NCERT Solutions for Class 11 Maths Chapter 6 Linear Inequalities

Ans : The given inequality is 24x < 100. 
\(\begin{array}{l}{24 x<100} \\ {\Rightarrow \frac{24 x}{24}<\frac{100}{24}} \\ {\Rightarrow x<\frac{25}{6}}\end{array} \quad[\text { Dividing both sides by same positive number] }\)
(i) It is evident that 1, 2, 3, and 4 are the only natural numbers less than \(\frac{25}{6}\).
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 \(\frac{25}{6}\) 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}. 

Ans : The given inequality is –12x > 30. 
\(\begin{array}{l}{-12 x >30} \\ {\Rightarrow \frac{-12 x}{-12}<\frac{30}{-12} \quad \quad[\text { Dividing both sides by same negative number }]} \\ {\Rightarrow x <-\frac{5}{2}}\end{array}\)
(i) There is no natural number less than \(\left(-\frac{5}{2}\right)\).
Thus, when x is a natural number, there is no solution of the given inequality.
(ii) The integers less than \(\left(-\frac{5}{2}\right)\) 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}. 

Ans : The given inequality is 5x– 3 < 7.
\(\begin{array}{l}{5 x-3<7} \\ {\Rightarrow 5 x-3+3<7+3} \\ {\Rightarrow 5 x<10} \\ {\Rightarrow \frac{5 x}{5}<\frac{10}{5}} \\ {\Rightarrow x<2}\end{array}\)
(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). 

Ans : The given inequality is 3x + 8 > 2.
\(\begin{array}{l}{3 x+8>2} \\ {\Rightarrow 3 x+8-8>2-8} \\ {\Rightarrow 3 x>-6} \\ {\Rightarrow \frac{3 x}{3}>\frac{-6}{3}} \\ {\Rightarrow x>-2}\end{array}\)
(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, ∞). 

Ans : 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, ∞).