Equal numbers may be added to (or subtracted from) both sides of an equation. Rule Both sides of an equation may be multiplied (or divided) by the same non-zero number. In the case of solving inequalities, we again follow the same rules except with a difference that in Rule , the sign of inequality is reversed (i.e., ‘<‘ becomes ‘>’, ≤ ’ becomes ‘ ≥ ’ and so on) whenever we multiply (or divide) both sides of an inequality by a negative number. It is evident from the facts that > while – < – , – < – while (– ) (– ) > (– ) (– ) , i.e., > .
LINEAR INEQUALITIES Thus, we state the following rules for solving an inequality: Rule Equal numbers may be added to (or subtracted from) both sides of an inequality without affecting the sign of inequality. Rule Both sides of an inequality can be multiplied (or divided) by the same positive number. But when both sides are multiplied or divided by a negative number, then the sign of inequality is reversed. Now, let us consider some examples.
Example Solve x < when (i) x is a natural number, (ii) x is an integer. Solution We are given x < or x < (Rule ), i.e., x < / . When x is a natural number, in this case the following values of x make the statement true. , , , , , .
The solution set of the inequality is { , , , , , }. When x is an integer , the solutions of the given inequality are ..., – , – , – , , , , , , , The solution set of the inequality is {...,– , – ,– , , , , , , , } Example Solve x – < x + when x is an integer, x is a real number. Solution We have, x – < x + or x – +