need help with another inequality

[abel muroi]

I was given another problem x/(x +1) > 3x

and so far here is the work that i did but i am stuck in a certain step.

x/(x+1) - 3x > 0 ((subtracted 3x from both sides))
x - 3x² - 3x/(x + 1) > 0 ((i multiplied 3x with (x + 1) which got me 3x² - 3x))
-2x - 3x²/(x +1) > 0 (( i combined like terms which got me -2x - 3x² ))

so this is as far as i got before i got stuck. do i have to factor -2x - 3x² or should i just find the "critical" numbers from the fraction?

 

[Steven G]

I was given another problem x/(x +1) > 3x

and so far here is the work that i did but i am stuck in a certain step.

x/(x+1) - 3x > 0 ((subtracted 3x from both sides))
x - 3x² - 3x/(x + 1) > 0 ((i multiplied 3x with (x + 1) which got me 3x² - 3x))
-2x - 3x²/(x +1) > 0 (( i combined like terms which got me -2x - 3x² ))

so this is as far as i got before i got stuck. do i have to factor -2x - 3x² or should i just find the "critical" numbers from the fraction?

The general answer to your question is if you can factor then factor. In this case you should factor -2x - 3x²

 

[HallsofIvy]

I was given another problem x/(x +1) > 3x

and so far here is the work that i did but i am stuck in a certain step.

x/(x+1) - 3x > 0 ((subtracted 3x from both sides))
x - 3x² - 3x/(x + 1) > 0 ((i multiplied 3x with (x + 1) which got me 3x² - 3x))
-2x - 3x²/(x +1) > 0 (( i combined like terms which got me -2x - 3x² ))

so this is as far as i got before i got stuck. do i have to factor -2x - 3x² or should i just find the "critical" numbers from the fraction?

Either way would work- in fact, they are pretty much the same thing. It is easy to factor -2x- 3x²= (-1)x(2+ 3x). Now a fraction is positive if and only if numerator and denominator have the same sign so, since -1 is negative, this fraction is positive if and only if
1) all three factors are negative
2) two of the factors are positive and the third negative.

And the simplest way to determine that is to look at the "critical numbers", the numbers that make each of those 0. Obviously, x= 0 makes -x equal to 0, x= -2/3 makes 2+ 3x 0, and x= -1 makes x+ 1 0. Also, where one of those is 0 is the only place the fraction can change sign. Writing them in increasing order, -1< -2/3< 0.
If x< -1, all three factors are negative
If -1< x< -2/3, x+ 1 is positive but the other two are negative.
If -2/3< x< 0, both x+ 1 and 2+ 3x are positive but x is negative.
if 0< x, all three factors are positive.