Solve the following linear inequations in R

Question 1: Solve: 12x < 50 when

i) x \in R      ii) x \in Z      iii) x \in N

Answer:

Given 12x < 50 \Rightarrow  x < \frac{25}{6}

i)        If \Rightarrow x \in R 

Therefore the solution of the given inequation is \Big( -\infty,   \frac{25}{6} \Big)

ii)      If x \in Z

Therefore the solution of the given inequation is \{ -\infty, \ldots , -3, -2, -1, 0 , 1, 2, 3, 4  \}

iii)     If x \in N

Therefore the solution of the given inequation is \{ 1, 2, 3, 4  \}

\\

Question 2: Solve: -4x > 30 when

i) x \in R      ii) x \in Z      iii) x \in N

Answer:

Given -4x > 30 \Rightarrow  x < - \frac{15}{2}

i)        If \Rightarrow x \in R 

Therefore the solution of the given inequation is \Big( -\infty,   \frac{-15}{2} \Big)

ii)      If x \in Z

Therefore the solution of the given inequation is \{ -\infty, \ldots , -9, -8  \}

iii)     If x \in N

Therefore the solution of the given inequation is \phi This is null set.

\\

Question 3: Solve: 4x-2 < 8 when

i) x \in R      ii) x \in Z      iii) x \in N

Answer:

Given 4x-2 < 8 \Rightarrow  4x < 10 \Rightarrow x < \frac{5}{2}

i)        If \Rightarrow x \in R 

Therefore the solution of the given inequation is \Big( -\infty,   \frac{5}{2} \Big)

ii)      If x \in Z

Therefore the solution of the given inequation is \{ -\infty, \ldots , -3, -2, -1, 0 , 1, 2  \}

iii)     If x \in N

Therefore the solution of the given inequation is \{ 1, 2  \}

\\

Question 4: 3x-7 > x + 1

Answer:

Given 3x-7 > x + 1 \Rightarrow 2x > 8 \Rightarrow x > 4

Therefore the solution of the given inequation is (4,  \infty)

\\

Question 5: x+5 > 4x-10

Answer:

Given x+5 > 4x-10 \Rightarrow 3x < 15 \Rightarrow x < 5

Therefore the solution of the given inequation is (-\infty , 5)

\\

Question 6: 3x+9 \geq -x+19

Answer:

Given 3x+9 \geq -x+19 \Rightarrow 4x \geq 10 \Rightarrow x \geq \frac{5}{2}

Therefore the solution of the given inequation is \Big[ \frac{5}{2} , -\infty  \Big)

\\

Question 7: 2( 3 - x) \geq \frac{x}{5} +4

Answer:

2( 3 - x) \geq \frac{x}{5} +4

\Rightarrow 6-2x \geq \frac{x}{5} + 4

\Rightarrow \frac{x}{5} + 2x \leq 2

\Rightarrow x \leq \frac{10}{11}

Therefore the solution of the given inequation is \Big( -\infty,   \frac{10}{11} \Big)

\\

Question 8: \frac{3x-2}{5} \leq \frac{4x-3}{2}

Answer:

\frac{3x-2}{5} \leq \frac{4x-3}{2}

\Rightarrow 6x-4 \leq 20x-15

\Rightarrow 14x \geq 11

\Rightarrow x \geq \frac{11}{14}

Therefore the solution of the given inequation is \Big[  \frac{11}{14} , \infty \Big)

\\

Question 9: -(x-3) +4 < 5 - 2x

Answer:

-(x-3) +4 < 5 - 2x

\Rightarrow -x+3+4 <5 -2x

\Rightarrow x < -2

Therefore the solution of the given inequation is (- \infty , -2)

\\

Question 10: \frac{x}{5} < \frac{3x-2}{4} - \frac{5x-3}{5}

Answer:

\frac{x}{5} < \frac{3x-2}{4} - \frac{5x-3}{5}

\frac{x}{5} < \frac{15x-10-20x+12}{20}

\Rightarrow 4x < -5x+2

\Rightarrow 9x < 2

\Rightarrow x < \frac{2}{9}

Therefore the solution of the given inequation is \Big[ - \infty,  \frac{2}{9} , \Big)

