document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=76935>Question 76935</A>: <I><SPAN STYLE=\"word-break: break-all;\"> can you help me solve this problem<BR>\n" );
document.write( "-2<2x+1< or = 6<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #55149 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=55149\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You can treat this somewhat like you do when you solve an equation ... whatever you do to<BR>\n" );
document.write( "one section of the inequality, you must also do to the other two sections.  And you also<BR>\n" );
document.write( "need to follow the rule that if you divide or multiply all the sections by a NEGATIVE<BR>\n" );
document.write( "quantity, you must reverse the directions of the inequality signs. And you should always<BR>\n" );
document.write( "check your answer by using some convenient values for each section. Let's apply these rules<BR>\n" );
document.write( "to your problem. You were given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%3C2x%2B1%3C=6 ALIGN=MIDDLE  ALT=\"-2%3C2x%2B1%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "We want to solve for x.  So let's start by eliminating the +1 in the middle section.<BR>\n" );
document.write( "We do that by subtracting 1 from the middle section, but when we do that we must also subtract<BR>\n" );
document.write( "1 from the other two sections of the inequality. The subtraction is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2+-1%3C2x+%2B+1+-1%3C=+6+-1 ALIGN=MIDDLE  ALT=\"-2+-1%3C2x+%2B+1+-1%3C=+6+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and this simplifies to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3+%3C+2x+%3C=+5 ALIGN=MIDDLE  ALT=\"-3+%3C+2x+%3C=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Since we are solving for x, we need to divide the center section by 2.  But then we must<BR>\n" );
document.write( "divide all three sections by 2. So the inequality becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3%29%2F2%3C2x%2F2%3C=5%2F2 ALIGN=MIDDLE  ALT=\"%28-3%29%2F2%3C2x%2F2%3C=5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and this simplifies to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3%2F2%29%3Cx%3C=%285%2F2%29 ALIGN=MIDDLE  ALT=\"%28-3%2F2%29%3Cx%3C=%285%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and if it helps you to visualize it on the number line better, you can convert the fractions<BR>\n" );
document.write( "to decimals and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1.5%3Cx%3C=2.5 ALIGN=MIDDLE  ALT=\"-1.5%3Cx%3C=2.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This says that on the number line x must be greater than -1.5 and less than or equal to <BR>\n" );
document.write( "+2.5<BR>\n" );
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document.write( "Since zero is a number within these two limits, let's return to the original problem and <BR>\n" );
document.write( "let x be zero. Then we can see if the inequality still holds true.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%3C2x%2B1%3C=6 ALIGN=MIDDLE  ALT=\"-2%3C2x%2B1%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substitute 0 for x and the inequality becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2+%3C+1+%3C+6 ALIGN=MIDDLE  ALT=\"-2+%3C+1+%3C+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Is this true?  Sure ... 1 is between -2 and +6 on the number line.  With this spot check<BR>\n" );
document.write( "we can tell that the inequality holds for at least one value of x between -1.5 and +2.5.<BR>\n" );
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document.write( "Let's now select a value of x outside the limits of our answer. For instance, let's chose<BR>\n" );
document.write( "x = -2.  This should NOT work in the original inequality.  Plug it in for x in the original<BR>\n" );
document.write( "inequality.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%3C2x%2B1%3C=6 ALIGN=MIDDLE  ALT=\"-2%3C2x%2B1%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%3C2%2A%28-2%29%2B1+%3C%2B+6 ALIGN=MIDDLE  ALT=\"-2%3C2%2A%28-2%29%2B1+%3C%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%3C-4%2B1%3C=6 ALIGN=MIDDLE  ALT=\"-2%3C-4%2B1%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%3C-3%3C=6 ALIGN=MIDDLE  ALT=\"-2%3C-3%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that this does NOT work because -3 is not greater than -2.<BR>\n" );
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document.write( "and then we need to do a similar check for x greater than +2.5.  Suppose we let x be +3.<BR>\n" );
document.write( "The original inequality would become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2+%3C+2%28%2B3%29%2B1+%3C=+6 ALIGN=MIDDLE  ALT=\"-2+%3C+2%28%2B3%29%2B1+%3C=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and the middle section becomes 6 + 1 so that the inequality is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2+%3C+7+%3C=+6 ALIGN=MIDDLE  ALT=\"-2+%3C+7+%3C=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is not true because 7 is not less than or equal to 6.<BR>\n" );
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document.write( "Our solution looks pretty good.  We did not have to multiply or divide all three sections<BR>\n" );
document.write( "of the inequality by a negative number so we did not have to reverse the direction of<BR>\n" );
document.write( "the inequality arrows.  But don't forget that rule.  In some problems you will have to make<BR>\n" );
document.write( "use of it.<BR>\n" );
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document.write( "Hope this helps you to understand inequalities a little better. </SPAN>\n" );
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