document.write( "Question 404526: solve: (2x/1-x) - (1/x) = 0 please show me step by step. i want to confirm my answer and if wrong i want to know where i went wrong. \n" ); document.write( "

Algebra.Com's Answer #285887 by jsmallt9(3759) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"%282x%29%2F%281-x%29+-+1%2Fx+=+0\"$
\n" ); document.write( "There are a couple of ways to solve an equation like this. The more general way would be to start by multiplying both sides of the equation by the Lowest Common Denominator (LCD) of the two denominators. The LCD of (1-x) and x is simply their product: (1-x)(x). Multiplying both sides by this we get:
\n" ); document.write( " $\"%28%282x%29%2F%281-x%29+-+1%2Fx%29%281-x%29%28x%29+=+%280%29%281-x%29%28x%29\"$
\n" ); document.write( "The right side, with the factor of zero is mindlessly easy to multiply. On the left side we must use the Distributive Property to multipy it correctly:
\n" ); document.write( " $\"%28%282x%29%2F%281-x%29%29%281-x%29%28x%29+-+%281%2Fx%29%281-x%29%28x%29+=+0\"$
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( "

\n" ); document.write( " $\"%28%282x%29%2F1%29%28x%29+-+%281%2F1%29%281-x%29+=+0\"$
\n" ); document.write( " $\"%282x%29%28x%29+-+%281-x%29+=+0\"$
\n" ); document.write( "(See how multiplying by the LCD makes the fractions go away?)
\n" ); document.write( " $\"2x%5E2+-+1+%2B+x+=+0\"$
\n" ); document.write( "Now we have a quadratic equation to solve. First let's put the terms in the \"right\" order:
\n" ); document.write( " $\"2x%5E2+%2B+x+-+1=+0\"$
\n" ); document.write( "Now we factor (or use the Quadratic Formula). This factors fairly easily:
\n" ); document.write( "(2x-1)(x+1) = 0
\n" ); document.write( "From the Zero Product Property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So:
\n" ); document.write( "2x-1 = 0 or x+1 = 0
\n" ); document.write( "Solving these we get:
\n" ); document.write( "x = 1/2 or x = -1

\n" ); document.write( "Another way to solve this will not always work on equations like yours. But with the zero on one side and two fractions, we can make a proportion out of the equation by just adding 1/x to both sides. (A proportion is where both sides of an equation are fractions.) Adding 1/x to each side of $\"%282x%29%2F%281-x%29+-+1%2Fx+=+0\"$
\n" ); document.write( " we get:
\n" ); document.write( " $\"%282x%29%2F%281-x%29+=+1%2Fx\"$
\n" ); document.write( "As you can see, this is a proportion. And proportions can be solved by cross multiplying:
\n" ); document.write( " $\"%282x%29%28x%29+=+%281-x%29%281%29\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"2x%5E2+=+1-x\"$
\n" ); document.write( "This is a quadratic equation. So we want one side to be zero. Adding x to and subtracting 1 from each side we get:
\n" ); document.write( " $\"2x%5E2+%2B+x+-+1+=+0\"$
\n" ); document.write( "This is the same equation we got the other way. And it will of course have the same solutions.

\n" ); document.write( "Not all equations like yours can be turned into a proportion. For those equations you should use the first method used above: Multiply both sides of the equation by the LCD of all the denominators. \n" ); document.write( "

\n" );