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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.
\n" ); document.write( "(x-4)(3x+1)(x+1) > 0
\n" ); document.write( "This inequality, in essence, says that the product of three numbers is greater than zero, IOW, the product of three numbers is positive.
\n" ); document.write( "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...
\n" ); document.write( "All three numbers are positive or two of the numbers are negative and the other one positive.
\n" ); document.write( "Now we just translate the above into inequalities and solve. Let's first deal with \"All three numbers are positive\". This translates into:
\n" ); document.write( "(x-4) > 0 and (3x+1) > 0 and (x+1) > 0
\n" ); document.write( "Solving these we get:
\n" ); document.write( "x > 4 and x > -1/3 and x > -1
\n" ); document.write( "We can \"condense\" these down to just
\n" ); document.write( "x > 4
\n" ); document.write( "because if x > 4 then it would also be greater than -1/3 and -1.
\n" ); document.write( "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:
\n" ); document.write( "((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)
\n" ); document.write( "Solving this we get:
\n" ); document.write( "(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)
\n" ); document.write( "Let's look at this, piece by piece. First:
\n" ); document.write( "(x < 4 and x < -1/3 and x > -1)
\n" ); document.write( "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:
\n" ); document.write( "(x < -1/3 and x > -1)
\n" ); document.write( "Next: (x < 4 and x > -1/3 and x < -1)
\n" ); document.write( "The first and last inequalities condense giving us:
\n" ); document.write( "(x > -1/3 and x < -1)
\n" ); document.write( "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.
\n" ); document.write( "Next: (x > 4 and x < -1/3 and x < -1)
\n" ); document.write( "The last two inequalities condense:
\n" ); document.write( "(x > 4 and x < -1)
\n" ); document.write( "This is also impossible. A number cannot be greater than 4 and less than -1 at the same time.
\n" ); document.write( "In summary, the solution to our problem is:
\n" ); document.write( "All three numbers are positive or two of the numbers are negative and the other one positive.
\n" ); document.write( "which translates into:
\n" ); document.write( "((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))
\n" ); document.write( "which solves to:
\n" ); document.write( "x > 4 or (x > -1/3 and x < -1)
\n" ); document.write( "In words, our solution is \"any number greater than 4 or any number between -1/3 and -1\". \n" ); document.write( "