\\

Question 11: \frac{2(x-1)}{5} \leq \frac{3(2+x)}{7}

Answer:

\frac{2(x-1)}{5} \leq \frac{3(2+x)}{7}

\Rightarrow \frac{2x-2}{5} \leq \frac{6+3x}{7}

\Rightarrow 14x-14 \leq 30+15x

\Rightarrow x \geq -44

Therefore the solution of the given inequation is [-44,  \infty) 

\\

Question 12: \frac{5x}{2} + \frac{3x}{4} \geq \frac{39}{4}

Answer:

\frac{5x}{2} + \frac{3x}{4} \geq \frac{39}{4}

\Rightarrow 10x + 3x \geq 39

\Rightarrow x \geq 3

Therefore the solution of the given inequation is [ 3,  \infty) 

\\

Question 13: \frac{x-1}{3} + 4 < \frac{x-5}{5} -2

Answer:

\frac{x-1}{3} + 4 < \frac{x-5}{5} -2

\Rightarrow \frac{x-1+12}{3} < \frac{x-5-10}{5}

\Rightarrow 5(x+11) < 3 ( x- 15)

\Rightarrow 5x + 55 < 3x - 45

\Rightarrow 2x < - 100

\Rightarrow x < - 50

Therefore the solution of the given inequation is ( -\infty, -50 )

\\

Question 14: \frac{2x+3}{4} - 3 < \frac{x-4}{3} -2

Answer:

\frac{2x+3}{4} - 3 < \frac{x-4}{3} -2

\Rightarrow \frac{2x+3-12}{4} < \frac{x-4-6}{3}

\Rightarrow \frac{2x-9}{4} < \frac{x-10}{3}

\Rightarrow 6x-27 < 4x-40

\Rightarrow 2x < -13

\Rightarrow x <  \frac{-13}{2}

Therefore the solution of the given inequation is \Big( -\infty, \frac{-13}{2} \Big)

\\

Question 15: \frac{5-2x}{3} < \frac{x}{6} -5

Answer:

\frac{5-2x}{3} < \frac{x}{6} -5

\Rightarrow 10-4x < x - 30

\Rightarrow 5x > 40

\Rightarrow x > 8

Therefore the solution of the given inequation is ( 8, \infty )

\\

Question 16: \frac{4+2x}{3} \geq \frac{x}{2} -3

Answer:

\frac{4+2x}{3} \geq \frac{x}{2} -3

\Rightarrow 8+4x \geq 3x-18

\Rightarrow x \geq -26

Therefore the solution of the given inequation is [ -26, \infty )

\\

Question 17: \frac{2x+3}{5} - 2 < \frac{3(x-2)}{5}

Answer:

\frac{2x+3}{5} - 2 < \frac{3(x-2)}{5}

\Rightarrow 2x+3-10 < 3x-6

\Rightarrow x > -1

Therefore the solution of the given inequation is ( -1, \infty )

\\

Question 18: x-2 \leq \frac{5x+8}{3}

Answer:

x-2 \leq \frac{5x+8}{3}

\Rightarrow 3x-6 \leq 5x + 8

\Rightarrow -14 \leq 2x

\Rightarrow x \geq -7

Therefore the solution of the given inequation is [ -7, \infty )

\\

Question 19: \frac{6x-5}{4x+1} < 0

Answer:

\frac{6x-5}{4x+1} < 0

Case I:  6x-5 > 0 \hspace{1.0cm} \& \hspace{1.0cm} 4x+1 < 0

\Rightarrow x > \frac{5}{6}                   \& \hspace{1.0cm}  x < \frac{-1}{4}

This is not possible

Case II: 6x-5 < 0 \hspace{1.0cm} \& \hspace{1.0cm} 4x+1 > 0

\Rightarrow x < \frac{5}{6}                   \& \hspace{1.0cm} x > \frac{-1}{4}

Therefore the solution set for the inequation is \Big( \frac{-1}{4} , \frac{5}{6} \Big)

\\

Question 20: \frac{2x-3}{3x-7} > 0

Answer:

