Linear Inequalities – Class XI – Exercise 6.1 [1 – 10]

  1. Solve 24x < 100, when

(i) x is a natural number

(ii) x is an integer

Solution:

We have the given inequality 24x < 100.

then, 24x/24 < 100/24

Thus, x < 25/6

(i) We know that 1, 2, 3, and 4 are the only natural numbers less than 25/6.

Thus, when x is a natural number, the solutions of the given inequality are 1, 2, 3, and 4.

Hence, in this case, the solution set is {1, 2, 3, 4}.

 

(ii) The integers less than 25/6 are …–3, –2, –1, 0, 1, 2, 3, 4.

Thus, when x is an integer, the solutions of the given inequality are …–3, –2, –1, 0, 1, 2, 3, 4.

Hence, in this case, the solution set is {…–3, –2, –1, 0, 1, 2, 3, 4}.


2: Solve –12x > 30, when

(i) x is a natural number

(ii) x is an integer

Solution:

The given inequality   –12x > 30.

Then -12x > 30

-12x/-12 < 30/-12

⇒ x < –5/2

(i) There is no natural number less than –5/2

Thus, when x is a natural number, there is no solution of the given inequality.

(ii) The integers less than –5/2 are …, –5, –4, –3.

Thus, when x is an integer, the solutions of the given inequality are …, –5, –4, –3.

Hence, in this case, the solution set is {…, –5, –4, –3}.


3: Solve 5x– 3 < 7, when

(i) x is an integer

(ii) x is a real number

Solution:

The given inequality is 5x – 3 < 7

⇒ 2x – 3 < 7

⇒ 5x – 3 + 3 < 7 + 3

⇒ 5x < 10

⇒ x < 2

(i)We know the integers less than 2 are …, –4, –3, –2, –1, 0, 1.

Thus, when x is an integer, the solutions of the given inequality are …, –4, –3, –2, –1, 0, 1.

Hence, in this case, the solution set is {…, –4, –3, –2, –1, 0, 1}.

(ii) When x is a real number, the solutions of the inequality 5x – 3 <7 are given by x < 2 i.e.,  all real numbers x which are less than 2.

Thus, the solution set of the given inequality is x € (–∞, 2).


4: Solve 3x + 8 > 2, when

(i) x is an integer

(ii) x is a real number

Solution:

The given inequality is 3x + 8 > 2

⇒ 3x + 8 > 2

⇒ 3x + 8 – 8 > 2 – 8

⇒ 3x > -6

3x/3 > -6/3

⇒ x > -2

(i)The integers greater than –2 are –1, 0, 1, 2, …

Thus, when x is an integer, the solutions of the given inequality are –1, 0, 1, 2 …

Hence, in this case, the solution set is {–1, 0, 1, 2, …}

(ii) When x is a real number, the solutions of the given inequality are all the real numbers, which are greater than –2.

Thus, in this case, the solution set is (– 2, ∞).


 5: Solve the given inequality for real x: 4x + 3 < 5x + 7

Solution:

4x + 3 < 5x + 7

⇒ 4x + 3 – 7 < 5x + 7 – 7

⇒ 4x – 4 < 5x

⇒ 4x – 4 – 4x < 5x – 4x

⇒ –4 < x

Thus, all real numbers x, which are greater than –4, are the solutions of the given inequality.

Hence, the solution set of the given inequality is (–4, ∞).


6: Solve the given inequality for real x: 3x – 7 > 5x – 1

Solution:

3x – 7 > 5x – 1

⇒ 3x – 7 + 7 > 5x – 1 + 7

⇒ 3x > 5x + 6

⇒ 3x – 5x > 5x + 6 – 5x

⇒ – 2x > 6

-2x/-2 < 6/-2

⇒ x < -3

Thus, all real numbers x, which are less than –3, are the solutions of the given inequality.

Hence, the solution set of the given inequality is (–∞, –3).


7: Solve the given inequality for real x: 3(x – 1) ≤ 2 (x – 3)

Solution:

We have the given inequality 3(x – 1) ≤ 2(x – 3)

⇒ 3x – 3 ≤ 2x – 6

⇒ 3x – 3 + 3 ≤ 2x – 6 + 3

⇒ 3x ≤ 2x – 3

⇒ 3x – 2x ≤ 2x – 3 – 2x

⇒ x ≤ – 3

Thus, all real numbers x, which are less than or equal to –3, are the solutions of the given inequality.

Hence, the solution set of the given inequality is (–∞, –3].


 8: Solve the given inequality for real x: 3(2 – x) ≥ 2(1 – x)

Solution:

We have the given inequality 3(2 – x) ≥ 2(1 – x)

⇒ 6 – 3x ≥ 2 – 2x

⇒ 6 – 3x + 2x ≥ 2 – 2x + 2x

⇒ 6 – x ≥ 2

⇒ 6 – x – 6 ≥ 2 – 6

⇒ –x ≥ –4

⇒ x ≤ 4

Thus, all real numbers x, which are less than or equal to 4, are the solutions of the given inequality.

Hence, the solution set of the given inequality is (–∞, 4].


  1. Solve the given inequality for real x: x + x/2 + x/3 < 11

Solution:

The given inequality is x + x/2 + x/3 < 11

x + x/2 + x/3 < 11

⇒ x(1 + 1/2 + 1/3) < 11

⇒ x(6+3+2/6) < 11

11x/6 < 11

x/6 < 11

⇒ x < 6

Thus, all real numbers x, which are less than 6, are the solutions of the given inequality.

Hence, the solution set of the given inequality is (–∞, 6).


  1. Solve the given inequality for real x: x/3 > x/2 + 1

Solution:

We have the given inequality x/3 > x/2 + 1

x/3 > x/2 + 1

x/3x/2 > 1

(2x – 3x)/6 > 1

⇒ –x/6 > 1

⇒ -x > 6

⇒ x < -6

Thus, all real numbers x, which are less than –6, are the solutions of the given inequality.

Hence, the solution set of the given inequality is (–∞, –6).


 

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