Solve each of the following system of inequations in R.

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

Answer:

\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

Answer:

\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

Answer:

\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) .

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

Answer:

\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

Answer:

\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

Answer:

\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) .

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

Answer:

\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} ) .

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

Answer:

\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) .

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

Answer:

\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) .

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

Answer:

\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] .

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

Answer:

\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.

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

Answer:

\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) .

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

Answer:

\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) .

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

Answer:

\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] .

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

Answer:

\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.

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

Answer:

\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

Answer:

\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 .

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

Answer:

\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) .

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

Answer:

\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 ] .

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

Answer:

\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) .

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

Answer:

\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] .