Solve each of the following system of inequations in R.

Question 1: $x+ 3 > 0, \hspace{0.3cm} 2x < 14$

Given: $x+ 3 > 0, \hspace{0.3cm} 2x < 14$

$x+ 3 > 0 \Rightarrow x > -3$   … … … … … i)

$2x < 14 \Rightarrow x < 7$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( -3, 7)$.

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Question 2: $2x - 7 > 5 - x, \hspace{0.3cm} 11 - 5x \leq 1$

Given: $2x - 7 > 5 - x, \hspace{0.3cm} 11 - 5x \leq 1$

$2x - 7 > 5 - x \Rightarrow 3x > 12 \Rightarrow x > 4$   … … … … … i)

$11 - 5x \leq 1 \Rightarrow 5x \geq 10 \Rightarrow x \geq 2$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( 4, \infty)$.

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Question 3: $x-2 > 0, \hspace{0.3cm} 3x < 18$

Given: $x-2 > 0, \hspace{0.3cm} 3x < 18$

$x-2 > 0 \Rightarrow x > 2$   … … … … … i)

$3x < 18 \Rightarrow x < 6$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( 2,6)$.

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Question 4: $2x+6 \geq 0, \hspace{0.3cm} 4x - 7 < 0$

Given: $2x+6 \geq 0, \hspace{0.3cm} 4x - 7 < 0$

$2x+6 \geq 0 \Rightarrow 2x \geq -6 \Rightarrow x \geq -3$   … … … … … i)

$4x - 7 < 0 \Rightarrow 4x < 7 \Rightarrow x <$ $\frac{7}{4}$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( -3,$ $\frac{7}{4}$ $)$.

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Question 5: $3x-6 > 0, \hspace{0.3cm} 2x - 5 > 0$

Given: $3x-6 > 0, \hspace{0.3cm} 2x - 5 > 0$

$3x-6 > 0 \Rightarrow 3x > 6 \Rightarrow x > 2$   … … … … … i)

$2x - 5 > 0 \Rightarrow 2x > 5 \Rightarrow x >$ $\frac{5}{2}$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $($ $\frac{5}{2}$ $, \infty )$.

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Question 6: $2x-3 < 7, \hspace{0.3cm} 2x > - 4$

Given: $2x-3 < 7, \hspace{0.3cm} 2x > - 4$

$2x-3 < 7 \Rightarrow 2x < 10 \Rightarrow x < 5$   … … … … … i)

$2x > - 4 \Rightarrow x > - 2$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( -2,5)$.

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Question 7: $2x+5 \leq 0, \hspace{0.3cm} x - 3 \leq 0$

Given: $2x+5 \leq 0, \hspace{0.3cm} x - 3 \leq 0$

$2x+5 \leq 0 \Rightarrow 2x \leq -5 \Rightarrow x \leq$ $\frac{-5}{2}$   … … … … … i)

$x - 3 \leq 0 \Rightarrow x \leq 3$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( - \infty ,$ $\frac{-5}{2}$ $)$.

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Question 8: $5x - 1 < 24, \hspace{0.3cm} 5x + 1 > - 24$

Given: $5x - 1 < 24, \hspace{0.3cm} 5x + 1 > - 24$

$5x - 1 < 24 \Rightarrow 5x < 25 \Rightarrow x < 5$   … … … … … i)

$5x + 1 > - 24 \Rightarrow 5x > - 25 \Rightarrow x > - 5$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( -5,5)$.

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Question 9: $3x-1 \geq 5, \hspace{0.3cm} x+2 > - 1$

Given: $3x-1 \geq 5, \hspace{0.3cm} x+2 > - 1$

$3x-1 \geq 5 \Rightarrow 3x \geq 6 \Rightarrow x \geq 2$   … … … … … i)

$x+2 > - 1 \Rightarrow x > - 3$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $[ 2, \infty)$.

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Question 10: $11-5x > - 4, \hspace{0.3cm} 4x + 13 \leq -11$

Given: $11-5x > - 4, \hspace{0.3cm} 4x + 13 \leq -11$

$11-5x > - 4 \Rightarrow 15 > 5x \Rightarrow x < 3$   … … … … … i)

$4x + 13 \leq -11 \Rightarrow 4x \leq - 24 \Rightarrow x \leq - 6$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( - \infty , -6]$.

