Linear Inequalities

 Linear Inequalities

Basic Concept:

•      Equation (Equality):

A mathematical statement containing equal sign (=) is called equation or equality. For eg. x = 4, 2x + 3y = 5, x – 6 = 3 etc.

•      Inequation (Inequality):

A mathematical statement containing smaller than (<), greater than (>), smaller than and equal to (≤), greater than or equal to (≥) or not equal to (≠) sign is called inequation or inequality. For eg. x > 2, 3x ≥ 5, x + y ≤ 2, y ≠ 3 etc.

•      The equation in the form of x = a is called linear equation with one variable x.

•      The equation in the form of ax + by = c is called linear equation in two variables x and y.

•      The equation in the form of ax + by < c, ax + by > c, ax + by ≤ c and ax + by ≥ c is called linear inequalities in two variables x and y.

•      The corresponding equation ax + by = c of above inequalities is called boundary line. 

 

Steps of drawing linear inequalities in one variable:

• Draw the graph of boundary line of given linear inequality.

• If linear inequality contains ≤ or ≥, draw solid boundary line. ___________
• If linear inequality contains < or >, draw dotted boundary line. - - - - - - - - -

• Determine the solution region or solution set.

• If x ≥ a or x > a, the solution set lies towards right from x = a.
• If x ≤ a or x < a, the solution set lies towards left from x = a.
• If y ≥ b or y > b, the solution set lies upward from y = b.
• If y ≤ b or y < b, the solution set lies downward from y = b. 

                                                             

Steps of drawing linear inequalities in two variables:

To draw graph of any linear inequalities in two variables i.e., ax + by < c, ax + by > c, ax + by ≤ c and ax + by ≥ c

• Draw the graph of boundary line ax + by = c.

 

• If inequality contains ≤ or ≥, draw solid boundary line.
• If inequality contains < or >, draw dotted boundary line.

• Determine the solution region or solution set.

• Choose testing point out of boundary line.
• Substitute the testing point in the given inequality.
• If it is true, the solution set lies towards the testing point.
• If it is false, the solution set lies opposite to the testing point.