Inequation: A statement involving variable(s) and the sign of inequality that is, is called an inequation or an inequality. Some examples of linear inequations are
Linear Inequation in One Variable: Let be a non-zero real number and be a variable. Then inequations of the form and are known as linear inequations in one variable .
For example, and are linear inequations in one variable.
Linear Inequation in Two Variable: Let be non-zero real numbers and be variables. Then inequations of the form and are known as linear inequations in two variables and .
For example, are linear inequations in two variables and .
Quadratic Inequation: Let a be a non-zero real number. Then an inequation of the form , or , or , or is known as a quadratic inequation.
For example, and are quadratic inequations.
Solving linear inequations in one variable: ln the process of solving an inequation, we use mathematical simplifications which are governed by the following rules:
- Same number may be added to (or subtracted from) both sides of an inequation without changing the sign of inequality.
- Both sides of an inequation can be multiplied (or divided) by the same positive real number without changing the sign of inequality . However, the sign of inequality is reversed when both sides of an inequation are multiplied or divided by a negative number.
- Any term of an inequation may be taken to the other side with its sign changed without affecting the sign of inequality.
How to solve a linear inequation?
- Obtain the linear inequation.
- Collect all terms involving the variable on one side of the inequation and the constant terms on the other side.
- Simplify both sides of inequality in their simplest forms to reduce the inequation in the form
- Solve the inequation obtained in step 3 by dividing both sides of the inequation by the coefficient of the variable.
- Write the solution set obtained in step IV in the form of an interval on the real line.
Inequations involving modulus of the variable
Case 1: If is a positive real number, then
i) ii) Case 2: If is a positive real number, then
Case 3: Let be a positive real number and be a fixed real number. Then,
Case 4: Let be positive real numbers. Then
Graphical solution of linear inequations in two variables:
- Convert the given inequations, say , into the equation which represents a straight line in plane.
- Put in the equation obtained in step 1 to get the point where the line meets with x-axis. Similarly, put to obtain a point where the line meets with y-axis.
- Join the points obtained in step 2 to obtain the graph of the line obtained from the given inequation. In case of a strict inequality i.e. or , draw the dotted line otherwise mark it thick line.
- Choose a point, if possible , not lying on this line : Substitute its coordinates in the inequations. If the inequation is satisfied, then shade the portion of the plane which contains the point. Otherwise, shade the portion of the plane that does not contain this point.
- Repeat step 4 for the other inequation also.
- The common shaded area obtained is the solution set for the two inequations.
- In case of the inequalities and , the points on the line are also part of the shaded region. While, in case of the inequalities and , the points on the line are also not part of the shaded region.