Inequations (Class 11, CBSE) Video Lecture

hi this is second video of the
inequalities and this we will discuss
about the V method to solve the
inequalities and plot think the numbers
on the number line the very first thing
about the solution of inequalities let
me tell you about X minus 2 greater than
equal this 0 X minus 2 is greater than 0
means means x is greater than 2 it means
take X is taking all the values which
I'll get our than to take a plot a
number line say this is 0 it says that X
greater than 2 is the solution of this
required we have started in the previous
video also right now how to read
solution of this problem he says that X
is greater than 2 means the writing way
is X belongs to this sign says belongs
to 2 to infinity all the values from 2
to infinity means X minus 2 is greater
than 0 yet we used this small brackets
small packets now keep this mind keep
this thing in your mind event to use a
small back when to use a square bracket
now how did beat the equation be X minus
2 greater than equal to 0 it means all
the values including two and gather them
two are the solution of this McCarthy
right says this is get is equal to two
now in this case when we have to show
that X includes the value of two also so
we say set X belongs to the square
bracket 2 to infinity so this is a
difference in writing the solutions has
come with this defense in the inequality
sign is a sign it fluids the value given
value we include the value in this way
this one bracket right so this is the
basic problem to writing the problem
solutions and plotting on the number
line let's take the more examples the
problem is of a sort X minus 2 less than
0 will say that X belongs to minus
infinity to 2 right I've had it pinned
this X minus 2 less than equal to 0
means X belongs to minus infinity to
minus 6 my 22 and this
it has to be square back adversity feuds
the value if the lizard's you now in all
the four cases we have never used
clothes back into the infinity because
infinity has it's not a fixed value we
cannot approach you can just approach to
infinity the exact infinity is not
defined just say this this small bracket
in front of the infinity while when we
half the value certain value fixed yes
it includes the value to also we include
the bracket oh right this is a way of
reading solution and now how did mean
the problems like X minus 2 into X minus
4 is greater than 0 and it says the X
minus 2 into X minus 4 is greater than 0
it means it's saying that X square minus
6x plus a is greater than 0 how to solve
this kind of in politics for example
this is in quality of the quadratic
equation in equations this is a quiet
equation and let's see how to solve it
first we split into the factors that is
X minus 2x minus four which is greater
than zero now solution use this is 0 1 2
3 4 minus one this one is nucleating
separately solve the critical points we
create this thing equal to 0 the
critical points is x equal to two week
wait this s20 then says it x equal to 4
remark this critical point on a number
line say this is X is 2 and again
mathematical point X now the learning
thing over here if you have marked this
2 & 4 here now assume the certain value
which is less than to say one for the
value 1 X minus 2 this is negative 4 11
minus 4 again- if you multipath in to
negative quantities the answer is
positive yes is no badami now if I take
a certain value which is bigger than
four let's say five if I take a 5-2 post
your quantity that is 35 minus 4 that is
positive quantity that is one what the
multiplying both the pasta quantity will
get the answers if we take some value
between two and four that it let's say
take value 3 for 3 minus 2 minus 1 and 3
minus 3 minus 2 positive 1 and 3 minus 4
negative 1 so negative into positive it
gets the negative I this is not in what
da me right
now it says that all the values in this
range gives a positive result all the
rallies in this range give the
negativism and all the values beyond
this will get a positive yourself right
so good so adding the solution X belongs
to the positive one is until infinity
that you would say infinity minus
infinity obviously I will not include
the square beggary up to to fight this
is 22 is not included because here is
the only greater in size sign here's the
equality equality is not included now we
got the solution all solution between
minus 22 minus infinity to 2 now we are
we have solution between 4 to infinity
he says this is Union you union means
including the solution for two so this
gives the solution of this impact this
way you present solution and in this way
we will present in the number right if
same problem is like like X minus 2 X
minus 4 is less than equal to 0 it means
i am saying that X square my x 6x plus 8
is less than equal to C right now again
we have splitted into the factors these
are the factors we've got down the
critical point that is exported to an X
equal to four critical points again I am
Telling is cutting back the marking the
values integer values to 0 right these
are critical points again I am drawn the
number line X equal to 2 is a critical
value X equal to 4 is a connected so
obviously less than 2 well we'll give
the positive this is minus and faster so
remember every time we start with from
the back end that is from the left-hand
side we start marking positive the
negative and positive so remember this
thing so the writing solution is because
we have the range of less than 0 means V
we are talking about this negative thing
this is less than zero different
solution is X belongs to two to four
right and now two and four are also
included because it's skills in quality
signs so we change the bracket here you
make
expect as a squire packet so solution of
business Garrett is X plums from two to
four and two n wat included and this is
way to write the solution next thing is
say if this equation becomes X minus two
X minus 3 and X plus 2 I ask you when
the solution X minus 2 into X minus 3
and Express to greater than zero
obviously we all multiply this week ever
get a cubic equation sometimes cubic
equation is given we have to split into
these factors select me to teach you the
outsourcing qualities if X minus 2 X
minus 3 in express look if we accept
separately we're 20 X minus 2 equal to 0
it gives a critical point to it gives a
critical point 3 q's a critical point
minus again the first step is draw the
number line say this is 0 1 2 3 minus 1
minus 2 now plot this critical points or
number line vs 2 and 3 then minus 2 now
we got French from this than this from
this again we do the same thing we start
on the left hand side we start marking
the positive negative positive and then
negative we have marked on them now we
check the problem is of greater than
zero it means all solution which has a
positive sign is the solution of this
