document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=73747>Question 73747</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you solve and graph:<BR>\n" );
document.write( " /c-4/ > 1\r<BR>\n" );
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document.write( "The c-4 is in absolute value lines.\r<BR>\n" );
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document.write( "Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #52734 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=52734\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28c-4%29%3E1 ALIGN=MIDDLE  ALT=\"abs%28c-4%29%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "solve and graph<BR>\n" );
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document.write( "One way to do absolute value problems is to solve them as two separate problems.  For the<BR>\n" );
document.write( "first problem, remove the absolute value signs and solve:<BR>\n" );
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document.write( "+(c-4)>1<BR>\n" );
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document.write( "You can treat this just as you would and equation ... except for one difference. That difference<BR>\n" );
document.write( "is that if you have to multiply or divide both sides by a negative number, you must then<BR>\n" );
document.write( "reverse the direction of the inequality sign. Don't forget that.<BR>\n" );
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document.write( "Now back to solving:<BR>\n" );
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document.write( "+(c-4)>1<BR>\n" );
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document.write( "The parentheses are preceded by a plus sign so you can just remove them and the inequality<BR>\n" );
document.write( "becomes:<BR>\n" );
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document.write( "+c - 4 > 1<BR>\n" );
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document.write( "Just as you would in an equation, eliminate the -4 on the left side by adding 4 to both<BR>\n" );
document.write( "sides to get:<BR>\n" );
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document.write( "+c > 5<BR>\n" );
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document.write( "That's the first limit on the value of c ... it must be greater than +5.<BR>\n" );
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document.write( "Now, on to the second equation.  We do the same thing as before ... remove the absolute<BR>\n" );
document.write( "value signs, except this time we put a minus sign in front of the quantity that was inside<BR>\n" );
document.write( "the absolute value signs.  When you do this you have:<BR>\n" );
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document.write( "-(c-4)>1<BR>\n" );
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document.write( "Because the quantity is preceded by a negative sign, when you remove the parentheses<BR>\n" );
document.write( "you must change the sign of every term that was inside the parentheses.  When you do<BR>\n" );
document.write( "that the inequality becomes:<BR>\n" );
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document.write( "-c + 4 >1<BR>\n" );
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document.write( "We need to solve this for +c.  Let's begin by eliminating the +4 on the left side by<BR>\n" );
document.write( "subtracting 4 from both sides.  This causes the inequality to become:<BR>\n" );
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document.write( "-c > -3<BR>\n" );
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document.write( "To change the equation so that the left side is +c, multiply both sides by negative 1.<BR>\n" );
document.write( "And when you do, do NOT forget to reverse the direction of the inequality sign to get:<BR>\n" );
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document.write( "+c < +3<BR>\n" );
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document.write( "This tells us that c must be less than +3<BR>\n" );
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document.write( "Combined, the two restrictions are that c can be any number less than 3 and it can also<BR>\n" );
document.write( "be any number greater than 5.  But it cannot be either 3 or 5 or any number between those<BR>\n" );
document.write( "two values.<BR>\n" );
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document.write( "Let's check that out with a trio of trials. Start with the problem:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28c-4%29%3E1 ALIGN=MIDDLE  ALT=\"abs%28c-4%29%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Let's set c equal to zero.  That's a number less than +3 so it should work. When you substitute<BR>\n" );
document.write( "0 for c you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%280-4%29%3E1 ALIGN=MIDDLE  ALT=\"abs%280-4%29%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which simplifies to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28-4%29%3E1 ALIGN=MIDDLE  ALT=\"abs%28-4%29%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28-4%29+=+%2B4 ALIGN=MIDDLE  ALT=\"abs%28-4%29+=+%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get +4 > 1.  That works!<BR>\n" );
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document.write( "Now let's set c equal to 4.  That should NOT work.  Substitute 4 for c and get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%284-4%29%3E1 ALIGN=MIDDLE  ALT=\"abs%284-4%29%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > this obviously becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%280%29%3E1 ALIGN=MIDDLE  ALT=\"abs%280%29%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the absolute value of 0 is 0.  Obviously 0 > 1 is NOT true so numbers <BR>\n" );
document.write( "between 3 and 4 probably all need to be excluded.  <BR>\n" );
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document.write( "Finally try c = 6.  That is greater than +5 and should work. Substitute +5 for c and get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%286-4%29%3E1 ALIGN=MIDDLE  ALT=\"abs%286-4%29%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%286-4%29%3E1 ALIGN=MIDDLE  ALT=\"abs%286-4%29%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%282%29%3E1 ALIGN=MIDDLE  ALT=\"abs%282%29%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "At this point it is obvious that 2 > 1 is true and that adds to the likelihood that numbers<BR>\n" );
document.write( "greater than 5 will work.<BR>\n" );
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document.write( "We have a good solution!<BR>\n" );
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document.write( "To graph, just make a number line and put dots at +3 and +5. Exclude those dots and shade<BR>\n" );
document.write( "the number line from just to the left of +3 all the way to the left and from just to <BR>\n" );
document.write( "the right of +5 all the way to the right.  c can take any value in the shaded region.<BR>\n" );
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document.write( "Hope this shows you a way to work absolute value inequalities.  Remember the two different<BR>\n" );
document.write( "equations to solve (one by using the absolute value quantity with a + sign and the other<BR>\n" );
document.write( "by using that quantity with a minus sign preceding it), the need to reverse the direction of <BR>\n" );
document.write( "the inequality sign if multiplying or dividing both sides by a negative number, and to <BR>\n" );
document.write( "always solve for the positive value of the variable. The rest is just algebraic manipulation.<BR>\n" );
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document.write( "Hope this works for you.<BR>\n" );
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document.write( " </SPAN>\n" );
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