Solve each of the following system of inequations in R.

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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)

Answer:

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

Answer:

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

Answer:

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  

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

\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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