document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=75867>Question 75867</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0078603900/algebrahomeworkh>Algebra I</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\">  Solve the open sentence and name and graph the solution set. Please and thank you, to I can have a clear understanding of what I am doing.\r<BR>\n" );
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document.write( "|6r+8|<-4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #54524 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=54524\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> |6r+8|<-4<BR>\n" );
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document.write( "This is probably a bad example for showing you a way to work absolute value problems.<BR>\n" );
document.write( "Why? Because this problem can be done by inspection.  Note that the left side of this<BR>\n" );
document.write( "inequality is entirely contained within the absolute value signs.  That means that whatever<BR>\n" );
document.write( "value you select for r, the left side of the inequality is always going to be positive.<BR>\n" );
document.write( "Now look at the right side of the inequality.  It is -4 and will always be negative.<BR>\n" );
document.write( "Now ask yourself, \"When is the positive quantity on the left side less than a negative<BR>\n" );
document.write( "quantity on the right side?\"  The answer is \"Cannot happen.\"  Therefore, this inequality <BR>\n" );
document.write( "is never true ... regardless of the value of r.<BR>\n" );
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document.write( "There is, however, a point to this problem, and that point is to always check whatever <BR>\n" );
document.write( "answer you get to make sure that you don't make a mistake in interpreting the results<BR>\n" );
document.write( "that you work out.<BR>\n" );
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document.write( "For example, let's work this problem in a way that you can use to solve absolute value equations<BR>\n" );
document.write( "or inequalities.  This way involves solving two separate inequality problems.  The first<BR>\n" );
document.write( "inequality problem is to write the inequality with out the absolute value signs and precede<BR>\n" );
document.write( "the quantity that was in the absolute value signs by a + sign.  For this problem that would<BR>\n" );
document.write( "result in:<BR>\n" );
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document.write( "+(6r + 8) < -4<BR>\n" );
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document.write( "Note that because a plus sign precedes the parentheses, you can just remove them without<BR>\n" );
document.write( "making changes to the terms within to get:<BR>\n" );
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document.write( "6r + 8 < -4<BR>\n" );
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document.write( "Now eliminate the 8 on the left side by subtracting 8 from both sides to get:<BR>\n" );
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document.write( "6r < -4 -8 which simplifies to:<BR>\n" );
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document.write( "6r < -12<BR>\n" );
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document.write( "Solve for +r by dividing both sides by 6 and you get:<BR>\n" );
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document.write( "r < -12/6 which simplifies to:<BR>\n" );
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document.write( "r < -2<BR>\n" );
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document.write( "This solution tells us that r must lie to the left of -2 on the number line, if it is to<BR>\n" );
document.write( "be a solution.<BR>\n" );
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document.write( "Now recall that we were setting up two inequalities.  The second one involves removing<BR>\n" );
document.write( "the absolute value signs and applying a negative sign to the quantity that was inside the<BR>\n" );
document.write( "absolute value signs.  For this problem that would result in:<BR>\n" );
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document.write( "-(6r + 8) < -4<BR>\n" );
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document.write( "Then we need to solve this inequality for +r.  As a start, we can multiply both sides of<BR>\n" );
document.write( "this inequality by -1 to eliminate the negative sign. However, recall that whenever<BR>\n" );
document.write( "you multiply or divide both sides of an inequality by a negative number you must reverse the<BR>\n" );
document.write( "direction of the inequality sign.  So when you multiply both sides of this inequality<BR>\n" );
document.write( "by -1 you end up with:<BR>\n" );
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document.write( "6r + 8 > 4<BR>\n" );
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document.write( "Subtract 8 from both sides to eliminate the 8 on the left side.  You get:<BR>\n" );
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document.write( "6r > -4<BR>\n" );
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document.write( "Then to solve for +r, divide both sides by 6 and you get:<BR>\n" );
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document.write( "r > -4/6 which simplifies to <BR>\n" );
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document.write( "r > -2/3<BR>\n" );
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document.write( "This tells you that another set of solutions involves all the values of r that lie to<BR>\n" );
document.write( "the right of -2/3 on the number line.<BR>\n" );
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document.write( "So far this looks like we're fine.  We have found two restrictions on r ...  valid<BR>\n" );
document.write( "solutions for r must be less than -2 and greater than -2/3.<BR>\n" );
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document.write( "This method of solving two inequalities (one with a + sign on the quantity inside the<BR>\n" );
document.write( "absolute value signs and another with a - sign on that same quantity) will work in general<BR>\n" );
document.write( "unless there is an unforeseen difficulty with the problem such as there was with this one.<BR>\n" );
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document.write( "So what we need to do is always check our answers. In this problem we can do that by <BR>\n" );
document.write( "picking a value for r that is less than -2 and plugging it into the original problem<BR>\n" );
document.write( "to see if it works.  Then pick a value of r that is greater than -2/3 and see if it works<BR>\n" );
document.write( "in the original problem. And finally pick a value of r in between -2 and -2/3 and see if <BR>\n" );
document.write( "it works.<BR>\n" );
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document.write( "Let's try r = -10.  That surely meets the criterion that r be less than -2.  If we substitute<BR>\n" );
document.write( "-10 for r in the original inequality we get:<BR>\n" );
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document.write( "|6*(-10) + 8 | < -4 and this simplifies to:<BR>\n" );
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document.write( "|-60 + 8| < -4 which further simplifies to:<BR>\n" );
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document.write( "| -52 | < -4<BR>\n" );
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document.write( "but the absolute value of -52 is +52 and it is not true that:<BR>\n" );
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document.write( "+52 < -4<BR>\n" );
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document.write( "This is a clue that something isn't correct here and we must find the reason.<BR>\n" );
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document.write( "You can try the same checking by letting r be 0 (which meets the requirement that r<BR>\n" );
document.write( "be greater than -2/3. If you let r be 0, the inequality becomes 8 < -4 and this is also<BR>\n" );
document.write( "not true.  <BR>\n" );
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document.write( "What happens if you let r = -1, which is a value between r = -2 and r = -2/3.  If you do<BR>\n" );
document.write( "this the original inequality becomes 2 < -4 and this also is not correct.  Nothing<BR>\n" );
document.write( "works.  So the original statement of the problem is based on an incorrect assumption<BR>\n" );
document.write( "which we have already noted ... that the absolute value of any quantity cannot be less<BR>\n" );
document.write( "than a negative number.<BR>\n" );
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document.write( "This is one of the things that you can learn from this problem ... the absolute value of<BR>\n" );
document.write( "a quantity cannot be less than zero or a negative number.  Another thing you can learn is<BR>\n" );
document.write( "a process for solving absolute value problems (use both + and - the quantity inside the<BR>\n" );
document.write( "absolute value signs). A third thing is that when you multiply or divide both sides of<BR>\n" );
document.write( "an inequality by a negative quantity, you reverse the direction of the inequality<BR>\n" );
document.write( "sign. And finally, always spot check your answer just to make sure that you haven't <BR>\n" );
document.write( "overlooked something, and that your solution does really work.  (In this case it did not<BR>\n" );
document.write( "and we found a reason for that.)<BR>\n" );
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document.write( "Hope this helps you understand inequalities and absolute values a little better.<BR>\n" );
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document.write( " </SPAN>\n" );
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