Woa, Woa, Woa.
Hold on.
You have to be really careful while multiplying with anything on both sides of an inequality.
Inequalities behave a little differently compared to an equation. Let's see how.
For e.g.
5 > 3
Lets multiply with a positive number on both sides (Let's say 2).
5*2 ? 3*2
What happens to the sign of the inequality?
15 is greater than 6. So, the sign of the inequality does not change.
5*2 > 3*2
Lets multiply with a negative number on both sides (Let's say -2).
5*(-2) ? 3*(-2)
What happens to the sign of the inequality?
-15 is smaller than -6. So, the sign of the inequality REVERSES.
5*(-2) < 3*(-2)
As we saw: (two simple guidelines)
(i) When we multiply both sides of an inequality with a positive quantity, the sign of the inequality DOES NOT CHANGE. Yes, you're right. NO EFFECT.
(ii) When we multiply both sides of an inequality with a negative quantity, the sign of an inequality REVERSES. It exactly reverses.
Do you see your mistake now?
You multiplied both sides by a quantity without changing the sign of the inequality.
That is, you assumed the quantity (x + 1) to be positive. And, we don't know if it is positive or negative.
Had (x + 1) been known to be positive, what you did was correct. But, since we don't, what you did is almost criminal.
This is an important concept and a very common mistake. Good that you made mistake, asked it and got it clarified in time.
Wait, This is getting interesting. Let's probe further?
We will solve Statement (1) by your method but we will consider both the possibilities.
Given: 1/(x + 1) < 1
If (x + 1) > 0, i.e. if x > -1,
(multiplying both sides by (x + 1))
1 < x + 1
or x > 0
What is the common solution of x > -1 and x > 0?
x > 0
However, if (x + 1) < 0, i.e. if x < -1,
(multiplying both sides by (x + 1) and reversing the sign)
1 > x + 1
or x < 0
What is the common solution of x < -1 and x < 0?
x < -1
So, what is the overall solution?
x > 0 OR x < -1
Is this the same solution that we got before also?
Yes. 'x' can be positive as well as negative.
Wow.
