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Class 11 NCERT Solutions- Chapter 6 Linear Inequalities – Exercise 6.1 | Set 1
  • Difficulty Level : Medium
  • Last Updated : 05 Apr, 2021

Question 1. Solve 24x < 100, when

(i) x is a natural number. (ii) x is an integer.

Solution:

(i) when x is a natural number.

Clearly x>0 because from definition (N =1,2,3,4,5,6…..)
Now we have to divide the inequation by 24 we get x<25/6
But x is a natural number that is the solution will be {1,2,3,4} 
Which is less than 25/6 and greater than 0.
Hence, {…, -2,-1, 0, 1, 2, 3, 4} is the solution set.

x={1,2,3,4}



(ii) Given 24x<100
Now we will divide the equation by 24 we get x<25/6
but according to question x is an integer then the solution 
less than 25/6 are…-2,-1, 0, 1, 2, 3, 4
Hence, {…, -2,-1, 0, 1, 2, 3, 4} is the solution set.

Question 2. Solve – 12x > 30, when

(i) x is a natural number.
(ii) x is an integer.

Solution:

(i) Given, – 12x > 30
Now by dividing the equation by -12 on both sides we get, x < -5/2(as per the rule if we divide by a negative integer then the inequality sign changes)
As per question when x is a natural integer then
It is clear that there is no natural number less than -2/5 since, the result of -2/5 will be negative and x is smaller than the result but the natural number contains only positive number 
Therefore, there would be  no any solution of the given equation when x is a natural number(x>0).

(ii) Given that, – 12x > 30
Now by dividing the equation by -12 on both sides we get, x < -5/2 (As explained earlier why sign changes)
As per question now x is an integer then
It is clear that the integer number less than -5/2 are…, -6.-5, -4, – 3
Thus, the solution of – 12x > 30 is …,-6,-5, -4, -3, when x is an integer.
Therefore, the solution set is {…,-6, -5, -4, -3}

Question 3. Solve 5x – 3 < 7, when

(i) x is an integer
(ii) x is a real number

Solution:

(i) Given 5x – 3 < 7
Now let us add 3 both side we get,
5x – 3 + 3 < 7 + 3
Above equation becomes
5x < 10
Again let us divide  both sides by 5 we get,
5x/5 < 10/5
x < 2
As per question x is an integer then
It is clear that the integer number less than 2 are…, -3, -2, -1, 0, 1.
Thus, the  solution of 5x – 3 < 7 is …, -3,-2, -1, 0, 1, when x is an integer.
Therefore, the solution set is {…, -3.-2, -1, 0, 1}



(ii) Given that, 5x – 3 < 7
Now let us add 3 both side we get,
5x – 3 + 3 < 7 + 3
Above equation becomes
5x < 10
Again let us divide both the sides by 5 we get,
5x/5 < 10/5
x < 2
As per question x is a real (x ∈  R) number then
It is clear that the solutions of 5x – 3 < 7 will be given by x < 2 i.e it states that all the real numbers that are less than 2.
Hence, the solution set is x ∈ (-∞, 2)

Question 4. Solve 3x + 8 > 2, when

(i) x is an integer.
(ii) x is a real number.

Solution:

(i) Given, 3x + 8 > 2
Now let us subtract 8 from both the sides we get,
3x + 8 – 8 > 2 – 8
The above equation becomes,
3x > – 6
Again let us divide both the sides by 3 we get,
3x/3 > -6/3
Hence, x > -2
As per question x is an integer then
It is clear that the integer number greater than -2 are -1, 0, 1, 2, 3,…
Thus, the solution of 3x + 8 > 2is -1, 0, 1, 2, 3,… when x is an integer.
Hence, the solution set is {-1, 0, 1, 2, 3,…}

(ii) Given3x + 8 > 2
Now let us subtract 8 from both sides we get,
3x + 8 – 8 > 2 – 8
The above equation becomes,
3x > – 6
Again let us divide both the sides by 3 we get,
3x/3 > -6/3
Hence, x > -2
As per question x is a real number.
It is clear that the solutions of 3x + 8 >2 will be given by x > -2 which states that all the real numbers that are greater than -2.
Therefore, the solution set is x ∈ (-2, ∞) 

Question 5. 4x + 3 < 5x + 7

Solution:

Given, 4x + 3 < 5x + 7
Now let us subtract 7 from both the sides, we get
4x + 3 – 7 < 5x + 7 – 7
The above equation becomes,
4x – 4 < 5
Again let us subtract 4x from both the sides, we get
4x – 4 – 4x < 5x – 4x
x > – 4