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Question 11: $4x -1 \leq 0, \hspace{0.3cm} 3-4x < 0$

Given: $4x -1 \leq 0, \hspace{0.3cm} 3-4x < 0$

$4x -1 \leq 0 \Rightarrow 4x \leq 1 \Rightarrow x \leq$ $\frac{1}{4}$   … … … … … i)

$3-4x < 0 \Rightarrow 4x > 3 \Rightarrow x >$ $\frac{3}{4}$   … … … … … ii)

There is no solution for the simultaneous inequations.

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Question 12: $x+5 > 2 ( x + 1), \hspace{0.3cm} 2 - x < 3 ( x+2)$

Given: $x+5 > 2 ( x + 1), \hspace{0.3cm} 2 - x < 3 ( x+2)$

$x+5 > 2 ( x + 1) \Rightarrow x+5 > 2x+2 \Rightarrow x < 3$   … … … … … i)

$2 - x < 3 ( x+2) \Rightarrow 2-x < 3x+6 \Rightarrow x > - 1$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( -1 , 3)$.

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Question 13: $2(x -6) < 3x-7, \hspace{0.3cm} 11 - 2x < 6 - x$

Given: $2(x -6) < 3x-7, \hspace{0.3cm} 11 - 2x < 6 - x$

$2(x -6) < 3x-7 \Rightarrow 2x-12 < 3x-7 \Rightarrow x > -5$   … … … … … i)

$11 - 2x < 6 - x \Rightarrow 2x-x > 11 - 6 \Rightarrow x > 5$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( 5 , \infty)$.

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Question 14: $5x-7 < 3 ( x + 3), \hspace{0.3cm} 1 -$ $\frac{3x}{2}$ $\geq x-4$

Given: $5x-7 < 3 ( x + 3), \hspace{0.3cm} 1 -$ $\frac{3x}{2}$ $\geq x-4$

$5x-7 < 3 ( x + 3) \Rightarrow 5x-7 < 3x+9 \Rightarrow x < 8$   … … … … … i)

$1 -$ $\frac{3x}{2}$ $\geq x-4 \Rightarrow x+$ $\frac{3x}{2}$ $\leq 5 \Rightarrow x \leq 2$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( - \infty , 2]$.

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Question 15: $\frac{2x-3}{4}$ $- 2 \geq$ $\frac{4x}{3}$ $- 6, \hspace{0.3cm} 2 ( 2x + 3) < 6(x-2)+10$

Given: $\frac{2x-3}{4}$ $- 2 \geq$ $\frac{4x}{3}$ $- 6, \hspace{0.3cm} 2 ( 2x + 3) < 6(x-2)+10$

$\frac{2x-3}{4}$ $- 2 \geq$ $\frac{4x}{3}$ $-6$

$\Rightarrow$ $\frac{2x-3-8}{4}$ $\geq$ $\frac{4x-18}{3}$

$\Rightarrow 3( 2x-11) \geq 4( 4x-18)$

$\Rightarrow 6x - 33 \geq 16x - 72$

$\Rightarrow 6x - 16 x \geq 33 - 72$

$\Rightarrow -10 x \geq - 39$

$\Rightarrow x \leq$ $\frac{39}{10}$    … … … … … i)

$2( 2x+3) < 6 ( x - 2) + 10$

$\Rightarrow 4x + 6 < 6x - 12 + 10$

$\Rightarrow 4x - 6x < - 12 - 6 + 10$

$\Rightarrow -2x < -8$

$\Rightarrow x > 8$    … … … … … ii)

There is no solution for the simultaneous inequations.

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Question 16: $\frac{7x-1}{2}$ $< -3, \hspace{0.3cm}$ $\frac{3x+8}{5}$ $+ 11 < 0$

Given: $\frac{7x-1}{2}$ $< -3, \hspace{0.3cm}$ $\frac{3x+8}{5}$ $+ 11 < 0$

$\frac{7x-1}{2}$ $< -3 \Rightarrow 7x-1 < -6 \Rightarrow x < \frac{-5}{7}$   … … … … … i)

$\frac{3x+8}{5}$ $+ 11 < 0 \Rightarrow 3x+63 < 0 \Rightarrow x < -21$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( - \infty , 21)$.

