< x + + (Rule ) or x < x + or x – x < x + – x (Rule ) or x < or x < (Rule ) When x is an integer, the solutions of the given inequality are ..., – , – , – , – , , When x is a real number , the solutions of the inequality are given by x < , i.e., all real numbers x which are less than . Therefore, the solution set of the inequality is x ∈ (– ∞ , ). We have considered solutions of inequalities in the set of natural numbers, set of integers and in the set of real numbers. Henceforth, unless stated otherwise, we shall solve the inequalities in this Chapter in the set of real numbers.
MATHEMATICS Example Solve x + < x + . Solution We have, x + < x + or x – x < x + – x or – x < or x > – i.e., all the real numbers which are greater than – , are the solutions of the given inequality. Hence, the solution set is (– , ∞). Example Solve – x – ≤ Solution We have – x – ≤ or ( – x ) ≤ x – .
or – x ≤ x – or – x ≤ – , i.e., x ≥ Thus, all real numbers x which are greater than or equal to are the solutions of the given inequality, i.e., x ∈ [ , ∞ ). Example Solve x + < x + . Show the graph of the solutions on number line. Solution We have x + < x + or x < or x < The graphical representation of the solutions are given in Fig .
. Fig . Example Solve