Algebra

Numbers — Algebra with Inequalities

Algebra….Algebra…Algebra….

I can understand the nightmares one gets while listening about Algebra. But is it so hard? Well, the concepts of Algebra to which we were introduced in fifth standard are still afresh in my mind. I still know the first formula to which we were introduced was BODMAS. I still remember the kind of difficulties the entire class was having in solving the questions through this formula except one mathematics genius girl in our class. She was really fast in holding on to new mathematical concepts and was very happy for being the only one answering the questions. Anyways, with time everyone learnt the concepts and moved on. But still I always remember her genius.

In this article, we are going to brush-up our knowledge of Numbers : Algebra with Inequalities concepts which we haven’t touched for long. These concepts are also important from the aspect of Data Science.

Algebra with equalities

We all know that 5 = 5

and 5+3 = 5+3 i.e.

8 = 8 (which is true)

So, the basic rule is :

If a = b , then a + c = b + c.

Example : x + 4 = 8

then (x + 4) - 5 = 8-5

i.e. x = 4

Similarly with multiplication,

i) 5 = 5

3 * 5 = 3 * 5

i.e. 15 = 15 (which is true).

ii) 3 = 3

(-5) * 3 = (-5) * 3

-15 = -15 (which is true).

So, the rule is :

If a, b, and c are numbers, and c is not equal to 0, and a = b, then c * a = c *b.

Example : -4x = 12

(-1/4) * (-4x) = (-1/4) * 12

x = -3

Algebra with inequalities

We know that 5 < 9

then 5 + 2 < 9 + 2

i.e. 7 < 11 (which is true)

Also, 5 < 9

then 5–2 < 9–2

i.e. 3 < 7 ( which is again true)

Rule : If a < b, then a + c < b + c.

Graph of R (the real numbers )

Example:

Let’s say, x + 4 < 11

then (x + 4)- 5 < 11-5

i.e. x < 7

Here, x is an element with value range between -infinity and 7

Test cases with multiplication:

As we know, 4 < 9

Now, 3 * 4 < 3 * 9

i.e. 12 < 27 ( which is true)

Similarly, 5 < 8

then (-1) * 5< (-1) * 8

i.e. -5 > -8 , so in case of multiplication with a negative integer reverses the rule.

In case of multiplication with -ve integer, the signs interchange

Rule :

Suppose a < b,

If c > 0, then a * c < b * c.

If c < 0, then a * c > b * c. (Keep this in mind)

So, in this article, we tried to brush up our knowledge of Numbers— Algebra with Inequalities concept.