# Simple Inequality

## Homework Statement

Solve : $$\frac {2}{x} < 3$$

The answers are $$\frac {2}{3} < x and x<0$$

## The Attempt at a Solution

$$\frac {2}{x} = 3$$
Multiplying by x and dividing by 3, I obtain $$\frac {2}{3} < x$$

Where did they obtain $$x<0$$ as an answer? Also, accounting that x can be a negative, $$\frac {2}{3} > x$$ also seems like a solution.

## Answers and Replies

Dick
Science Advisor
Homework Helper
Think about it. If x>0, then you get, as you say, x>2/3 AND x>0. Here the x>0 is superfluous since if x>2/3 it's automatically bigger than 0. If x<0 then you reverse the inequality when you multiply by x, so 2>3x. Or 2/3>x AND x<0. Here the 2/3>x is superfluous, since if x<0 it's automatically less than 2/3. Try some numbers if you don't believe me.

It just means x either has to be bigger than 2/3 or smaller than 0. Makes sense, for example 1/2 is smaller than 2/3, plug it in and u get 4, which is not smaller than 3. Now try 2 which is bigger than 2/3, u get 1 which is smaller than three. This is why 2/3 > x doesn't make sense.

Now obviously, if the number on the left was negative it would be smaller than 3 however infinitely small (-1000000) or -0.0001 it is. So anything smaller than 0 would make solve the inequality.

symbolipoint
Homework Helper
Education Advisor
Gold Member
Dick, you want also to emphasise that x must not be 0; 0 can not be solution in the example.

Dick
Science Advisor
Homework Helper
Dick, you want also to emphasise that x must not be 0; 0 can not be solution in the example.

I hereby emphasize I do not endorse x=0 as a solution. In fact, I denounce anyone who supports me who would say x=0. Because they would be a terrorist, since 2/0 is not defined. How's that?