Complex Numbers and Quadratic Equations Notes Class 11 Maths Chapter 6

Important Definitions: (a) A statement involving variable(s) an a sign of inequality, viz. ,>, <, ≥ or ≤ is called an inequality.
(b) Linear inequality : An inequality involving one or two variables in linear power (in single power) is known as linear inequality.
(c) Quadratic inequality: An inequality involving one or two variables in quadratic powers (i.e., in second degree) is known as quadratic inequality.

Important Rules:

Rule 1: Equal numbers may be added to (or subtracted from) both sides of an inequality without affecting the sigh of inequality.
Rule 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 sign of inequality is reversed.

Solution of System of Linear Inequalities in One Variable
Procedure adopted :

(a) Find all the values of the variables satisfying each of the given inequalities.
(b) Find the values of the variable, which are common to the given inequalities.
These common values of the variables are the solutions of the given inequalities.

Note : Range of the values of the solutions of the given inequalities is known as the solution interval of the given inequality.

Graphical Solution of Linear Inequalities in Two Variables

Important definitions : The region containing all the solutions of an inequality is called the solution region.

Note: If an equation involves sign of equality, then the points on the line are also included in the solution. In other cases, these points are not included in the solution. In this case, the line is drawn dotted or broken.

Procedure to Draw Graph of an Inequality

1. First draw the graph of the equation corresponding to the given inequality. To draw the graph of an equation, find the point on the x-axis (by putting y = 0 in the equation) and on they-axis (by putting x = 0 in the equation). Join these two points.

2. This line will divide the coordinate plane in two half plane regions, viz.,
(a) Half plane region I below the line.
(b) Half plane region II above the line.

3. To identify the half plane represented by the inequality, we take a point (a, b) not lying on the line. If putting the point (a, 6) in the inequality, the inequality is satisfied, then the point is situated in the region represented by the inequality otherwise not.

Note : Generally take (0, 0) as the arbitrary point. Sometimes we take the point (1, 1) also.

Solutions of System of Linear Inequalities in Two Variables

Procedure adopted : (a) Draw the graph of the inequalities following the method learnt as stated above reference to the same coordinate axes.
(b) Shade the relevant solution region of each inequality.
(c) Multishaded region will be the solution region of the given inequalities.

Applications : Here in this section, knowledge of solving inequalities will be utilized in solving problems from different fields such as economics, science, mathematics, psychology, etc.