LINEAR INEQUALITIES LINEAR INEQUALITIES Since in this case the total amount spent may be upto ` . Note that the statement ( ) consists of two statements x + y < ... ( ) and x + y = ... ( ) Statement ( ) is not an equation, i.e., it is an inequality while statement ( ) is an equation.
Definition Two real numbers or two algebraic expressions related by the symbol ‘<’, ‘>’, ‘ ≤ ’ or ‘ ≥ ’ form an inequality . Statements such as ( ), ( ) and ( ) above are inequalities. < ; > are the examples of numerical inequalities while x < ; y > ; x ≥ , y ≤ are some examples of literal inequalities . < < (read as is greater than and less than ), < x < (read as x is greater than or equal to and less than ) and < y < are the examples of double inequalities .
Some more examples of inequalities are: ax + b < ... ( ) ax + b > ... ( ) ax + b ≤ ... ( ) ax + b ≥ ...
( ) ax + by < c ... ( ) ax + by > c ... ( ) ax + by ≤ c ... ( ) ax + by ≥ c ...
( ) ax + bx + c ≤ ... ( ) ax + bx + c > ... ( ) Inequalities ( ), ( ), ( ), ( ) and ( ) are strict inequalities while inequalities ( ), ( ), ( ), ( ), and ( ) are slack inequalities . Inequalities from ( ) to ( ) are linear inequalities in one variable x when a ≠ , while inequalities from ( ) to ( ) are linear inequalities in two variables x and y when a ≠ , b ≠ .
Inequalities ( ) and ( ) are not linear (in fact, these are quadratic inequalities in one variable x when a ≠ ) . In this Chapter, we shall confine ourselves