SOLUTION: (x-4)(3x+1)(x+1)>0

Question 614843: (x-4)(3x+1)(x+1)>0
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
This has nothing to do with logarithms. Posting your problems in the correct category will improve the chances and speed of a response.

(x-4)(3x+1)(x+1) > 0
This inequality, in essence, says that the product of three numbers is greater than zero, IOW, the product of three numbers is positive.

The solution to this will be based on our understanding how this happens, how the product of three numbers turns out to be a positive number. With a little thought you should be able to figure out that this happens only if...

All three numbers are positive or two of the numbers are negative and the other one positive.

Now we just translate the above into inequalities and solve. Let's first deal with "All three numbers are positive". This translates into:
(x-4) > 0 and (3x+1) > 0 and (x+1) > 0
Solving these we get:
x > 4 and x > -1/3 and x > -1
We can "condense" these down to just
x > 4
because if x > 4 then it would also be greater than -1/3 and -1.

Now we deal with the "two of the numbers are negative and the other one positive" possibility. This is a bit involved because we don't know which two will be the negative ones. So we have to include all possibilities:
((x-4) < 0 and (3x+1) < 0 and (x+1) > 0) or ((x-4) < 0 and (3x+1) > 0 and (x+1) < 0) or ((x-4) > 0 and (3x+1) < 0 and (x+1) < 0)

Solving this we get:
(x < 4 and x < -1/3 and x > -1) or (x < 4 and x > -1/3 and x < -1) or (x > 4 and x < -1/3 and x < -1)

Let's look at this, piece by piece. First:
(x < 4 and x < -1/3 and x > -1)
The first two inequalities condense down to x < -1/3 because if this one is true then the other one would be, too. Now we're down to:
(x < -1/3 and x > -1)

Next: (x < 4 and x > -1/3 and x < -1)
The first and last inequalities condense giving us:
(x > -1/3 and x < -1)
But this is impossible. x cannot be greater than -1/3 and less than -1 at the same time. So there is no solution to this part.

Next: (x > 4 and x < -1/3 and x < -1)
The last two inequalities condense:
(x > 4 and x < -1)
This is also impossible. A number cannot be greater than 4 and less than -1 at the same time.

In summary, the solution to our problem is:

All three numbers are positive or two of the numbers are negative and the other one positive.

which translates into:
((x-4) > 0 and (3x+1) > 0 and (x+1) > 0) or (((x-4) < 0 and (3x+1) < 0 and (x+1) > 0) or ((x-4) < 0 and (3x+1) > 0 and (x+1) < 0) or ((x-4) > 0 and (3x+1) < 0 and (x+1) < 0))

which solves to:
x > 4 or (x > -1/3 and x < -1)

In words, our solution is "any number greater than 4 or any number between -1/3 and -1".
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