Sr no: 000 class: XI sub: Maths chap: 6 Inequalities lesson: 01
The name of the chapter says it all.
***If we look at any two things, we note that:
One thing may be smaller than the other
It may be bigger , OR
It may be equal to other thing.
We can make a general statement: X can be <, = or > than some value if we are sure of ourselves.
***If we are not sure then we can say:
X may be less than Y or equal to Y but I cannot say with confidence.
X may be equal to or bigger than Y but I am not sure.
***Or we can even place a condition:
Mother tells us “ek kilo se jyaada aaloo nahin laana” meaning “kam chhahe le aana but 1 kg se jyaada nahin. Meaning ≤ 1 kg
***Or “kam se kam 1 kg mango laana “ meaning 1 kg yaa jyaada ie ≥ I kg
We can think of more similar examples.
*** Simply put, if A ≠ B, it is called an IN-equality. That is all. That is if X is NOT equal to Y then it is IN-equal.
*** In real life we compare more than two things. We come across mathematical equations also. Simple linear equations or higher order equations starting with quadratic equations. They can have a variety of solutions for their variables.
*** An equation also tells us that LHS = RHS,
it is an equality like
X + Y = 7 could be TRUE for x= 1,2,3,4,5,6,7 and y=0
OR x=2, y=5 and so on. We will go mad solving even such an easy equation.
Therefore, here also we introduce concept of IN-equality. Like saying find solutions for Y >OR = 5
In this case we now know that Y can have only values as 5,6,7. And then X can be 2,1,0
We could have also said Y is <3 in which case ie 2,1,0
Solutions for X would be 5,6,7
***These are called solutions.
SOLUTION: Those set of values of variable for which the given condition is TRUE.
*** We know how to solve equations.
We need to eliminate some variable by manipulations so that we can find value of the single remaining variable.
Given an equation we can
1. Multiply each element on both sides of eqn by some number
2. Divide each element on both sides of eqn by some number
3. Subtract something from both sides, add something to both sides
Since it’ s an eqn it does not matter whether integer used is positive or negative like 3 OR -3.
***But suppose we use INeq like
2X + 3Y < 9
Here in case of ineq REMEMBER small things like: -1 is bigger than -9; 0 is bigger than -100.
Therefore, although we have been doing the same thing by transferring one element of eqn from LHS to RHS and changing the sign, we do this, in case of ineq by ADDING or SUBTRACTING something from BOTH SIDES. For avoiding mistakes only.
Multiplication or division by a positive integer makes no difference.
***BUT multiplication or division by a NEGATIVE qty changes the signs of ineq OPPOSITE; < becomes >; just like -7 is less than -2 etc.
There is no problem in removing denominators.—Multiply by LCM.
TRY to keep LHS in its original sign. Keep sign of variable POSITIVE for ease.
EXAMPLE: suppose we solve an equation that leads to a solution like
X >= 1; This means that solution is x=1,2,3,4………. Infinity
You can draw this on x-axis.
See Fig 6.2 of your book.
Suppose solution comes to
X <2 then it means that the ineq is true for all values of x = -3,-4,-5,………. Minus infinity
EX: et us try to make an ineq Q25:
Let us call longest side as L, shortest as S and third side as T
Given that: L = 3S
T = L – 2
L + S + T => 61
Min Length of S ??? Means S ≥
PUT value of L and T in third eq, giving s ≥ 9 answer.
***ON GRAPHICAL SOLUTIONS:
Once you know the basics you can do a variety of things with the results, like presenting them in graphic form.
We know solutions can be :
1. One variable like x ≥ 5
2. In two variables, like X ≥ 8 and Y ≥ 3
Presenting them on graph is easy.
First presentation is like example above. A single line diagram, say, along x-axis.
For two dimensions we draw a graph in a plane.
Procedure is simple:
1. Put value to x = 0; find value of second variable. Mark these points on x-y coordinates
2. Plot value of Y for x = 0. One point
3. Now put y = 0 giving value of x; with an ineq. Mark Y = 0 and mark value of y. Second point on graph.
4. Connect these two points using a scale.
5. You can do same thing if second ineq is also given for the same problem.
6. It is so easy now. Just keep in mind the ranges like < , > .
***If we are sure that A is less than, = or > than something then we call it a STRICT ineq.
But if we are not sure and say A is (= or>) Or A is (< =) the it is called SLACK ineq
NOW you can and will be able to solve any solved and unsolved problems of your Text book.