document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=318717>Question 318717</A>: <I><SPAN STYLE=\"word-break: break-all;\"> given that 1<x<4, write |x+7|-|x-9| without absolute value symbols.  Is the answer 16? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #228152 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=228152\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You may be tempted to think that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%29=x ALIGN=MIDDLE  ALT=\"abs%28x%29=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but that is not true for all 'x'. It is only true if 'x' is positive. This is where you made a mistake. You're thinking that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x-9%29=x-9 ALIGN=MIDDLE  ALT=\"abs%28x-9%29=x-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but with the given domain, this is not true.\r<BR>\n" );
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document.write( "Recall that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%29=x ALIGN=MIDDLE  ALT=\"abs%28x%29=x\"   DISABLED_style= \"max-width:90%; height:auto;\" > if 'x' is positive. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%2B7%29=x%2B7 ALIGN=MIDDLE  ALT=\"abs%28x%2B7%29=x%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" > if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B7%3E0 ALIGN=MIDDLE  ALT=\"x%2B7%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Solve for 'x' to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E-7 ALIGN=MIDDLE  ALT=\"x%3E-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3Cx%3C4 ALIGN=MIDDLE  ALT=\"1%3Cx%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E-7 ALIGN=MIDDLE  ALT=\"x%3E-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > is most certainly true and that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%2B7%29=x%2B7 ALIGN=MIDDLE  ALT=\"abs%28x%2B7%29=x%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also true.\r<BR>\n" );
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document.write( "On the flip side, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%29=-x ALIGN=MIDDLE  ALT=\"abs%28x%29=-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C0 ALIGN=MIDDLE  ALT=\"x%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie x is negative). Notice that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x-9%29=-%28x-9%29 ALIGN=MIDDLE  ALT=\"abs%28x-9%29=-%28x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > only because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-9%3C0 ALIGN=MIDDLE  ALT=\"x-9%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C9 ALIGN=MIDDLE  ALT=\"x%3C9\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is certainly true since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3Cx%3C4 ALIGN=MIDDLE  ALT=\"1%3Cx%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" >). So once again <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x-9%29=-%28x-9%29 ALIGN=MIDDLE  ALT=\"abs%28x-9%29=-%28x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To recap, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%2B7%29=x%2B7 ALIGN=MIDDLE  ALT=\"abs%28x%2B7%29=x%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x-9%29=-%28x-9%29 ALIGN=MIDDLE  ALT=\"abs%28x-9%29=-%28x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If this makes no sense at all why these two equations hold, just graph the two functions with the domain <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3Cx%3C4 ALIGN=MIDDLE  ALT=\"1%3Cx%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Let's now use this info to simplify the expression.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%2B7%29-abs%28x-9%29 ALIGN=MIDDLE  ALT=\"abs%28x%2B7%29-abs%28x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B7%29-%28-%28x-9%29%29 ALIGN=MIDDLE  ALT=\"%28x%2B7%29-%28-%28x-9%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%2B7%29=x%2B7 ALIGN=MIDDLE  ALT=\"abs%28x%2B7%29=x%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x-9%29=-%28x-9%29 ALIGN=MIDDLE  ALT=\"abs%28x-9%29=-%28x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B7%2Bx-9 ALIGN=MIDDLE  ALT=\"x%2B7%2Bx-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate the negative\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-2 ALIGN=MIDDLE  ALT=\"2x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%2B7%29-abs%28x-9%29=2x-2 ALIGN=MIDDLE  ALT=\"abs%28x%2B7%29-abs%28x-9%29=2x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3Cx%3C4 ALIGN=MIDDLE  ALT=\"1%3Cx%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So for example, if <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;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%2B7%29-abs%28x-9%29=abs%282%2B7%29-abs%282-9%29=abs%289%29-abs%28-7%29=9-7=2 ALIGN=MIDDLE  ALT=\"abs%28x%2B7%29-abs%28x-9%29=abs%282%2B7%29-abs%282-9%29=abs%289%29-abs%28-7%29=9-7=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and also <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-2=2%282%29-2=4-2=2 ALIGN=MIDDLE  ALT=\"2x-2=2%282%29-2=4-2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which confirms that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%2B7%29-abs%28x-9%29=2x-2 ALIGN=MIDDLE  ALT=\"abs%28x%2B7%29-abs%28x-9%29=2x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <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;\" >. We can use other numbers in the domain <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3Cx%3C4 ALIGN=MIDDLE  ALT=\"1%3Cx%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to reconfirm our answer.\r<BR>\n" );
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document.write( "A better confirmation would be to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=abs%28x%2B7%29-abs%28x-9%29 ALIGN=MIDDLE  ALT=\"y=abs%28x%2B7%29-abs%28x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x-2 ALIGN=MIDDLE  ALT=\"y=2x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to see that the two graphs match up completely over the domain <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3Cx%3C4 ALIGN=MIDDLE  ALT=\"1%3Cx%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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