\frac{2x-3}{3x-7} > 0

Case I:  2x-3 > 0 \hspace{1.0cm} \& \hspace{1.0cm} 3x-7 > 0

\Rightarrow x > \frac{3}{2}                   \& \hspace{1.0cm}  x > \frac{7}{3}

\Rightarrow x > \frac{7}{3}

Case II:   2x-3 < 0 \hspace{1.0cm} \& \hspace{1.0cm} 3x-7 < 0

\Rightarrow x < \frac{3}{2}                    \& \hspace{1.0cm}  x < \frac{7}{3}

\Rightarrow x < \frac{3}{2}

Therefore the solution set for the inequation is \Big( -\infty, \frac{3}{2} \Big) \cup \Big( \frac{7}{3} , \infty \Big)

\\

Question 21: \frac{3}{x-2} < 1

Answer:

\frac{3}{x-2} < 1

\Rightarrow \frac{3}{x-2} - 1 < 0 \ \   \Rightarrow \frac{3-x+2}{x-2} < 0   \ \ \Rightarrow \frac{-x+5}{x-2} < 0 

Case I:  -x+5 > 0 \hspace{1.0cm} \& \hspace{1.0cm} x-2 < 0

\Rightarrow x < 5                    \& \hspace{1.0cm}  x < 2

\Rightarrow x < 2

Case II:   -x+5 < 0 \hspace{1.0cm} \& \hspace{1.0cm} x-2 > 0

\Rightarrow x > 5                    \& \hspace{1.0cm}  x > 2

\Rightarrow x > 5

Therefore the solution set for the inequation is ( - \infty, 2 ) \cup ( 5, \infty)

\\

Question 22: \frac{1}{x-1} \leq 2

Answer:

\frac{1}{x-1} \leq 2

\Rightarrow \frac{1}{x-1} - 2 \leq 0 \ \   \Rightarrow \frac{1-2x+2}{x-1} \leq 0   \ \ \Rightarrow \frac{-2x+3}{x-1} \leq 0 

Case I:  -2x+3 \geq 0 \hspace{1.0cm} \& \hspace{1.0cm} x-1 < 0

\Rightarrow x \leq \frac{3}{2}                   \& \hspace{1.0cm}  x < 1

\Rightarrow x < 1

Case II:  -2x+3 \leq 0 \hspace{1.0cm} \& \hspace{1.0cm} x-1 > 0

\Rightarrow x \geq \frac{3}{2}                   \& \hspace{1.0cm}  x > 1

\Rightarrow x \geq \frac{3}{2}

Therefore the solution set for the inequation is ( - \infty, 1 ) \cup \Big[ \frac{3}{2} , \infty \Big)

\\

Question 23: \frac{4x+3}{2x-5} < 6

Answer:

\frac{4x+3}{2x-5} < 6

\Rightarrow \frac{4x+3}{2x-5} - 6 < 0 \ \   \Rightarrow \frac{4x+3-12x+30}{2x-5} < 0   \ \ \Rightarrow \frac{-8x+33}{2x-5} < 0 \ \ \Rightarrow \frac{8x-33}{2x-5} > 0 

Case I:  8x-33 < 0 \hspace{1.0cm} \& \hspace{1.0cm} 2x-5 < 0

\Rightarrow x < \frac{33}{8}                   \& \hspace{1.0cm}  x < \frac{5}{2}

\Rightarrow x < \frac{5}{2}

Case II: 8x-33 > 0 \hspace{1.0cm} \& \hspace{1.0cm} 2x-5 > 0

\Rightarrow x > \frac{33}{8}                   \& \hspace{1.0cm}  x > \frac{5}{2}

\Rightarrow x > \frac{33}{8}

Therefore the solution set for the inequation is \Big( - \infty, \frac{5}{2} \Big) \cup \Big( \frac{33}{8} , \infty \Big)

\\

Question 24: \frac{5x-6}{x+6} < 1

Answer:

\frac{5x-6}{x+6} < 1

\Rightarrow \frac{5x-6}{x+6} - 1 < 0 \ \   \Rightarrow \frac{5x-6-x-6}{x+6} < 0   \ \ \Rightarrow \frac{4x-12}{x+6} < 0 

Case I:  4x-12 < 0 \hspace{1.0cm} \& \hspace{1.0cm} x+6 > 0

\Rightarrow x < 3                    \& \hspace{1.0cm}  x > -6

Case II:  4x-12 > 0 \hspace{1.0cm} \& \hspace{1.0cm} x+6 < 0

\Rightarrow x > 3                    \& \hspace{1.0cm}  x < -6

This is not possible.