Thus, solution of the given equation is defined by all the real numbers greater than -4.
Required solution set is (-4, ∞)

Question 6. 3x – 7 > 5x – 1

Solution:

Given, 3x – 7 > 5x – 1
Now let us add 7 to both the sides, we get
3x – 7 +7 > 5x – 1 + 7
3x > 5x + 6
Again let us subtract 5x from both the sides,
3x – 5x > 5x + 6 – 5x
-2x > 6
Now let us divide both the  sides by -2 to simplify we get
-2x/-2 < 6/-2
x < -3
Solutions of the given inequality are defined by all the real numbers less than -3.
Hence, the required solution set is (-∞, -3)

Question 7. 3(x – 1) ≤ 2 (x – 3)

Solution:

Given, 3(x – 1) ≤ 2 (x – 3)
After multiplying, the above inequation can be written as
3x – 3 ≤ 2x – 6
Now let us add 3 to both the sides, we get
3x – 3+ 3 ≤ 2x – 6+ 3
3x ≤ 2x – 3
Again let us subtract 2x from both the sides,
3x – 2x ≤ 2x – 3 – 2x
x ≤ -3
Therefore, the solutions of the given equation is defined by all the real numbers less than or equal to -3.
Hence, the required solution set is (-∞, -3]

Question 8. 3 (2 – x) 2 (1 – x)

Solution:

Given, 3 (2 – x) ≥ 2 (1 – x)
After multiplying, the above equation can be written as
6 – 3x ≥ 2 – 2x
Now let us add 2x to both the sides,
6 – 3x + 2x ≥ 2 – 2x + 2x
6 – x ≥ 2
Again let us subtract 6 from both the sides, we get
6 – x – 6 ≥ 2 – 6
– x ≥ – 4
Now multiplying the equation by negative sign we get
x ≤ 4
Thus, solutions of the given equation is defined by all the real numbers greater than or equal to 4.
Hence, the required solution set is (- ∞, 4]

Question 9. x + x/2 + x/3 < 11

Solution:

Given, x + x/2 + x/3 < 11
Now taking 6 as the lcm we will simplify the equation,

(6x+3x+2x)/6 <11
11x/6<11
Now let us multiply 6 at both the sides
11x<66
Now let us divide the equation by 11 at both the sides we get,
x<6

Thus, the solution of the given equation is defined by all the real numbers less than 6.
Hence, the solution set is (-∞, 6)

Question 10. x/3 > x/2 + 1

Solution:

Given x/3 > x/2 + 1
Firstly, we will move all the terms containing x to the left-hand side we get,
x/3-x/2>1
Now taking 6 as the lcm we get
(2x-3x)/6>1
-x/6>1
Now multiplying 6 at both the sides we get,
 -x>6
Now multiplying by -1 at both the ends
x<6
Thus, the solution of the given equation is defined by all the real numbers less than – 6.
Hence, the required solution set is (-∞, -6)

Question 11. 3(x – 2)/5 5 (2 – x)/3

Solution:

Given, 3(x – 2)/5 ≤ 5 (2 – x)/3
Now by cross – multiplying the denominators, we get
9(x- 2) ≤ 25 (2 – x)
9x – 18 ≤ 50 – 25x
Now let us add 25x both the sides,
9x – 18 + 25x ≤ 50 – 25x + 25x
34x – 18 ≤ 50
Let us add 25x both the sides,
34x – 18 + 18 ≤ 50 + 18
34x ≤ 68
Dividing both the sides by 34,
34x/34 ≤ 68/34
x ≤ 2
Thus, the solution of the given equation is defined by all the real numbers less than or equal to 2.
Hence, solution set is (-∞, 2]

Question 12. 1/2(3x/5+4)>=1/3(x-6)

Solution:

Given, 1/2(3x/5+4)>=1/3(x-6)
Now let us cross multiply,
3(3x/5+4)>=2(x-6)
Now multiply the respective terms at both the sides
9x/5+12>=2x-12
Now subtracting 12 at both the sides
9x/5+12-12 >= 2x-12-12
9x/5>=2x-24
Now multiplying by 5 both the sides,
9x>=10x-120
Now subtracting 10x both the sides.
-x>=-120
Now multiplying with -1 both the sides
x <= 120
Thus, the solutions of the given equation is defined by all the real numbers less than or equal to 120.
Thus, (-∞, 120] is the required solution set.

 Chapter 6 Linear Inequalities – Exercise 6.1 | Set 2

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