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Question 17: $\frac{2x+1}{7x-1}$ $> 5, \hspace{0.3cm}$ $\frac{x+7}{x-8}$ $> 2$

Given: $\frac{2x+1}{7x-1}$ $> 5, \hspace{0.3cm}$ $\frac{x+7}{x-8}$ $> 2$

$\frac{2x+1}{7x-1}$ $> 5 \Rightarrow$ $\frac{2x+1}{7x-1}$ $- 5 > 0 \Rightarrow$ $\frac{2x+1 - 35x + 5}{7x-1}$ $> 0 \Rightarrow$ $\frac{-33x+6}{7x-1}$ $> 0$

Case I:  $-33x+6 > 0 \hspace{1.0cm} \& \hspace{1.0cm} 7x-1 > 0$

$\Rightarrow x <$ $\frac{6}{33}$                   $\& \hspace{1.0cm} x >$ $\frac{1}{7}$

Case II: $-33x+6 < 0 \hspace{1.0cm} \& \hspace{1.0cm} 7x-1 > 0$

$\Rightarrow x >$ $\frac{6}{33}$                   $\& \hspace{1.0cm} x <$ $\frac{1}{7}$

Therefore the solution set for the inequation is $($ $\frac{1}{7}$ $,$ $\frac{6}{33}$ $)$

$\frac{x+7}{x-8}$ $> 2 \Rightarrow$ $\frac{x+7}{x-8}$ $- 2 > \Rightarrow$ $\frac{x+7-2x+16}{x-8}$ $\Rightarrow$ $\frac{-x+23}{x-8}$ $> 0$

Case I:  $-x+23 > 0 \hspace{1.0cm} \& \hspace{1.0cm} x-8 > 0$

$\Rightarrow x < 23$                    $\& \hspace{1.0cm} x > 8$

Case II: $-x+23 < 0 \hspace{1.0cm} \& \hspace{1.0cm} x-8 < 0$

$\Rightarrow x > 23$                    $\& \hspace{1.0cm} x < 8$

Therefore the solution set for the inequation is $( 8, 23 )$

We see that there is no common solution between $($ $\frac{1}{7}$ $,$ $\frac{6}{33}$ $)$ and $( 8, 23 )$.

Therefore we cans say that there is no solution for the simultaneous inequations $\frac{2x+1}{7x-1}$ $> 5, \hspace{0.3cm}$ $\frac{x+7}{x-8}$ $> 2$ .

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Question 18: $0 <$ $\frac{-x}{2}$ $< 3$

Given: $0 <$ $\frac{-x}{2}$ $< 3$

$\frac{-x}{2} > 0 \Rightarrow x < 0$   … … … … … i)

$\frac{-x}{2} < 3 \Rightarrow -x < 6 \Rightarrow x > - 6$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( - 6 , 0)$.

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Question 19: $10 \leq -5(x-2) <20$

Given: $10 \leq -5(x-2) <20$

$-5(x-2) \geq 10 \Rightarrow -5x+10 \geq 10 \Rightarrow -5x \geq 0 \Rightarrow x \leq 0$   … … … … … i)

$-5(x-2) < 20 \Rightarrow -5x+10 < 20 \Rightarrow -5x < 10 \Rightarrow x > -2$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( - 2 , 0 ]$.

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Question 20: $-5 <2x - 3 < 5$

Given: $-5 <2x - 3 < 5$

$-5 < 2x-3 \Rightarrow -2 < 2x \Rightarrow x > -1$    … … … … … i)

$2x-3 < 5 \Rightarrow 2x<8 \Rightarrow x < 4$   … … … … … ii)

From i) and ii) solution set for the simultaneous inequations is $( - 1 , 4)$.

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Question 21: $\frac{4}{x+1}$ $\leq 3 \leq$ $\frac{6}{x+1}$ $, x > 0$

$\frac{4}{x+1}$ $\leq 3 \leq$ $\frac{6}{x+1}$ $, x > 0$

$\Rightarrow 4 \leq 3(x+1) \leq 6 , x > 0$

$\Rightarrow 4 \leq 3x +3 \leq 6$

$\Rightarrow 1 \leq 3x \leq 3$

$\Rightarrow$ $\frac{1}{3}$ $\leq x \leq 1$

Therefore solution set for the simultaneous inequations is $\Big[$ $\frac{1}{3}$ $, 1 \Big]$.

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