document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=8495>Question 8495</A>: <I><SPAN STYLE=\"word-break: break-all;\"> It's been awhile since I've done absolute value and I'm having trouble with |2x-1|-|x+5|=3. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #4716 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=4716\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  |2x-1|-|x+5|=3\r<BR>\n" );
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document.write( " The root of 2x-1 =0 is x= 1/2<BR>\n" );
document.write( " and the root of x+5 =0 is x= -5.<BR>\n" );
document.write( " As<BR>\n" );
document.write( "  -oo        -5              1/2            +oo<BR>\n" );
document.write( "   -----------*---------------*---------------\r<BR>\n" );
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document.write( " Note, -5 < 1/2 and these two roots divide the real line into 3 parts<BR>\n" );
document.write( " (-oo,-5) , [-5,1/2] and (1/2,+oo).\r<BR>\n" );
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document.write( " Case (i) if x is in (-oo,-5) , since x+5 < 0 and 2x-1 <0 <BR>\n" );
document.write( "      then  |2x-1| = -(2x-1) = -2x+1, and |x+5| = -x-5.<BR>\n" );
document.write( "      So,|2x-1|-|x+5|=3 is equivalent to -2x+1 -x-5 =3<BR>\n" );
document.write( "      and we get -3x = 7, x = -7/3 .<BR>\n" );
document.write( "      But -7/3> -5 , so it is an invalid answer.\r<BR>\n" );
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document.write( " Case (ii) if x is in [-5,1/2] , since x+5 >= 0 and 2x-1 <= 0 <BR>\n" );
document.write( "      then  |2x-1| = -(2x-1) = -2x+1, and |x+5| = x+5.<BR>\n" );
document.write( "      So,|2x-1|-|x+5|=3 is equivalent to -2x+1 +x+5 =3<BR>\n" );
document.write( "      and we get -x = -3,  x = 3.<BR>\n" );
document.write( "      But -3 is not in [-5,1/2] , so it is an invalid answer.<BR>\n" );
document.write( " <BR>\n" );
document.write( " Case (iii) if x is in (1/2,+oo) , since x+5 < 0 and 2x-1 > 0 <BR>\n" );
document.write( "      then  |2x-1| = 2x-1, and |x+5| = -x-5.<BR>\n" );
document.write( "      So,|2x-1|-|x+5|=3 is equivalent to 2x-1 -x-5 =3<BR>\n" );
document.write( "      and we get x = 9.<BR>\n" );
document.write( "      But 9 is in (1/2,+oo) , so it is a valid answer.<BR>\n" );
document.write( " <BR>\n" );
document.write( " We claim that there is only one solution x = 9.\r<BR>\n" );
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document.write( " If you have  trouble understanding ,try to review the definition about<BR>\n" );
document.write( " the absolute value, think carfully and draw  <BR>\n" );
document.write( " a diagram on the real line. Sorry ,I won't give you further explanations.\r<BR>\n" );
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document.write( " Kenny\r<BR>\n" );
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document.write( " </SPAN>\n" );
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