Linear Inequalities Class 11 Notes Maths Chapter 5

 Page 1


 
KEY POINTS
? Two real numbers or two algebraic expressions related by the 
symbol 
'<', '>', ' ?' or ' ?' form an inequality. 
? The inequality containing < or > is called strict inequality. 
? The inequality containing ? or ? is called slack inequality. 
? 6 < 8 statements 7.2 > –1 are examples of numerical 
inequalities and 3x + 7 > 8, x + 3 ? 7, 
3
2
y ?
> 2y + 2 are
examples of litral inequalities. 
? The inequalities of the form ax + b > 0, ax + b < 0, ax + b ? 0, 
ax + b ? 0 ; a ? 0 are called linear inequalities in one variable x 
? The inequalities of the form ax + by + c > 0, ax + by + c < 0, 
ax + by + c ? 0, ax + by + c ? 0, a ? 0, b ? 0 are called linear 
inequalities 
in two variables x and y. 
? Rules for solving inequalities : 
(i) a ? b then a ± k ? b ± k where k is any real number. 
(ii) but if a ? b then ka is not always ? kb 
Page 2


 
KEY POINTS
? Two real numbers or two algebraic expressions related by the 
symbol 
'<', '>', ' ?' or ' ?' form an inequality. 
? The inequality containing < or > is called strict inequality. 
? The inequality containing ? or ? is called slack inequality. 
? 6 < 8 statements 7.2 > –1 are examples of numerical 
inequalities and 3x + 7 > 8, x + 3 ? 7, 
3
2
y ?
> 2y + 2 are
examples of litral inequalities. 
? The inequalities of the form ax + b > 0, ax + b < 0, ax + b ? 0, 
ax + b ? 0 ; a ? 0 are called linear inequalities in one variable x 
? The inequalities of the form ax + by + c > 0, ax + by + c < 0, 
ax + by + c ? 0, ax + by + c ? 0, a ? 0, b ? 0 are called linear 
inequalities 
in two variables x and y. 
? Rules for solving inequalities : 
(i) a ? b then a ± k ? b ± k where k is any real number. 
(ii) but if a ? b then ka is not always ? kb 
 
If k > 0 (i.e. positive) then a ? b ? ka ? kb 
If k < 0 (i.e. negative) then a ? b ? ka ? kb 
Thus always reverse the sign of inequality while multiplying or 
dividing both sides of an inequality by a negative number. 
? Procedure to solve a linear inequality in one variables. 
(i) Simplify both sides by removing graph symbols and 
collecting like terms. 
(ii) Remove fractions (or decimals) by multiplying both sides 
by appropriate factor (L.C.M of denomination or a power 
of 10 in case of decimals.) 
(iii) Isolate the variable on one side and all constants on the 
other side. Collect like terms whenever possible. 
(iv) Make the coefficient of the variable. 
(v) Choose the solution set from the replacement set. 
? Replacement Set: The set from which values of the variable 
(involved in the inequality) are chosen is called replacement set. 
? Solution Set: A solution to an inequality is a number which 
when substituted for the variable, makes the inequality true. The 
set of all solutions of an inequality is called the solution set. It is 
obviously a subset of replacement set. 
? The graph of the inequality ax + by > c is one of the half planes 
and is called the solution region. 
? When the inequality involves the sign = or = then the points on 
the line are included in the solution region but if it has the sign < 
Page 3


 
KEY POINTS
? Two real numbers or two algebraic expressions related by the 
symbol 
'<', '>', ' ?' or ' ?' form an inequality. 
? The inequality containing < or > is called strict inequality. 
? The inequality containing ? or ? is called slack inequality. 
? 6 < 8 statements 7.2 > –1 are examples of numerical 
inequalities and 3x + 7 > 8, x + 3 ? 7, 
3
2
y ?
> 2y + 2 are
examples of litral inequalities. 
? The inequalities of the form ax + b > 0, ax + b < 0, ax + b ? 0, 
ax + b ? 0 ; a ? 0 are called linear inequalities in one variable x 
? The inequalities of the form ax + by + c > 0, ax + by + c < 0, 
ax + by + c ? 0, ax + by + c ? 0, a ? 0, b ? 0 are called linear 
inequalities 
in two variables x and y. 
? Rules for solving inequalities : 
(i) a ? b then a ± k ? b ± k where k is any real number. 
(ii) but if a ? b then ka is not always ? kb 
 
If k > 0 (i.e. positive) then a ? b ? ka ? kb 
If k < 0 (i.e. negative) then a ? b ? ka ? kb 
Thus always reverse the sign of inequality while multiplying or 
dividing both sides of an inequality by a negative number. 
? Procedure to solve a linear inequality in one variables. 
(i) Simplify both sides by removing graph symbols and 
collecting like terms. 
(ii) Remove fractions (or decimals) by multiplying both sides 
by appropriate factor (L.C.M of denomination or a power 
of 10 in case of decimals.) 
(iii) Isolate the variable on one side and all constants on the 
other side. Collect like terms whenever possible. 
(iv) Make the coefficient of the variable. 
(v) Choose the solution set from the replacement set. 
? Replacement Set: The set from which values of the variable 
(involved in the inequality) are chosen is called replacement set. 
? Solution Set: A solution to an inequality is a number which 
when substituted for the variable, makes the inequality true. The 
set of all solutions of an inequality is called the solution set. It is 
obviously a subset of replacement set. 
? The graph of the inequality ax + by > c is one of the half planes 
and is called the solution region. 
? When the inequality involves the sign = or = then the points on 
the line are included in the solution region but if it has the sign < 
 
or > then the points on the line are not included in the solution 
region and it has to be drawn as a dotted line. 
? The common values of the variable form the required solution of 
the given system of linear inequalities in one variable. 
? The common part of coordinate plane is the required solution of 
the system of linear inequations in two variables when solved by 
graphical method.