document.write( "Question 1100461: How am i to get the solution to the equation |x-3|=2x+1?
\n" ); document.write( "It says I am to end up with a extraneous solution, but I do not know what that means or how to get it. \n" ); document.write( "

Algebra.Com's Answer #714946 by KMST(5328) $\"\"$ $\"About$

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Maybe $\"x-3%3E=0\"$ , and then $\"abs%28x-3%29=x-3\"$ ,
\n" ); document.write( "or maybe $\"x-3%3C0\"$ and then $\"abs%28x-3%29=-%28x-3%29\"$ .
\n" ); document.write( "
\n" ); document.write( "If $\"x-3%3C0\"$ ,
\n" ); document.write( "the equation simplifies to
\n" ); document.write( " $\"-%28x-3%29=2x%2B1\"$ <--> $\"-x%2B3=2x%2B1\"$ <--> $\"3-1=2x%2Bx\"$ <--> $\"2=3x\"$ <--> $\"x=2%2F3\"$ .
\n" ); document.write( "That is a solution of $\"-%28x-3%29=2x%2B1\"$ ,
\n" ); document.write( "but is it a solution of the original equation?
\n" ); document.write( "Let's check.
\n" ); document.write( "If $\"x=2%2F3\"$ , $\"x-3=2%2F3-3=-7%2F3\"$ and $\"abs%28x-3%29=7%2F3\"$ .
\n" ); document.write( "At the same time, if $\"x=2%2F3\"$ , $\"2x%2B1=2%2A%282%2F3%29%2B1=4%2F3%2B1=7%2F3\"$ .
\n" ); document.write( "Clearly, $\"x=2%2F3\"$ makes $\"abs%28x-3%29=2x%2B1\"$ true,
\n" ); document.write( "so $\"highlight%28x=2%2F3%29\"$ is a solution of $\"abs%28x-3%29=2x%2B1\"$ .
\n" ); document.write( "
\n" ); document.write( "What about the case when $\"x-3%3E=0\"$ ?
\n" ); document.write( "Will that lead to another solution?
\n" ); document.write( "If $\"x-3%3E=0\"$ ,
\n" ); document.write( "the equation simplifies to
\n" ); document.write( " $\"x-3=2x%2B1\"$ <--> $\"-3-1=2x-x\"$ <--> $\"-4=x\"$ .
\n" ); document.write( " $\"x=-4\"$ is a solution to $\"x-3=2x%2B1\"$ ,
\n" ); document.write( "but it is not a solution to the original equation.
\n" ); document.write( "It makes $\"x-3=-7\"$ , $\"abs%28x-3%29=7\"$ , and
\n" ); document.write( " $\"2x%2B1=2%2A%28-4%29%2B1=-8%2B1=-7\"$ .
\n" ); document.write( "Clearly, if $\"x=-4\"$ makes $\"abs%28x-3%29=7\"$ , and $\"2x%2B1=-7\"$ ,
\n" ); document.write( "it is not a solution to $\"abs%28x-3%29=2x%2B1\"$ .
\n" ); document.write( "we call it an extraneous solution.
\n" ); document.write( " \n" ); document.write( "

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