document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1003018>Question 1003018</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I don't understand inequalities. how does one translate this into simple English.\r<BR>\n" );
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document.write( "-2x&#8804;g(x)&#8804;2x\r<BR>\n" );
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document.write( "Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #619951 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=619951\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Starting simple, suppose you have 2 quantities,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+ ALIGN=MIDDLE  ALT=\"+a+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b+ ALIGN=MIDDLE  ALT=\"+b+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then<BR>\n" );
document.write( "(1) <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;\" > means:<BR>\n" );
document.write( "[ quantity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > ] is GREATER than [ quantity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > ]<BR>\n" );
document.write( "(2) <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:<BR>\n" );
document.write( "[ quantity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > ] is LESS than [ quantity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > ]<BR>\n" );
document.write( "-------------------------------------------<BR>\n" );
document.write( "Now if you add on the equals sign, you have:<BR>\n" );
document.write( "(3) <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;\" > means:<BR>\n" );
document.write( "[ quantity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > ] is GREATER than OR EQUAL to  [ quantity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > ]<BR>\n" );
document.write( "(4) <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:<BR>\n" );
document.write( "[ quantity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > ] is LESS than OR EQUAL to [ quantity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > ]<BR>\n" );
document.write( "--------------------------------------------------<BR>\n" );
document.write( "You have to keep in mind that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-3+%3E+-7+ ALIGN=MIDDLE  ALT=\"+-3+%3E+-7+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a true statement<BR>\n" );
document.write( "The quantities can be any number on the number line<BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "A potential sticky point is the following:<BR>\n" );
document.write( "If you divide both sides by a negative number, then the<BR>\n" );
document.write( "inequality sign has to be reversed.<BR>\n" );
document.write( "Example:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+10+%3E5+ ALIGN=MIDDLE  ALT=\"+10+%3E5+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-5+ ALIGN=MIDDLE  ALT=\"+-5+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+10%2F%28-5%29+%3E+5%2F%28-5%29+ ALIGN=MIDDLE  ALT=\"+10%2F%28-5%29+%3E+5%2F%28-5%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-2+%3E+-1+ ALIGN=MIDDLE  ALT=\"+-2+%3E+-1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > oh-oh, this is NOT TRUE, but if<BR>\n" );
document.write( "I reverse the inequality sign,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-2+%3C+-1+ ALIGN=MIDDLE  ALT=\"+-2+%3C+-1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > OK now<BR>\n" );
document.write( "---------------------<BR>\n" );
document.write( "As for your example, it reads:<BR>\n" );
document.write( "The function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+g%28x%29+ ALIGN=MIDDLE  ALT=\"+g%28x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is LESS THAN OR EQUAL to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2x+ ALIGN=MIDDLE  ALT=\"+2x+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+g%28x%29+ ALIGN=MIDDLE  ALT=\"+g%28x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also GREATER THAN OR EQUAL to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-2x+ ALIGN=MIDDLE  ALT=\"+-2x+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "------------------------------------------------<BR>\n" );
document.write( "You don't have to put this on one line<BR>\n" );
document.write( "You can break it up into 2 statements:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-2x+%3C=+g%28x%29+ ALIGN=MIDDLE  ALT=\"+-2x+%3C=+g%28x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+g%28x%29+%3C=+2x+ ALIGN=MIDDLE  ALT=\"+g%28x%29+%3C=+2x+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "Also note that you can spin the inequality around and<BR>\n" );
document.write( "it's still true:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+g%28x%29+%3E=+-2x+ ALIGN=MIDDLE  ALT=\"+g%28x%29+%3E=+-2x+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is perfectly valid also<BR>\n" );
document.write( "------------------<BR>\n" );
document.write( "You can treat inequalities just like equations by adding,<BR>\n" );
document.write( "subtracting, multiplying, dividing both sides by the<BR>\n" );
document.write( "same quantity EXCEPT the special case of dividing by<BR>\n" );
document.write( "a negative number just noted.<BR>\n" );
document.write( "Hope this helps </SPAN>\n" );
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