Good afternoon class, welcome everyone.

Today, we will study the chapter Function. In functions we will study about many different kinds of

functions, modulus functions, greatest integer functions, fractional part functions, logarithmic

function. There are infinite types of different functions that we can study on. But to study all these,

first we should know how to solve the inequalities, because if we are able to solve basic inequalities

first only then we will be able to solve other questions. So, to study basic inequalities, first thing is

we should know about different inequalities.

First thing is basic inequalities. Basic Inequalities, if a is greater than b, if any number is greater than

the other number then a plus c is always greater than b plus c. a minus c is always greater than b

minus c. a into c will be greater than b into c. Is it always true, no, it will be true if and only if c is

positive, that is if in an inequality we multiple a positive number on both the sides inequality remains

same. And a into c will be less than b into c, if and only if c is less than negative and if we multiple

both the numbers with negative then what will be the sign of inequality, it will change. So, one, very

important thing that we learnt here is, in one inequality if we add or subtract from both the sides,

there will be no change in inequality and we are multiplying the same number on both the sides, it

will remain same if the number is positive and change if the number is negative. This is what we

learnt.

Same goes for division. Same goes for division, a by c, b by c, a is greater than b. a is greater than b.

So, a by c and b by c, what will be the relation? a by c will be greater than b by c, inequality will

remain same if c is positive and it will change if c is negative. Inequality remains same if c is positive

and it changes if c is negative. These are few basic inequalities that you should know.

Next thing, when the multiplication between two numbers will be positive. By multiplying two

numbers together when will it be positive. If a and b are of same sign. I hope everybody agrees, yes.

And when will multiplication of the two numbers be negative, if a and b are of opposite sign. If both

the numbers sign are opposite then the multiplication of both the numbers will be negative.

Let us use this concept to solve few questions. Say, we need to solve a question that, find x such that

x minus 1 into x minus 2 is greater than zero. We need to solve this question, we need to find x for

which x minus 1 into x minus 2 will be greater than zero. For example, let us put x equal to 3. 3

minus 1 is 2 and 3 minus 2 is 1, 2 into 1 is 2, which is greater than zero, so x equals to 3 satisfies this

in equation. So we have to find all other x like x equal to 3, which satisfy this in equation. What we

will do, we will find x, for which x minus 1 is zero, x minus 1 is zero for x equals to 1. Similarly, x

minus 2 is zero for x equals to 2. We will plot them on a straight line. Now, when x is greater than 2,

when x is greater than 2, x minus 2 is positive. If you subtract a number bigger than 2, so you will get

one positive number, for example 3 minus 2 is 1 positive. Like that only, when x is greater than 2,

number bigger than 2, x minus 1 is also positive. Both of them will be positive, positive into positive

will be positive. If x is bigger than 2, so x minus 2 into x minus 1 will be positive. Similarly, when x lies

between 1 and 2, when x is between 1 and 2, x minus 2. For example put 1 point 5 here, 1 point 5

minus 2 is minus point 5 it is negative and x minus 1 will be, between 1 and 2, x minus 1 is positive

because x is bigger than 1, so, bigger than 1 is positive. Negative into positive is negative. Similarly

here x minus 2 will be negative and x minus 1 will also be negative and negative into negative

becomes positive. Now, what we wanted to show is, when is it positive when both the numbers are

multiplied. Where they are positive, you can see that it is positive for all x greater than 2 and it is

positive for all x is less than 2. And the answer is x is less than 1 or x is greater than 2, which we can

write in set form as minus infinite to 1, union 2 to infinite. All these x, what will they do to this

inequality, satisfy. For all these x, this inequality will satisfy means what will happen x minus 1 into x

minus 2, positive. It is a very length process, so we will not solve all questions like this.

So will develop a short cut, what is that short cut. That short cut is called as Wavy Curve method. It is

applied when multiplication or division of 3, 4, 5 as many factors are possible is less than zero or

greater than zero. Say, question we have been asked x minus 1 into x minus 2 into x minus 3 is less

than zero. Find the value of x which satisfies this. How to solve? We will apply all the rule of x

factors. x minus 1 will be zero when x is equal to 1, x minus 2 will be zero when x is equal to 2, x

minus will be zero when x is equal to 3. When x is greater than 3, all the x factors will be positive and

product will be positive and after that negative, positive, negative, positive. You write it alternately.

What you have to write in the beginning, positive, on the right positive and then negative, positive

write alternately and whatever is asked in the question, answer according to that. What we have

been asked is, when will multiplication be less than zero. When it is negative, when x is less than 1 or

when x lies between 2 and 3. If we write it in set form minus infinite to 1 union 2 to 3. Everybody got

it, yes, very good.

Let us solve one more question. Solve for x, such that x minus 2 into x minus 3 into x minus 5 is less

than or equal to zero. First step.