Question 614843
This has nothing to do with logarithms. Posting your problems in the correct category will improve the chances and speed of a response.<br>
(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.<br>
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...<br>
All three numbers are positive or two of the numbers are negative and the other one positive.<br>
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.<br>
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)<br>
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)<br>
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)<br>
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.<br>
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.<br>
In summary, the solution to our problem is:<br>
All three numbers are positive or two of the numbers are negative and the other one positive.<br>
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))<br>
which solves to:
x > 4 or (x > -1/3 and x < -1)<br>
In words, our solution is "any number greater than 4 or any number between -1/3 and -1".