inequality now let's write the solution
here X plus 2 minus infinity to 2 right
22 then we close the bracket now these
are the closing bracket would be a small
bag or square method obviously it's a
square bracket obviously it's a square
bracket because it's a sign of equality
also will c square bracket then union
then again a square bracket that is 223
this is a solution of this inequality
right what if so what is the solution 14
quality X minus two X minus 3 next
cluster the lens then see if I solve
this thing means the domain points are
negative 1 this is less than zero here
is not in equality to make the solution
is
x belongs to we start my left hand side
this is positive do not take it this is
negative we include this minus 2 2 to
both have the small bracket because
values are not included because you're
here is it it's not a sign of equality
here then Union 32 in so for this
equation solution is this for this the
solutions right next another important
thing while solving this equation is
pick the question is like X minus 1 X
minus 2 for X minus 3
again there is same method if it is in
the division if it's multiplication does
not matter again we plot the number line
0 1 2 and 3 kids apostasy again with
mark the critical points X minus one
equal to zero critical point from here
is one critical point from here is two
critical point from here it's three you
mark the point one two and three that's
Carol insane amount of the reasons this
is positive than negative and positive
then negative now here design is
positive solution is of course x belongs
to minus infinity to 1 then Union this
is negative with left it then 2 to 3 is
a solution right now the important point
over here is n had it been the solution
included here p if solution be if the
equation might have signed this X cannot
equal to 0 right the solution will
become obviously the number line is seen
give me right solution X plums from
minus infinity to 1 obviously we include
this bracket yet greater than sign you
need two to three another thing is yell
free isn't denominator if x is equal to
3 this home value is not defined because
X cannot be three because denominator
cannot be zero so we include will not
include the three just recruited to this
is the right solution will not include
the three so when entering half the
denominator value when's 20 will not
include that value because number zero
is not a fine so whenever we have the
critical points in the denominator will
not include that points therefore we
have not included it's fine this thing
will always be a closed bracket or small
break up and break it next solution of
the quantity x squared plus 1 into X
minus 2 in X plus 3 plus NZ now again we
mark the critical points here the
critical point is to hear the critical
for it is minus what about this this
quantity Kelly now
p 0 x squared plus 1 equal to 0 this not
possible for all the values of X with a
negative positive this quantity square
is positive and again one is bostic so
adding if you add to Basra quantitative
you will never get the values you so
this is this never become faster than
zero now if you put any value of the
expo co- this thing from fasting and
fasting plus one is always first to say
this always remain positive right we
have learned that multiplying any
positive bonded to the equation the
equation will not change so now this
equation will turn up like this X minus
2 into X plus 3 is less right now we can
write the solution is X minus 2 to plus
3 is less than 0 therefore the solution
is minus 3 to do you plot the number
line minus C and this is to this is
positive this is negative and again this
is possible so we're range is minus 32
so always remember the critical things
first check your denominator if it is
denominator cannot be 0 second thing
whether the quality is already a
positive one so we'll eliminate the
pasta quantity and solve the remaining
problem final solutions equation x
squared plus 1 into x square plus X plus
1 less than 0 now x square plus 1 this
is always be a positive quantity it fits
x squared plus X plus 1 if you check it
d D that is discriminant of this
quadratic equation is negative so this
has the not this does not have any
positive roots and of course this Monty
is scattered then the value of efficient
of x squared is positive so the value of
this expression is always positive if
you draw the graph of this is something
above the x-axis the album the value of
this expression is always positive this
is positive this is positive multiplying
the two positive quantity will always
get a pass to monthly therefore this
quantity can never be a negative one so
the solution here is x5
this null set that is no value which is
satisfied this quantity right next
question X square plus 1 X square minus
X 1 X square plus X plus 1 fix an X
minus Rho this lesson 0 if we solve this
problem again we mount the past critical
points here critical part series this
quantity is positive Naga tickle points
critical points here X square minus X
critical point size 0 and 11 X equal to
0 become 0 and 4 1 this is 0 this
quantity is always positive this makes
this less than equal to 0 this is 0 and
this is one right now let's plot this
critical price the critical points are
zero and zero and Polizei equal to 0 for
up this quantity also 0 and what these
are the critic for critical points to RM
to R 3 and C now if you've made in the
past or negative points this is this
positive again the value between 0 and 0
is negative which is nothing that costed
then negative them first here we see
that all the valances is positive except
zero and forth between something's you n
0 there is nothing quantity where it is
negative and against negative 1 so this
is an important example you X cannot be
0 because the denominator and X cannot
be 1 because again it is into
denominator so to say the solution of
this is X belongs to one there is no
solution of this equation because for
every value it is positive except the
value zero and but 40 and what it is not
defined right solution is x belongs to
Phi and there is no solution of this
quantity hadn't been the problem we it
is greater than equal to 0 a degree in
the problem is X square than equal to 0
it means the value is equating value
right so toleration what is x belongs to
TL lambert's any value
minus zero and will not include the
value zero and because in the
denominator more ever you analyze the
problems x square minus X this is also X
square minus six we can cancel out but
we'll are not before cancelling mark the
critical points on show because there
are important into the problems right
you can cancel the things but mark the
critical points and then solve the
problem again magnetical points and said
that the value instead of inter geometry
0 or not this is a good example of
solving the box thank you have a nice
day