Linear Inequations in One Variable:
We know that equations are mathematical statements having two sides connected by a sign of equality. In an inequation, the two sides of the statement are connected by a sign of inequality. The signs of inequality are <, >, ≤, ≥, ≠.
Statements like 2x < 3, 12x – 7 ≤ 2, (x – 2)/3 > 2x – 1, 3x – 2 < x + 1 etc., are known as linear inequations in one variable. In general, a linear inequation in one variable may be written as, ax + b > 0, ax + b < 0, ax + b ≥ 0, ax + b ≤ 0, where a, b ∈ R.
Solving a linear inequation of one variable is more or less similar to solving a linear equation in one variable as most of the basic rules are applicable except one exception discussed below. The permissible rules are-
(i) If the same expression is added to or subtracted from both sides of an inequation, the solution of the inequation remains unchanged.
(ii) If both the sides of an inequation are multiplied or divided by the same positive number, the solution of the resulting inequation remains unaltered.
(iii) If both the sides of an inequation are multiplied or divided by the same negative number, the resulting inequation has the same solution provided the sign of inequality is reversed.
Thus, the basic difference, between solving a linear equation and solving an inequation concerns the multiplication or division of both sides by a negative number.
The set from which the values of the variable involved in the inequation are chosen is generally known as the replacement set. For example, if the replacement set is the set of natural numbers, then from the values obtained after solving the inequation, only the natural numbers are to be taken as the solution.
Steps to Solve a Linear Inequation in One Variable (Simple Inequation):
(i) Both sides are simplified by removing the group symbols and collecting the like terms.
(ii) Fractions and decimals are removed (if necessary) after multiplying both sides by an appropriate factor.
(iii) Terms containing the variable are transposed to the left side and the constant terms are kept on the right side.
(iv) Both the sides are divided by the coefficient of the variable to get the values of the unknown.
(v) The solution set is, then, chosen from the replacement set.
Graphs of Inequations Involving One Variable:
We know that the solution set of the inequality x < 3 consists of all numbers less than 3, i.e., all the numbers lying to the left of 3 on the number line. We draw a picture of this set of numbers by shading the number line to the left of 3. The statement that the set extends up to infinity to the left of 3 is indicated by an arrowhead. We use an open circle at 3 to indicate that the number 3 is not included in the solution set. This picture representing the solution set is called the graph of the inequation (Fig. a).
The graph of the inequality x ≥ -1/2 is drawn by shading the number line to the right of -1/2. In this case, a filled circle is drawn at x = -1/2 to indicate that x = -1/2 is included in the solution set (Fig. b).