Therefore the solution set for the inequation is ( -6, 3 )

\\

Question 25: \frac{5x+8}{4-x} < 2

Answer:

\frac{5x+8}{4-x} < 2

\Rightarrow \frac{5x+8}{4-x} - 2 < 0 \ \   \Rightarrow \frac{5x+8-8+2x}{4-x} < 0   \ \ \Rightarrow \frac{7x}{4-x} < 0 

Case I:  7x > 0 \hspace{1.0cm} \& \hspace{1.0cm} 4-x < 0

\Rightarrow x > 0                    \& \hspace{1.0cm}  x > 4

\Rightarrow x > 4

Case II: 7x < 0 \hspace{1.0cm} \& \hspace{1.0cm} 4-x > 0

\Rightarrow x < 0                    \& \hspace{1.0cm}  x < 4

\Rightarrow x < 0

Therefore the solution set for the inequation is ( - \infty, 0 ) \cup ( 4, \infty)

\\

Question 26: \frac{x-1}{x+3} > 2

Answer:

\frac{x-1}{x+3} > 2

\Rightarrow \frac{x-1}{x+3} - 2 > 0 \ \   \Rightarrow \frac{x-1-2x-6}{x+3} > 0   \ \ \Rightarrow \frac{-x-7}{x+3} > 0 

Case I:  -(x+7) > 0 \hspace{1.0cm} \& \hspace{1.0cm} x+3 > 0

\Rightarrow x < -7                    \& \hspace{1.0cm}  x > -3

This is not possible.

Case II:  -(x+7) < 0 \hspace{1.0cm} \& \hspace{1.0cm} x+3 < 0

\Rightarrow x > -7                    \& \hspace{1.0cm}  x < -3

Therefore the solution set for the inequation is ( - 7, -3 )

\\

Question 27: \frac{7x-5}{8x+3} > 4

Answer:

\frac{7x-5}{8x+3} > 4

\Rightarrow \frac{7x-5}{8x+3} - 4 > 0 \ \   \Rightarrow \frac{7x-5-32x-12}{8x+3} > 0   \ \ \Rightarrow \frac{-25x-17}{8x+3} > 0 

Case I:  -(25x+17) > 0 \hspace{1.0cm} \& \hspace{1.0cm} 8x+3 > 0

\Rightarrow x < \frac{-17}{25}                   \& \hspace{1.0cm}  x > \frac{-3}{8}

This is not possible.

Case II: -(25x+17) < 0 \hspace{1.0cm} \& \hspace{1.0cm} 8x+3 < 0

\Rightarrow x > \frac{-17}{25}                   \& \hspace{1.0cm}  x < \frac{-3}{8}

Therefore the solution set for the inequation is \Big( \frac{-17}{25} , \frac{-3}{8} \Big)

\\

Question 28: \frac{x}{x-5} > \frac{1}{2}

Answer:

\frac{x}{x-5} > \frac{1}{2}

\Rightarrow \frac{x}{x-5} - \frac{1}{2} > 0 \ \   \Rightarrow \frac{2x-x+5}{2x-10} > 0   \ \ \Rightarrow \frac{x+5}{2x-10} > 0 

Case I:  x+5 > 0 \hspace{1.0cm} \& \hspace{1.0cm} 2x-7 > 0

\Rightarrow x > -5                    \& \hspace{1.0cm}  x > 5

\Rightarrow x > 5

Case II:  x+5 < 0 \hspace{1.0cm} \& \hspace{1.0cm} 2x-10 < 0

\Rightarrow x < -5                    \& \hspace{1.0cm}  x < 5

\Rightarrow x < -5

Therefore the solution set for the inequation is ( - \infty, -5 ) \cup ( 5, \infty)