document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=484807>Question 484807</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help with inequalities problems, I don't know a thing!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #331640 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=331640\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> THIS SIGN < means is less than..  \r<BR>\n" );
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document.write( "This sign > means is greater than.  \r<BR>\n" );
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document.write( "In each case, the sign opens towards the larger number.\r<BR>\n" );
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document.write( "For example,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+%3C5 ALIGN=MIDDLE  ALT=\"2+%3C5\"   DISABLED_style= \"max-width:90%; height:auto;\" > (\"2 is less than 5\").  Equivalently, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+5+%3E+2 ALIGN=MIDDLE  ALT=\"+5+%3E+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >} (\"5 is greater than 2\").\r<BR>\n" );
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document.write( "These are the two senses of an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=inequality ALIGN=MIDDLE  ALT=\"inequality\"   DISABLED_style= \"max-width:90%; height:auto;\" >:  <  and  > , \r<BR>\n" );
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document.write( "and you also might have  &#8804;, \"is less than or equal to\" or &#8805;, \"is greater than or equal to\"\r<BR>\n" );
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document.write( "On the number line,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C+b ALIGN=MIDDLE  ALT=\"a+%3C+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >  means:  a falls to the left of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "ex.:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%3C+5 ALIGN=MIDDLE  ALT=\"2%3C+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "&#8722;1 < 2\r<BR>\n" );
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document.write( "a compound inequality:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E+1 ALIGN=MIDDLE  ALT=\"x+%3E+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and   x &#8804; 5.\r<BR>\n" );
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document.write( "It is a compound sentence whose conjunction is \"and.\"  It says that x takes on values that are greater than 1  and  less than or equal to 5. \r<BR>\n" );
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document.write( "It is within that interval that x takes its values.  The endpoint 1 is not included.  x is definitely greater than 1; we indicate that by placing a parenthesis \"(\".  The endpoint 5 is included; we indicate that by placing a bracket \"]\".\r<BR>\n" );
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document.write( "A continued inequality\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C+x+%3C+b ALIGN=MIDDLE  ALT=\"a+%3C+x+%3C+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "That is called a continued inequality.  It means\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C+x ALIGN=MIDDLE  ALT=\"a+%3C+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C+b ALIGN=MIDDLE  ALT=\"x+%3C+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "A continued inequality always implies the conjunction and.  The sense is always <  or  &#8804;.\r<BR>\n" );
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document.write( "The continued inequality means:\r<BR>\n" );
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document.write( "x falls in the interval between a and b.\r<BR>\n" );
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document.write( "\"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C+x+%3C+b ALIGN=MIDDLE  ALT=\"a+%3C+x+%3C+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\" illustrates that\r<BR>\n" );
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document.write( "Some theorems of inequalities\r<BR>\n" );
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document.write( "To prove a statement whose predicate is \"is greater than,\" we must have a definition of \"is greater than.\"  We shall adopt the following.  We shall define \"a is greater than b\" to mean: a &#8722; b is positive.  \r<BR>\n" );
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document.write( "Algebraically:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3E+b ALIGN=MIDDLE  ALT=\"a+%3E+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >  if and only if  a &#8722; b > 0\r<BR>\n" );
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document.write( "Theorem 1.    We may add the same number to both sides of an inequality, and the sense will not change.\r<BR>\n" );
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document.write( "Theorem 1 as follows:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3E+b ALIGN=MIDDLE  ALT=\"a+%3E+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "                      means<BR>\n" );
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document.write( "a &#8722; b > 0\r<BR>\n" );
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document.write( "upon adding and subtracting c :<BR>\n" );
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document.write( "a &#8722; b + c &#8722; c > 0<BR>\n" );
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document.write( "(a + c) &#8722; (b + c) > 0\r<BR>\n" );
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document.write( "Theorem 2.   We may multiply both sides of an inequality by the same positive number, and the sense will not change.\r<BR>\n" );
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document.write( "If 	 <BR>\n" );
document.write( "  	a  > b,   and c > 0,<BR>\n" );
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document.write( "then 	 <BR>\n" );
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document.write( "  	ca>cb.\r<BR>\n" );
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document.write( "For example:\r<BR>\n" );
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document.write( "  	4 > &#8722;5\r<BR>\n" );
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document.write( "If we now multiply both sides by 3, for example, then<BR>\n" );
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document.write( "  	12 > &#8722;15.\r<BR>\n" );
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document.write( "The sense does not change.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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