Linear Inequalities

In an inequality, there are two real numbers or expressions that are separated by the symbols <, >, ≤, ≥ which indicate how one expression is related to the other expression. Numerical, linear, double, literal, strict and slack inequalities are the six types of inequalities. Solutions of an inequality are those values of the variable in the inequality which makes it a true statement. A solution set of an inequality consists of only real numbers, as the terms ‘less than or greater than’ are not defined for complex numbers. The orientation of an inequality sign does not change when the same number is added to or subtracted from both the sides of the inequality. The orientation of the inequality sign does not change if both sides of the inequality are multiplied or divided by the same positive number. The orientation of the inequality sign is reversed if both sides of the inequality are multiplied or divided by the same negative number. The solution set of an inequality can be visualised with the help of a graph. This means that on a number line, the portion that is included in the graph is highlighted. Inequalities of the form x < a (or x > a) are represented on a number line by marking an open circle on the number ‘a’ and darkening the line towards the left (or right) of the number a. Inequalities of the form x ≤ a (or x ≥ a) are represented on a number line by marking a closed circle on the number ‘a’ and darken the line towards the left (or right) of the number x. The regions represented by ax + by ≤ c and ax + by ≥ c are known as the closed half spaces. The regions represented by ax + by < c and ax + by > c are known as the open half spaces. The graph of an inequality is one of the half planes (called solution region) and represented by shading the corresponding half plane. Solution of a system of linear inequalities in two variables is the area where the shadings of the two inequalities overlap.

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