Real Number and Inequality

Class XI

Chapter 6 – Linear Inequalities

Maths

Compiled By : OP Gupta [[phone] | [phone]]
Exercise 6.1

Question 1:
Solve 24x < 100, when (i) x is a natural number (ii) x is an integer
Answer
The given inequality is 24x < 100.

(i) It is evident that 1, 2, 3, and 4 are the only natural numbers less than

.

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

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.

(i) There is no natural number less than

.

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By OP Gupta [[phone]]

Class XI

Chapter 6 – Linear Inequalities

Maths

Thus, when x is a natural number, there is no solution of the given inequality.
(ii) The integers less than

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.

(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.

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