inequalities (i)

4/x < 1/3 means that 4/x - 1/3 < 0. Bring both fraction to their common denominator to get:

(12 - x)/3x < 0.

This is why you need the intervals for which 12 - x and 3x have a different sign. Start by splitting the number axis into three parts: x < 0, x is between 0 and 12, x is greater than 12. We'll analyze each part separately:

a. x < 0 means that 3x is negative as well. But since x < 0, then -x > 0, with 12 - x being also greater than zero. This means that this interval is a keeper.

b. x is between 0 and 12 means that 3x > 0. 12 - x will be also greater than zero, since if you subtract anything less than 12 from 12 you'll get a positive value. Since 3x and 12 - x are both positive, this case is no good.

c. x is greater than 12 means that 3x > 0. But 12 - x will be negative: when you subtract something more than 12 from 12, then you'll surely end up with a negative values. Since 3x and 12 - x have different signs, this one's a keeper too.

So the final answer will indeed be the one pointed out by you.

i see it this way:

4/x < 1/3
or
x/4 > 3
x>12

since 4/x is there so x can not be zero as we know value of x from 0 to 111 will violate the equalities x<0 and so <0 or x>12

Thanks, DanaJ.

Although I understand how to get x > 12, I don't understand how x < 0.

Would appreciate your clarification, thanks.

DanaJ wrote:4/x < 1/3 means that 4/x - 1/3 < 0. Bring both fraction to their common denominator to get:

(12 - x)/3x < 0.

This is why you need the intervals for which 12 - x and 3x have a different sign. Start by splitting the number axis into three parts: x < 0, x is between 0 and 12, x is greater than 12. We'll analyze each part separately:

a. x < 0 means that 3x is negative as well. But since x < 0, then -x > 0, with 12 - x being also greater than zero. This means that this interval is a keeper.

b. x is between 0 and 12 means that 3x > 0. 12 - x will be also greater than zero, since if you subtract anything less than 12 from 12 you'll get a positive value. Since 3x and 12 - x are both positive, this case is no good.

c. x is greater than 12 means that 3x > 0. But 12 - x will be negative: when you subtract something more than 12 from 12, then you'll surely end up with a negative values. Since 3x and 12 - x have different signs, this one's a keeper too.

So the final answer will indeed be the one pointed out by you.

You basically "split" the number line into three bits and the "rupture points" are the ones where either the denominator or the numerator of the fraction change their signs.
12 - x is positive up to 12, but negative from there on.
3x is negative up to zero and positive for value grater than 0.
So you basically just use these two "landposts" for your calculations (try to look at it like a little table):

x 0 12
3x negative 0 positive positive
12-x positive positive 0 negative

Getting to x>12 is easy.

But you have to remember that if x is -1 then the inequality is still true. So there is a whole range of negative numbers for which the inequality is true.

you can use brute force to find the point where 4/x is -ve...
try x=1 which makes inequality not true coz 4/1 > 1/3.
Try x =0 is undefined.
Try x = -1 makes the inequyality true coz the answer is -ve.
Then try x= -0.0000000001 still makes the inequality true coz the answer is -ve so you can be confident that when x<0 the inequality is true.

But DanaJ's way is more elegant.

The trick is realizing that there are -ve values for x that you have to consider. If it were a multiple choice question, the answers might give you a clue that you were not considering something...