# Question: 2's, x's and an Inequality

## Comment on 2's, x's and an Inequality

### can we work on quant A and B

can we work on quant A and B

first taking the reciprocals then multiplying by 2 then subscribing 2 then multiplying by -1 to get (1 for A) and (2-2/x for B) we already know that (2-2/x < 0) so A=1 is bigger

### Sorry, can you please

Sorry, can you please elaborate on your solution? I'm not sure what you are doing in each step.

Cheers,
Brent

### Hi Brent, why can’t we

Hi Brent, why can’t we subtract 2 from both sides and be left with -2/x<-2, then divide the entire equation by -2?

### That approach also works.

That approach also works.

GIVEN: 2 - 2/x < 0
Subtract 2 from both sides to get: -2/x < -2
Divide both sides by -2 to get: 1/x > 1
IMPORTANT: If 1/x > 1, then we know that x is POSITIVE
So, we can safely multiply both sides by x to get: 1 > x
This means Quantity A is greater.

Cheers,
Brent

### why can't you just start by

why can't you just start by multiplying both sides by x. Whether or not x is positive does not affect 0. Doing so gives us 2x-x<0. Which = x<0 and A is still greater.

### Regardless of whether one

Regardless of whether one quantity is zero, we can't multiply both quantities by a variable unless we are 100% certain that the variable is POSITIVE

Consider this example:
QUANTITY A: 1/x
QUANTITY B: 0

Multiply both quantities by x to get:
QUANTITY A: 1
QUANTITY B: 0

Quantity A is greater. So, the answer is A (or is it?).
See the problem here?
-----------------------------------
Consider the two cases where x is either positive or negative

If x is POSITIVE, then we get:
QUANTITY A: 1/x = 1/POSITIVE = POSITIVE
QUANTITY B: 0
In this case, Quantity A is greater.

If x is NEGATIVE, then we get:
QUANTITY A: 1/x = 1/NEGATIVE = NEGATIVE
QUANTITY B: 0
In this case, Quantity B is greater.

So, the correct answer is actually D
-----------------------------------

Does that help?

Cheers,
Brent

### How can we multiply both

How can we multiply both sides by x if we don't know if x is positive or negative??

I was comparing 2 < 2/x and reasoned that x must be less than 2 for the inequality be true, therefore A is the answer. Is my reasoning correct here?

### Starting at 0:50 in the video

Starting at 0:50 in the video, I explain how we know that x is positive. It goes like this.

Given: 2 - 2/x < 0
Add 2/x to both sides: 2 < 2/x
If 2/x is greater than 2, we know that 2/x is POSITIVE

If 2/x is POSITIVE, then x must be POSITIVE

Cheers,
Brent