Class 11 Maths Revision Notes for Linear Inequalities of Chapter 6 – Free PDF Download

Free PDF download of Class 11 Maths revision notes & short key-notes for Linear Inequalities of Chapter 6 to score high marks in exams, prepared by expert mathematics teachers from latest edition of CBSE books.

CBSE Class 11 Maths Revision Notes for Linear Inequalities of Chapter 6

• Inequalities: Two real numbers or two algebraic expressions related by the symbols <, >, ≤ or ≥ form an inequality. For example: 3x<20, 4x+y<12.
• Equal numbers may be  added to (or subtracted from) both sides of an inequality.
• Both sides of an inequality can be  multiplied (or divided) by the same  positive number. But when both sides are multiplied (or divided) by a negative number, then the inequality is reversed.
• Linear Inequality: An inequality is said to be linear, if each variable occurs in first degree only and there is no term involving the product of the variables. For example: ax+b<0,ax+b<0, ax+b≤0,ax+b≤0, ax+by+c>0,ax+by+c>0, ax+by+c≥0ax+by+c≥0.
• Closed Interval: If aa and bb are real numbers, such that a<b,a<b, then the set of all real numbers xx such that a≤x≤ba≤x≤b is called a closed interval and is denoted by [a,b].[a,b]. Therefore, [a,b]={x:a≤x≤b,x∈< /span>R}[a,b]={x:a≤x≤b,x∈R}.
• Open Interval: If aa and bb are real numbers, such that a<b,a<b, then the set of all real numbers xx such that a<x<ba<x<b is called a open interval and is denoted by (a,b)(a,b) or ]a,b[]a,b[. Therefore, (a,b)={x:a<x<b,x∈R}(a,b)={x:a<x<b,x∈R}.
• Solution of an Inequality: The values of x, which make an inequality a true statement, are called solutions of the inequality.
• To represent x <  a (or x >  a) on a number line, put a circle on the number and dark line to the left (or right) of the number a.
• To represent x ≤ a (or  x ≥ a) on a number line, put a dark circle on the number and dark the line to the left (or right) of the number x.
• If an inequality is having ≤ or ≥ symbol, then the points on the line are also included in the solutions of the inequality and the graph of the inequality lies left (below) or right (above) of the graph of the equality represented by dark line that satisfies an arbitrary point in that part.
• If an inequality is having < or > symbol, then the points on the line are not included in the solutions of the inequality and the graph of the inequality lies tothe left (below) or right (above) of the graph of the corresponding equality represented by dotted line that satisfies an arbitrary point in that part.
• To remove the denominator when the sign of the value of the denominator is unknown, may be +ve or -ve: We multiply by the square of the denominator. Square of the denominator is always positive. On multiplication of an inequation by a positive number the sign of inequality does not change.
• Solution of a system of Linear Inequality: The solution set of a system of linear inequality in one variable is defined as the intersection of the solution sets of the linear inequality of the system. For example: If we have two solution sets x>4x>4 and x>6,x>6, then solution of the system is the intersection of x>4x>4 and x>6,x>6, i.e., x>6.x>6.
• Properties of absolute values:
  (i) |x|<a⇔−a<x<a|x|<a⇔−a<x<a i.e., x∈(−a,a)x∈(−a,a)
  (ii) |x|≤a⇔−a≤x≤a|x|≤a⇔−a≤x≤a i.e., x∈[−a,a]x∈[−a,a]
  (iii) |x|>a⇔x<−a|x|>a⇔x<−a or x>ax>a i.e., x∈(−∞,−a)∪(a,∞)x∈(−∞,−a)∪(a,∞)
  (iv) |x|≥a⇔x≤−a|x|≥a⇔x≤−a or x≥ax≥a i.e., x∈(−∞,−α)∪(a,∞)x∈(−∞,−α)∪(a,∞)
• Triangle Inequality: |x+y|≤|x|+|y||x+y|≤|x|+|y|, |x−y|≥|x|−|y|

 
Inequation
A statement involving variables and the sign of inequality viz. >, <, ≥ or ≤ is called an inequation or an inequality.
Numerical Inequalities
Inequalities which do not contain any variable is called numerical inequalities, e.g. 3 < 7, 2 ≥ -1, etc. Literal Inequalities Inequalities which contains variables are called literal inequalities e.g. x – y > 0, x > 5, etc.
Linear Inequation of One Variable
Let a be non-zero real number and x be a variable. Then, inequalities of the form ax + b > 0, ax + b < 0, ax + b ≥ 0 and ax + b ≤ 0 are known as linear inequalities in one variable.
Linear Inequation of Two Variables
Let a, b be non-zero real numbers and x, y be variables. Then, inequation of the form ax + by < c, ax + by > c, ax + by ≤ c and ax + by ≥ c are known as linear inequalities in two variables x and y.
Solution of an Inequality
The value(s) of the variable(s) which makes the inequality a true statement is called its solutions. The set of all solutions of an inequality is called the solution set of the inequality.
Solving Linear Inequations in One Variable
Same number may be added (or subtracted) to 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.
Representation of Solution of Linear Inequality in One Variable on a Number Line
To represent the solution of a linear inequality in one variable on a number line. We use the following algorithm.
If the inequality involves ‘>’ or ‘<‘ we draw an open circle (O) on the number line, which indicates that the number corresponding to the open circle is not included in the solution set.
If the inequality involves ‘≥’ or ‘≤’ we draw a dark circle (•) on the number line, which indicates the number corresponding to the dark circle is included in the solution set.
Graphical Representation of the Solution of Linear Inequality in One or Two Variables
To represent the solution of linear inequality in one or two variables graphically in a plane, we use the following algorithm.
If the inequality involves ‘<’ or ‘>’, we draw the graph of the line as dotted line to indicate that the points on the line are not included from the solution sets.
If the inequality involves ‘≥’ or ‘≤’, we draw the graph of the line as a dark line to indicate the points on the line is included from the solution sets.
Solution of a linear inequality in one variable can be represented on number line as well as in the plane but the solution of a linear inequality in two variables of the type ax + by > c, ax + by ≥ c,ax + by < c or ax + by ≤ c (a ≠ 0, b ≠ 0) can be represented in the plane only.
Two or more inequalities taken together comprise a system of inequalities and the solution of the system of inequalities are the solution common to all the inequalities comprising the system.