Solve each of the following system of inequations in R.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

There is no solution for the simultaneous inequations.

$\displaystyle \\$

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

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

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

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

$\displaystyle \Rightarrow -10 x \geq - 39$

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

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

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

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

$\displaystyle \Rightarrow -2x < -8$

$\displaystyle \Rightarrow x > 8$ … … … … … ii)

There is no solution for the simultaneous inequations.

$\displaystyle \\$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\displaystyle \\$

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

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

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

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

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

$\displaystyle \\$

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

$\displaystyle \text{Given: } 10 \leq -5(x-2) <20$

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

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

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

$\displaystyle \\$

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

$\displaystyle \text{Given: } -5 <2x - 3 < 5$

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

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

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

$\displaystyle \\$

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

$\displaystyle \frac{4}{x+1} \leq 3 \leq \frac{6}{x+1} , x > 0$
$\displaystyle \Rightarrow 4 \leq 3(x+1) \leq 6 , x > 0$
$\displaystyle \Rightarrow 4 \leq 3x +3 \leq 6$
$\displaystyle \Rightarrow 1 \leq 3x \leq 3$
$\displaystyle \Rightarrow \frac{1}{3} \leq x \leq 1$
$\displaystyle \text{Therefore solution set for the simultaneous inequations is } \Big[ \frac{1}{3} , 1 \Big] .$