document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207733>Question 1207733</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the absolute value equation. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Solve |x + |3x-2|| = 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: Here we have an absolute value inside another absolute value.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845779 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=845779\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%2Babs%283x-2%29%29=2 ALIGN=MIDDLE  ALT=\"abs%28x%2Babs%283x-2%29%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "If we think of this as a function, we know the behavior of the graph changes when (3x-2) is 0 -- i.e., at x=2/3.  So let's solve two separate absolute value equations, considering only values of x greater than or equal to 2/3 in one equation and considering only values of x less than 2/3 in the other.<br><BR>\n" );
document.write( "(1) If x is greater than or equal to 2/3, then |3x-2| is equal to 3x-2:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%2B%283x-2%29%29=2 ALIGN=MIDDLE  ALT=\"abs%28x%2B%283x-2%29%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%284x-2%29=2 ALIGN=MIDDLE  ALT=\"abs%284x-2%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(1a) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x-2=2 ALIGN=MIDDLE  ALT=\"4x-2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x=4 ALIGN=MIDDLE  ALT=\"4x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "x=1 satisfies our condition that x is greater than or equal to 2/3, so x=1 is a solution.<br><BR>\n" );
document.write( "OR<br><BR>\n" );
document.write( "(1b) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x-2=-2 ALIGN=MIDDLE  ALT=\"4x-2=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x=0 ALIGN=MIDDLE  ALT=\"4x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "x=0 does NOT satisfy our condition that x is greater than or equal to 2/3, so x=0 is not a solution.<br><BR>\n" );
document.write( "(2) If x is less than 2/3, then |3x-2| is equal to -(3x-2) = -3x+2:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%2B%28-3x%2B2%29%29=2 ALIGN=MIDDLE  ALT=\"abs%28x%2B%28-3x%2B2%29%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28-2x%2B2%29=2 ALIGN=MIDDLE  ALT=\"abs%28-2x%2B2%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(2a) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%2B2=2 ALIGN=MIDDLE  ALT=\"-2x%2B2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x=0 ALIGN=MIDDLE  ALT=\"-2x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "x=0 satisfies our condition that x is less than 2/3, so x=0 is a solution.<br><BR>\n" );
document.write( "OR<br><BR>\n" );
document.write( "(2b) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%2B2=-2 ALIGN=MIDDLE  ALT=\"-2x%2B2=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x=-4 ALIGN=MIDDLE  ALT=\"-2x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "x=2 does NOT satisfy our condition that x is less than 2/3, so x=2 is not a solution.<br><BR>\n" );
document.write( "We found two solutions above, in cases (1a) and (2a).<br><BR>\n" );
document.write( "ANSWER: x=1 or x=0<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );