document.write( "Question 1069303: Solve the following inequality. Write the answer using interval notation.
\n" ); document.write( "1+2/(x+1)≤2x \n" ); document.write( "

Algebra.Com's Answer #684525 by KMST(5328) $\"\"$ $\"About$

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$\"1%2B2%2F%28x%2B1%29%3C=2x\"$ <--> $\"2%2F%28x%2B1%29%3C=2x-1\"$
\n" ); document.write( "After that we have to think.
\n" ); document.write( "We know that $\"x%2B1=0\"$ <--> $\"x=-1\"$ is forbidden.
\n" ); document.write( "so, either $\"x%2B1%3E0\"$ <--> $\"x%3E-1\"$ ,
\n" ); document.write( "or $\"x%2B1%3E0\"$ <--> $\"x%3E-1\"$ .
\n" ); document.write( "We also know that if we multiply both sides of an inequality times a positive number
\n" ); document.write( "(or divide by a positive number),
\n" ); document.write( "the relationship (greater than, or lesser than) remains the same,
\n" ); document.write( "but if we multiply times or divide by a negative number,
\n" ); document.write( "the resulting inequality has the opposite sign.
\n" ); document.write( "So, for $\"x%2B1%3E0\"$ <--> $\"x%3E-1\"$ ,
\n" ); document.write( " $\"2%2F%28x%2B1%29%3C=2x-1\"$ is equivalent to
\n" ); document.write( " $\"2%3C=%282x-1%29%28x%2B1%29\"$ <--> $\"0%3C=%282x-1%29%28x%2B1%29-2\"$ <--> $\"0%3C=%282x%5E2%2Bx-1%29-2\"$ <--> $\"2x%5E2%2Bx-3%3E=0\"$ ,
\n" ); document.write( "and for $\"x%2B1%3C0\"$ <--> $\"x%3C-1\"$ ,
\n" ); document.write( " $\"2%2F%28x%2B1%29%3C=2x-1\"$ is equivalent to
\n" ); document.write( " $\"2%3E=%282x-1%29%28x%2B1%29\"$ <--> $\"0%3E=%282x-1%29%28x%2B1%29-2\"$ <--> $\"0%3E=%282x%5E2%2Bx-1%29-2\"$ <--> $\"2x%5E2%2Bx-3%3C=0\"$ .
\n" ); document.write( "So, the solutions to $\"1%2B2%2F%28x%2B1%29%3C=2x\"$ will be
\n" ); document.write( "the solutions to $\"system%28x%3E-1%2C2x%5E2%2Bx-3%3E=0%29\"$
\n" ); document.write( "and the solutions to $\"system%28x%3C-1%2C2x%5E2%2Bx-3%3C=0%29\"$ .
\n" ); document.write( "We can factor $\"2x%5E2%2Bx-3\"$ ,
\n" ); document.write( "which is better than having to find the zeros any other way,
\n" ); document.write( "makes it easier to see when when it is positive, negative, or zero.
\n" ); document.write( " $\"2x%5E2%2Bx-3=%28x-1%29%282x%2B3%29=0\"$ for $\"x=-3%2F2=-1.5\"$ and for $\"x=1\"$ .
\n" ); document.write( " $\"2x%5E2%2Bx-3=%28x-1%29%282x%2B3%29%3C=0\"$ for $\"-3%2F2=-1.5%3C=x%3C=1\"$ (in between the zeros).
\n" ); document.write( " $\"2x%5E2%2Bx-3=%28x-1%29%282x%2B3%29%3E=0\"$ for $\"x%3C-3%2F2=-1.5\"$ or for $\"x%3E=1\"$ .
\n" ); document.write( "The solutions to $\"system%28x%3C-1%2C2x%5E2%2Bx-3%3C=0%29\"$
\n" ); document.write( "are the solutions to $\"system%28x%3C-1%2C-3%2F2=-1.5%3C=x%3C=1%29\"$ ,
\n" ); document.write( "and that is $\"-3%2F2=-1.5%3Cx%3C-1\"$ or $\"highlight%28%22%5B+-1.5+%2C+-1+%29%22%29\"$ .
\n" ); document.write( "The solutions to $\"system%28x%3E-1%2C2x%5E2%2Bx-3%3E=0%29\"$
\n" ); document.write( "are the solutions to $\"system%28x%3E-1%2Cx%3E=1%29\"$ ,
\n" ); document.write( "and that is $\"x%3E=1\"$ or $\"highlight%28matrix%281%2C3%2C%22%5B+1+%2C%22%2Cinfinity%2C%22%29%22%29%29\"$ . \n" ); document.write( "

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