Linear Inequalities:

Previously we read about equations which denote equality. We take an example of the number of student two classes. If the number of student in one class is same as that in another class then this involves equality. But in reality, we see that the number of students in each class varies in number. Thus, if the number of students is different in different classes we represent it by linear inequalities. The number of student in one class may be either greater than or less than another class.

We denote a greater symbol as ‘>’ and lesser symbol as ‘<’ when compared to other quantity.

Inequality:

Two real numbers or two algebraic expressions related by the symbol ‘<’, ‘>’, ‘≤’ or ‘≥’ this form an inequality.

Terminology related to Linear Inequalities:

Numerical inequalities: When only numerals are compared then it is numerical inequalities.
3 < 5; 7 > 5 are the examples of numerical inequalities.

Literal inequality: When literals or variables are compared with numerical or literals then it is Literal inequalities.
x < 5; y > 2; x ≥ 3, y ≤ 4 are some examples of literal inequalities.

Strict Inequalities: When two quantities are compared and one is either greater than or less than the other but not equal to other is strict inequality.
Example: ax + b < 0, ax + by > c etc.

Slack Inequalities: When two quantities are compared and one is either greater than and equal to or less than and equal to the other than it is slack inequality.
Example: ax + b ≤ 0, ax + by ≥ c

Linear inequality in one variable: Inequation containing only one variable is linear inequalities in one variable.
Example: ax + b < 0, ax + b ≤ 0, ax + b ≥ 0 etc.

Linear inequality in two variable: Inequation containing two variables is linear inequalities in two variable.
Example: ax + by < c, ax + by > c, ax + by ≥ c etc.

Rules of Linear Inequalities:

1. Equal numbers may be added to (or subtracted from) both sides of an inequality.
2. 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 inequality is reversed.
3. The values of x, which make an inequality a true statement, are called solutions of the inequality.
4. To represent x < a (or x > a) on a number line, put a circle on the number a, and dark line to the left (or right) of the number a.
5. To represent x < a (or x > a) on a number line, put a circle on the number a and dark line to the left (or right) of the number a.
6. For Graphical representation of linear inequality having sign ≤ or ≥ then points on the line is also included in calculating its shading and graph of the inequality lies left (below) or right (above) of the graph.
7. For Graphical representation of linear inequality having sign < or > graph of the inequality lies only left (below) or right (above) of the graph not on the line.
8. The solution region of a system of inequalities is the region which satisfies all the given inequalities in the system simultaneously.