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:

    1. Convert the inequality to slope-intercept form (y = mx + b) if it is not already in that form.
    2. 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 >.
    3. 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 programming problems:

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

• Half-plane theorems and their applications:

  • A theorem that states that a line divides the plane into two half-planes.
  • 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.

• Non-linear inequalities (quadratic, exponential, logarithmic, etc.):

  • Inequalities involving non-linear 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.