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Exercise 6.1 - Chapter 6 Linear Inequalities class 11 ncert solutions Maths - SaraNextGen [2024-2025]


Updated By SaraNextGen
On April 24, 2024, 11:35 AM

 

Question 1:

Solve 24x < 100, when (i) x is a natural number (ii) x is an integer

Answer:

The given inequality is 24x < 100.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/5549/Chapter%206_html_72bab6e0.gif

(i) It is evident that 1, 2, 3, and 4 are the only natural numbers less thanhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/5549/Chapter%206_html_14ccb9fb.gif .

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 https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/5549/Chapter%206_html_14ccb9fb.gif  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}.

Question 2:

Solve –12x > 30, when

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

Answer:

The given inequality is –12x > 30.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4153/CHAPTER%206_html_16c44d4.gif

(i) There is no natural number less thanhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4153/CHAPTER%206_html_m31b07fb4.gif .

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

(ii) The integers less than https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4153/CHAPTER%206_html_m31b07fb4.gif  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}.

Question 3:

Solve 5x– 3 < 7, when

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

Answer:

The given inequality is 5x– 3 < 7.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4154/CHAPTER%206_html_m133c740.gif

(i) 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 given inequality are given by x < 2, that is, all real numbers x which are less than 2.

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

Question 4:

Solve 3x + 8 > 2, when

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

Answer:

The given inequality is 3x + 8 > 2.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4155/CHAPTER%206_html_2e4a8d5.gif

(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, ∞).

Question 5:

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

Answer:

4x + 3 < 5x + 7

⇒ 4x + 3 – 7 < 5x + 7 – 7

⇒ 4x – 4 < 5x

⇒ 4x – 4 – 4x < 5– 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, ∞).

Question 6:

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

Answer:

3x – 7 > 5x – 1

⇒ 3x – 7 + 7 > 5x – 1 + 7

⇒ 3x > 5x + 6

⇒ 3x – 5x > 5x + 6 – 5x

⇒ – 2> 6

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4157/CHAPTER%206_html_m2d11356b.gif

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).

Question 7:

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

Answer:

3(x – 1) ≤ 2(x – 3)

⇒ 3x – 3 ≤ 2x – 6

⇒ 3x – 3 + 3 ≤ 2x – 6 + 3

⇒ 3x ≤ 2x – 3

⇒ 3x – 2≤ 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].

Question 8:

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

Answer:

3(2 – x) ≥ 2(1 – x)

⇒ 6 – 3x ≥ 2 – 2x

⇒ 6 – 3x + 2x ≥ 2 – 2+ 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].

Question 9:

Solve the given inequality for real xhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4160/CHAPTER%206_html_738694e4.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4160/CHAPTER%206_html_fbd3f68.gif

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).

Question 10:

Solve the given inequality for real xhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4161/CHAPTER%206_html_20c56f3b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4161/CHAPTER%206_html_m6772349.gif

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).

Question 11:

Solve the given inequality for real xhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4162/CHAPTER%206_html_m7d4049fd.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4162/CHAPTER%206_html_1afeff1f.gif

Thus, all real numbers x,which are less than or equal to 2, are the solutions of the given inequality.

Hence, the solution set of the given inequality is (–∞, 2].

Question 12:

Solve the given inequality for real xhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4163/CHAPTER%206_html_2f2bb7f4.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4163/CHAPTER%206_html_1e1dec1a.gif

Thus, all real numbers x,which are less than or equal to 120, are the solutions of the given inequality.

Hence, the solution set of the given inequality is (–∞, 120].

Question 13:

Solve the given inequality for real x: 2(2x + 3) – 10 < 6 (x â€“ 2)

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4164/CHAPTER%206_html_32c88a8a.gif

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

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

Question 14:

Solve the given inequality for real x: 37 ­– (3x + 5) ≥ 9x – 8(– 3)

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4165/CHAPTER%206_html_m799fbc14.gif

Thus, all real numbers x,which are less than or equal to 2, are the solutions of the given inequality.

Hence, the solution set of the given inequality is (–∞, 2].

Question 15:

Solve the given inequality for real xhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4166/CHAPTER%206_html_m4b06b148.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4166/CHAPTER%206_html_6999607d.gif

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, ∞).

Question 16:

Solve the given inequality for real xhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4167/CHAPTER%206_html_5d7019d.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4167/CHAPTER%206_html_46668295.gif

Thus, all real numbers x,which are less than or equal to 2, are the solutions of the given inequality.

Hence, the solution set of the given inequality is (–∞, 2].

Question 17:

Solve the given inequality and show the graph of the solution on number line: 3x – 2 < 2x +1

Answer:

3x – 2 < 2x +1

⇒ 3– 2x < 1 + 2

⇒ x < 3

The graphical representation of the solutions of the given inequality is as follows.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4168/CHAPTER%206_html_264ff3af.jpg

Question 18:

Solve the given inequality and show the graph of the solution on number line: 5x – 3 ≥ 3x – 5

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4169/CHAPTER%206_html_m2a98634.gif

The graphical representation of the solutions of the given inequality is as follows.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4169/CHAPTER%206_html_m10fbaef9.jpg

Question 19:

Solve the given inequality and show the graph of the solution on number line: 3(1 – x) < 2 (x + 4)

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4170/CHAPTER%206_html_416d235.gif

The graphical representation of the solutions of the given inequality is as follows.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4170/CHAPTER%206_html_m1ace8ad5.jpg

Question 20:

Solve the given inequality and show the graph of the solution on number line: https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4171/CHAPTER%206_html_md13eb7f.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4171/CHAPTER%206_html_5bee3ddb.gif

The graphical representation of the solutions of the given inequality is as follows.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4171/CHAPTER%206_html_m2d114e50.jpg

Question 21:

Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third test to have an average of at least 60 marks.

Answer:

Let x be the marks obtained by Ravi in the third unit test.

Since the student should have an average of at least 60 marks,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4172/CHAPTER%206_html_m6f5c714d.gif

Thus, the student must obtain a minimum of 35 marks to have an average of at least 60 marks.

Question 22:

To receive Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita’s marks in first four examinations are 87, 92, 94 and 95, find minimum marks that Sunita must obtain in fifth examination to get grade ‘A’ in the course.

Answer:

Let x be the marks obtained by Sunita in the fifth examination.

In order to receive grade ‘A’ in the course, she must obtain an average of 90 marks or more in five examinations.

Therefore,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4173/CHAPTER%206_html_m35542617.gif

Thus, Sunita must obtain greater than or equal to 82 marks in the fifth examination.

Question 23:

Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.

Answer:

Let x be the smaller of the two consecutive odd positive integers. Then, the other integer is x + 2.

Since both the integers are smaller than 10,

x + 2 < 10

⇒ x < 10 – 2

⇒ x < 8 … (i)

Also, the sum of the two integers is more than 11.

x + (x + 2) > 11

⇒ 2x + 2 > 11

⇒ 2x > 11 – 2

⇒ 2x > 9

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/5390/chapter%206_html_4d7391ac.gif

From (i) and (ii), we obtain 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4174/CHAPTER%206_html_4d7391ac.gif

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4175/CHAPTER%206_html_m4c228eed.gif

 

 

 

 

 

 

 

 

 

 

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4176/CHAPTER%206_html_73df4d33.gif

 

 

 

 

 

 

 

 

 

 

 

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/166/4177/CHAPTER%206_html_m76294ef5.gif

 

 

 

 

 

 

 

Also Read : Exercise-6.2-Chapter-6-Linear-Inequalities-class-11-ncert-solutions-Maths

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