Linear Inequality
LINEAR INEQUALITY (JEE and CBSE):
Key Points:

Linear inequalities in one variable:
 Inequalities involving a single variable, x, such as x > 5 or x ≤ 3.
 Can be represented on a number line with open or closed circles to indicate the allowed values of x.

Linear inequalities in two variables:
 Inequalities involving two variables, x and y, such as 3x + 4y ≥ 12 or 2x – y < 5.
 Can be represented graphically by shading the feasible region (the area satisfying the inequality) on a coordinate plane.

Graphical representation of linear inequalities:

To graph a linear inequality in two variables, follow these steps:
 Convert the inequality to slopeintercept form (y = mx + b) if it is not already in that form.
 Plot the boundary line (the line that represents the equation) using a solid line if the inequality is ≤ or ≥, and a dashed line if the inequality is < or >.
 Shade the feasible region above or below the boundary line, depending on the inequality symbol.


Solving linear inequalities algebraically:
 To solve a linear inequality algebraically, isolate the variable on one side of the inequality sign.

Applications of linear inequalities:
 Linear inequalities can be used to:
 Find feasible regions: the set of points that satisfy a system of linear inequalities.
 Perform optimization: finding the maximum or minimum value of a function subject to given constraints (linear inequalities).
 Linear inequalities can be used to:

Linear programming problems:
 A type of optimization problem where the objective function and constraints are all linear.
 Can be solved using graphical or algebraic methods.

Halfplane theorems and their applications:
 A theorem that states that a line divides the plane into two halfplanes.
 Can be used to determine whether a point lies in the feasible region of a system of linear inequalities.

Systems of linear inequalities:
 A set of two or more linear inequalities.
 Can be solved graphically or algebraically.

Nonlinear inequalities (quadratic, exponential, logarithmic, etc.):
 Inequalities involving nonlinear functions, such as x^2  4x + 3 ≤ 0 or e^x > 5.
 Can be solved graphically or algebraically, depending on the complexity of the function.