4$ $\Rightarrow 3x<5$ $\Rightarrow x<\frac{5}{3}$ $Solution \ Set = \{1 \}$ $v)$ $\frac{3}{2}-\frac{x}{2}>-1$ $\Rightarrow \frac{x}{2}<\frac{5}{2}$ $\Rightarrow x<5$ $Solution \ Set = \{1,2,3,4 \}$ $vi)$ $\frac{-1}{4} \leq \frac{1}{2} - \frac{2}{3}$ $\Rightarrow \frac{-3}{4} \leq \frac{-x}{3}$ $\Rightarrow \frac{x}{3} \leq \frac{3}{4}$ $\Rightarrow x \leq \frac{9}{4}$ $Solution \ Set = \{1,2 \}$

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

 $i)$ $9x-7 \leq 25+3x$ $\Rightarrow 6x \leq 32$ $\Rightarrow x \leq \frac{32}{6}$ $Solution \ Set = \{...-3, -2, -1, 0, 1, 2, 3, 4, 5...\}$ $ii)$ $-17<9x-8$ $\Rightarrow 9x>-9$ $\Rightarrow x>-1$ $Solution \ Set =\{0, 1, 2, 3...\}$ $iii)$ $-4(x+5)>10$ $\Rightarrow -4x>30$ $\Rightarrow x<\frac{(-15)}{2}$ $Solution \ Set = \{...-10, -9, -8 \}$ $iv)$ $4-3x<13+x$ $\Rightarrow 4x>-9$ $\Rightarrow x>\frac{(-9)}{4}$ $Solution \ Set = \{-2, -1, 0, 1, 2, 3, ...\}$ $v)$ $5-4x<10-x$ $\Rightarrow -5<3x$ $\Rightarrow x >\frac{(-5)}{3}$ $Solution \ Set = \{-1, 0, 1, 2, ...\}$ $vi)$ $10-2(1+4x)<20$ $\Rightarrow 10-2-20<8x$ $\Rightarrow 8x>-12$ $\Rightarrow x>-3/2$ $Solution \ Set = \{-1, 0, 1, 2, ...\}$

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

$\ \ i)\ \ \ 1-4x\geq{}-1, x \in N$

$\Rightarrow{} 4x\leq{}2$

$\Rightarrow{} x\leq{} \frac{1}{2}$

$Solution \ Set = \varnothing{}$

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

$\Rightarrow{} -4\leq{}4x<8$

$\Rightarrow{} -1\leq{}x<2$

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

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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}

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

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

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

$\Rightarrow{} -4

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

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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}

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

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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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