document.write( "Question 204386: Please help me solve 2x^2-x>3 and condense and simplify 2logx -log5-log(x+1).\r
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Algebra.Com's Answer #154298 by jsmallt9(3758)\"\" \"About 
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a) Solve \"2x%5E2+-+x+%3E+3\"
\n" ); document.write( "This can be solved in a manner similar to solving the related quadratic equation. In other words, get one side equal to zero and then factor the other side. So we'll start by subtracting 3 from both sides:
\n" ); document.write( "\"2x%5E2+-+x+-+3+%3E+0\"
\n" ); document.write( "Then factor:
\n" ); document.write( "\"2x+-+3%29%28x+%2B+1%29+%3E+0\"
\n" ); document.write( "What we have now is a product (multiplication) which is greater than 0. In other words, we have a product that is positive. And how do we multiply two numbers and end up with a positive result? It should be clear that the two numbers must be both positive or both negative. So we just state this in the form of appropriate inequalities:
\n" ); document.write( "(2x - 3 > 0 and x + 1 > 0) or (2x - 3 < 0 and x + 1 < 0)
\n" ); document.write( "The first two state that the factors are positive and the last two state that the factors are negative and since either of these will produce a positive result we separate the two pairs of inequalities by an \"or\".
\n" ); document.write( "Now we just solve these four inequalities:
\n" ); document.write( "(2x > 3 and x > -1) or (2x < 3 and x < -1)
\n" ); document.write( "(x > 3/2 and x > -1) or (x < 3/2 and x < -1)
\n" ); document.write( "The left pair of inequalities says that x must be greater than 3/2 and it must be greater than -1. With some thought it should be clear that only numbers greater than 3/2 would fit both. So we will replace the pair with just x > 3/2.
\n" ); document.write( "The right pair of inequalities says that x must be less than 3/2 and it must be less than -1. With some thought it should be clear that only numbers less than -1 would fit both. So we will replace the pair with just x < -1.:
\n" ); document.write( "x > 3/2 or x < -1
\n" ); document.write( "And this is our solution. Any number that is greater than 3/2 or less than -1 will work.

\n" ); document.write( "b) Simplify \"2+log%28x%29+-+log%28%285%29%29+-+log%28%28x%2B1%29%29\"
\n" ); document.write( "To simplify logarithms we often use the following properties:
  • \"log%28a%2C+%28x%2Ay%29%29+=+log%28a%2C+%28x%29%29+%2B+log%28a%2C+%28y%29%29\"
  • \"log%28a%2C+%28x%2Fy%29%29+=+log%28a%2C+%28x%29%29+-+log%28a%2C+%28y%29%29\"
  • \"log%28a%2C+%28x%5Ey%29%29+=+y%2Alog%28a%2C+%28x%29%29\"

\n" ); document.write( "Any of these propoerties may be used (in either direction) and in this problem we will use two of the three.
\n" ); document.write( "On the first term we will use the third property (from right to left) to \"move\" the coefficient (the \"2\") from in front of the log into the argument:
\n" ); document.write( "\"log%28%28x%5E2%29%29+-+log%28%285%29%29+-+log%28+%28x%2B1%29%29\"
\n" ); document.write( "Now, since we have subtraction of logarithms of the same base, we can use the second property (from right to left) to combine the logarithms:
\n" ); document.write( "\"log%28%28%28x%5E2%29%2F5%29%29+-+log%28%28x%2B1%29%29\"
\n" ); document.write( "\"log%28%28%28%28x%5E2%29%2F5%29%2F%28x%2B1%29%29%29\"
\n" ); document.write( "Now we can simplify:
\n" ); document.write( "\"log%28%28%28x%5E2%29%2F%285%28x%2B1%29%29%29%29\"
\n" ); document.write( "\"log%28%28%28x%5E2%29%2F%285x%2B5%29%29%29\"\r
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