Class 8: Linear Inequations – Exercise 25 – ICSE / ISC / CBSE Mathematics Portal for K12 Students

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

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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) $\displaystyle \frac{3}{2}-\frac{x}{2}>-1$ vi) $\displaystyle \frac{-1}{4} \leq \frac{1}{2} - \frac{2}{3}$

Answer:

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

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

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

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

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