Let’s begin with inequalities first. Inequalities is more like a concept than a chapter. Let’s go through some basic concepts;

Let’ say that x>y (this basically means that x is greater than y) and if I add or substract a constant on both sides of inequality would the sign of inequality change?? Didn’t get it? No worries, lets take some numbers.

Let’s talk about two numbers 10 and 7. We all know that 10 > 7, right? Now let’s add a constant on both sides of inequalities, say, for example let’s add 5 on both sides or let’s add 0.1 on both sides. If I add 5 on both sides, LHS will become 15 and RHS will become 12 and we all know that 15 > 12 and if I add 0.1 on both sides, then 10.1 > 7.1. Hence we can say that sign of inequality would not change.

Similarly, if we substract 5 or 0.1 on both sides, we’ll get 5 > 2 or 9.9 > 6.9. Again we can say that sign of inequality will not change.

Hence, two important conclusions, whenever we add or substract the sign of inequality will not change.

Now, let’s talk about multiplying and dividing by a positive constant. Let’s first multiply 10 and 7 by 5, we’ll get 50 and 35 and we all know that 50 > 35, hence the sign of inequality will not change, similarly let’s divide 10 and 7 with 5, hence we’ll get 2 and 1.4 and we know that 2 > 1.4.

Hence, we can again say that, whenever we multiply or divide by positive constants the sign of inequality will not change.

Now, let’s talk about multiplying and dividing by a negative constant. Let’s first multiply 10 and 7 by -5, we’ll get -50 and -35 and we all know that -50 < -35, hence the sign of inequality will change (or reverses), similarly let’s divide 10 and 7 with -5, hence we’ll get -2 and -1.4 and we know that -2 < -1.4.

Hence, we can again say that, whenever we multiply or divide by positive constants the sign of inequality will change (or reverses).

Now, how do we use them.

Lets start with an easy question first of all. If 4a – 6b = 0 and b < 5, then find the real values of a satisfying the inequality.

4a =6b ; a = 1.5b ; a = 1.5(<5) ; a < 7.5

Let’s try another one. If -0.25p > 4, then find the range of values of p, satisfying the above inequality.

Since we have negative sign in front of p, so what we’ll do is, we’ll multiply by -1 on both sides of inequality, which will change the sign of inequality and hence inequality will become 0.25p < -4, now we’ll multiply both sides by 0.25 and since 0.25 is a positive quantity hence sign of inequality will not change. Final result will become p < -1.

In order to have in-depth understanding of inequality go through the below video.

Modulus

Most of you would be knowing that, result of |±5| = 5. Ever thought why ?? Whenever I ask this question, mostly student says that its because mod always gives positive result or it gives the absolute result, but that is not the correct explanation. The explanation of this lies in the definition of modulus.

So what is modulus or what is meant by |±5|. When we write |±5|, it basically means, if you are standing on 0 on a number line whether you walk to +5 or you walk to -5, the distance covered is 5 units, hence the answer is 5.

Hence to conclude, modulus basically means distance covered. And since distance covered is always non-negative, hence the result of modulus is never negative.

Let’s take this concept forward, if I say |p – 4| = 7, then find the value of p. Before you figure out the value of p, first try and understand on how to read this expression, because that is important. The equation above can be read as distance between p and 4 on a number line is 7. Hence which all numbers are at a distance of 7 from 4. The two numbers are 11 and -3. And that is your answer.

This is the concept which is often tested from modulus.

So remember two things. First is result of modulus is always non-negative and second is the distance concept.

In order to have in-depth understanding of inequality go through the below video.

Try the below questions once you have gone through the above video;