PPT: Linear Inequalities

 Page 1


Linear 
Inequalities
Page 2


Linear 
Inequalities
I n t r o d u c t i o n
 An inequality involving linear expressions (e.g., 
, ) is called a Linear Inequality. a x + b > 0 3 x + 7 f 5
For 
Example: 
Page 3


Linear 
Inequalities
I n t r o d u c t i o n
 An inequality involving linear expressions (e.g., 
, ) is called a Linear Inequality. a x + b > 0 3 x + 7 f 5
For 
Example: 
Some Examples of Inequalities:
Numerical inequalities
3 < 5
Literal inequalities
Inequalities which involves 
variables: 
x >=2
Double inequalities
3 < 5 < 7
Strict inequalities
Inequalities that have either '<' or '>' in the equation: 
ax + b < 0; ax + b > 0; 
ax + by < c; ax + by > c.
Slack inequalities
Inequalities that have a 'f' or 'g' sign: 
ax + b f 0; ax + b g 0; 
ax + by f c; ax + by g c.
Page 4


Linear 
Inequalities
I n t r o d u c t i o n
 An inequality involving linear expressions (e.g., 
, ) is called a Linear Inequality. a x + b > 0 3 x + 7 f 5
For 
Example: 
Some Examples of Inequalities:
Numerical inequalities
3 < 5
Literal inequalities
Inequalities which involves 
variables: 
x >=2
Double inequalities
3 < 5 < 7
Strict inequalities
Inequalities that have either '<' or '>' in the equation: 
ax + b < 0; ax + b > 0; 
ax + by < c; ax + by > c.
Slack inequalities
Inequalities that have a 'f' or 'g' sign: 
ax + b f 0; ax + b g 0; 
ax + by f c; ax + by g c.
Types of Linear Inequalities
Linear inequalities in one variable
'x' where a b0: 
ax + b >0; ax + b f 0.
Linear inequalities in two variables
'x' and 'y' where a b 0 and b b 0: 
ax + by < c; ax + by g c.
R e m a r k : ax² + bx + c < 0 is an example of a quadratic inequa lity in 'x' where a b 0.
Page 5


Linear 
Inequalities
I n t r o d u c t i o n
 An inequality involving linear expressions (e.g., 
, ) is called a Linear Inequality. a x + b > 0 3 x + 7 f 5
For 
Example: 
Some Examples of Inequalities:
Numerical inequalities
3 < 5
Literal inequalities
Inequalities which involves 
variables: 
x >=2
Double inequalities
3 < 5 < 7
Strict inequalities
Inequalities that have either '<' or '>' in the equation: 
ax + b < 0; ax + b > 0; 
ax + by < c; ax + by > c.
Slack inequalities
Inequalities that have a 'f' or 'g' sign: 
ax + b f 0; ax + b g 0; 
ax + by f c; ax + by g c.
Types of Linear Inequalities
Linear inequalities in one variable
'x' where a b0: 
ax + b >0; ax + b f 0.
Linear inequalities in two variables
'x' and 'y' where a b 0 and b b 0: 
ax + by < c; ax + by g c.
R e m a r k : ax² + bx + c < 0 is an example of a quadratic inequa lity in 'x' where a b 0.
Rules for Solving Equations
While solving linear equations, we followed the following rules:
Rule 1
Equal numbers may be added to (or subtracted from) both 
sides of an equation.
E x a m p l e :
2x + 3 = 7
Subtracting 3 from both sides gives:
2x = 4
Rule 2
Both sides of an equation may be multiplied (or divided) 
by the same non-zero number.
E x a m p l e :
2x = 6
Divide both sides by 2:
x = 3