document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192806>Question 1192806</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the inequalities for the range of x if any.<BR>\n" );
document.write( "|x|+|2-x|<2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824725 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</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=824725\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The other tutor uses a method that I would not choose; there are two other standard solution methods that I think are easier.<br><BR>\n" );
document.write( "And his final statement that the inequality is a contradiction is not mathematically correct; his statement should be that the solution set to the inequality is the empty set.<br><BR>\n" );
document.write( "First alternative method....<br><BR>\n" );
document.write( "|x-a| can be interpreted as the distance between x and a.  So<br><BR>\n" );
document.write( "|x| = |x-0|<BR>\n" );
document.write( "|2-x| = |x-2|<br><BR>\n" );
document.write( "Then the inequality can be read as \"the sum of the distances from x to 0 and from x to 2 is less than 2\".  Picturing points on a number line, it is easy to see that |x|+|2-x| is EQUAL to 2 for 0 < x < 2; and |x|+|2-x| is greater than 2 everywhere else.  So there is no solution the inequality; its solution is the empty set.<br><BR>\n" );
document.write( "Second alternative method....<br><BR>\n" );
document.write( "The terms |x| and |2-x| mean the behavior of the expression changes at x=0 and x=2.  So analyze the three cases for x less than 0, for x between 0 and 2, and for x greater than 2.<br><BR>\n" );
document.write( "(1) If x is less than 0, then |x| = -x and |2-x| = 2-x.  The inequality is then<BR>\n" );
document.write( "(-x)+(2-x)<2<BR>\n" );
document.write( "-2x+2<2<BR>\n" );
document.write( "-2x<0<BR>\n" );
document.write( "x>0<br><BR>\n" );
document.write( "But that \"solution\" is not in the range of x values we are working with (x less than 0).  So there is no solution for x less than 0.<br><BR>\n" );
document.write( "(2) If x is between 0 and 2, then |x| = x and |2-x| = 2-x; the inequality is then<BR>\n" );
document.write( "(x)+(2-x)<2<BR>\n" );
document.write( "2<2<br><BR>\n" );
document.write( "That statement is never true; so there is no solution for x between 0 and 2.<br><BR>\n" );
document.write( "(3) If x is greater than 2, then |x| = x and |2-x| = x-2; the inequality is then<BR>\n" );
document.write( "(x)+(x-2)<2<BR>\n" );
document.write( "2x-2<2<BR>\n" );
document.write( "2x<4<BR>\n" );
document.write( "x<2<br><BR>\n" );
document.write( "But that \"solution\" is again not in the range of x values we are working with (x greater than 2).  So there is no solution for x greater than 2.<br><BR>\n" );
document.write( "This shows that there is no solution to the inequality in any of the three intervals; therefore, there is no solution to the inequality -- the solution is the empty set.<br><BR>\n" );
document.write( " </SPAN>\n" );
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