# Class 11 RD Sharma Solutions – Chapter 15 Linear Inequations – Exercise 15.2 | Set 2

### Question 11. Solve each of the following system of equations in R: 4x – 1 â‰¤ 0, 3 – 4x < 0

Solution:

Let the first equation be 4x – 1 â‰¤ 0

â‡’ 4x â‰¤ 1

â‡’ x â‰¤ 1/4

and the second equation be 3 – 4x < 0

â‡’ 4x > 3

â‡’ x > 3/4

According to first equation, x lies in range ( -âˆž, 1/4 ] and according to second equation, x lies in range ( 3/4, -âˆž )

Calculating the intersection of these two intervals we get no value for x. Therefore, given set of inequations has no solution.

### Question 12. Solve each of the following system of equations in R: x + 5 > 2(x+1), 2 – x < 3(x+2)

Solution:

Let the first equation be x + 5 > 2(x+1)

â‡’ x + 5 > 2x + 2

â‡’ x < 3

and the second equation be 2 – x < 3(x+2)

â‡’ 2 – x < 3x + 6

â‡’ -4 < 4x

â‡’ x > -1

Hence using above equations, we know x lies in range (-1,3)

### Question 13. Solve each of the following system of equations in R: 2(x – 6) < 3x – 7, 11 – 2x < 6 – x

Solution:

Let the first equation be 2(x – 6) < 3x – 7

â‡’ 2x – 12 < 3x – 7

â‡’ -x < 5

â‡’ x > -5

and the second equation be 11 – 2x < 6 – x

â‡’ -x < -5

â‡’ x > 5

Hence using above equations, we know x lies in range (5,âˆž)

### Question 14. Solve each of the following system of equations in R: 5x – 7 < 3 (x + 3),  1 –  â‰¥ x – 4

Solution:

Let the first equation be 5x – 7 < 3 (x + 3)

â‡’ 5x-7 < 3x + 9

â‡’ 2x < 16

â‡’ x < 8

and the second equation be1 – â‰¥ x – 4

â‡’ x +  â‰¤ 5

â‡’  < 5

â‡’ \frac{x}{2} â‰¤ 1

â‡’ x â‰¤ 2

Hence using above equations, we know x lies in range [-âˆž, 2 ]

### Question 15. Solve each of the following system of equations in R:  -2 â‰¥  – 6, 2(2x + 3) < 6(x – 2) + 10

Solution:

Let the first equation be  -2 â‰¥  – 6

â‡’  –  â‰¥ – 6 + 2

â‡’  â‰¥ -4

â‡’ 6x – 9 -16x â‰¥ -48

â‡’ 10x â‰¤ 39

â‡’ x â‰¤

and the second equation be 2(2x + 3) < 6(x – 2) + 10

â‡’ 4x + 6 < 6x -12 + 10

â‡’ -2x < -8

â‡’ x > 4

According to first equation, x lies in range ( -âˆž, 39/10 ] and according to second equation, x lies in range ( 4, âˆž )

Calculating the intersection of these two intervals we get no value for x. Therefore, given set of inequations has no solution.

### Question 16. Solve each of the following system of equations in R:  < -3,  + 11 < 0

Solution:

Let the first equation be  < -3

â‡’ 7x – 1 < -6

â‡’ 7x < -5

â‡’ x < -5/7

and the second equation be  + 11 < 0

â‡’ 3x + 8 < -55

â‡’ 3x < -63

â‡’ x < -21

Hence using above equations, we know x lies in range ( -âˆž, -21)

### Question 17. Solve each of the following system of equations in R:  > 5, > 2

Solution:

Let the first equation be  > 5

â‡’ 2x + 1 > 5 (7x -1)

â‡’ 2x – 35x > -6

â‡’  – 33x > -6

â‡’ x < 2/11

Also, 7x -1 > 0 â‡’ x > 1/7

using first equation we get, x lies in range

and let the second equation be  > 2

â‡’ x + 7 >  2x – 16

â‡’ 23 >  x

â‡’ x < 23

Also, x – 8 > 0 â‡’ x > 8

Hence using above equations, we know x lies in range ( 8, -23 )

Calculating the intersection of the two intervals we get after solving equation 1 and equation 2 we get no value for x. Therefore, given set of inequations has no solution.

### Question 18. Solve each of the following system of equations in R: 0 <  < 3

Solution:

Using the equation, 0 <  < 3

â‡’ 0 < -x < 6

â‡’ 0 > x > -6

â‡’ x > -5

Hence using above equation, we know x lies in range ( -6, 0 )

### Question 19. Solve each of the following system of equations in R: 10 â‰¤ -5 (x – 2) < 20

Solution:

Let the first equation be 10 â‰¤ -5 (x – 2) < 20

â‡’ 10 â‰¤ -5x + 10 < 20

â‡’ 0 â‰¤ -5x < 10

â‡’ 0 â‰¤ x < -2

Hence using above equations, we know x lies in range ( -2, 0 ]

### Question 20. Solve each of the following system of equations in R: -5 < 2x -3 < 5

Solution:

Using the equation -5 < 2x -3 < 5

â‡’ -2 < 2x < 8

â‡’ -1 < x < 4

Hence using above equations, we know x lies in range ( -1, 4 )

### Question 21. Solve each of the following system of equations in R:  â‰¤ 3 â‰¤ , x>0

Solution:

Using the equation  â‰¤ 3 â‰¤ , x>0

â‡’ 4 â‰¤ 3 (x+1) â‰¤ 6

â‡’ 4 â‰¤ 3x + 3 â‰¤ 6

â‡’ 4 â‰¤ 3x + 3 â‰¤ 6

â‡’ 1 â‰¤ 3x â‰¤ 3

â‡’ 1/3 â‰¤ x â‰¤ 1

Hence using above equations, we know x lies in range

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