document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=92975>Question 92975</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help! <BR>\n" );
document.write( "Solve each absolute value inequality and graph the solution set.\r<BR>\n" );
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document.write( "1>1/2 6-x -3/4 <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  the /6-x/is in straight up & down line brackets  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #68282 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=68282\">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=1%3E%281%2F2%29%2Aabs%286-x%29+-+%283%2F4%29 ALIGN=MIDDLE  ALT=\"1%3E%281%2F2%29%2Aabs%286-x%29+-+%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Our general plan for solving this is to work on it just as we would an equation, with one<BR>\n" );
document.write( "exception ... any time we need to multiply or divide both sides of the inequality by a <BR>\n" );
document.write( "negative quantity, we also need to reverse the direction of the inequality.<BR>\n" );
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document.write( "We solve the above inequality for the quantity in the absolute value signs and then we<BR>\n" );
document.write( "set up two inequalities, one having the quantity in the absolute value signs preceded by <BR>\n" );
document.write( "a positive sign, and one having the quantity in the absolute value signs preceded by a negative<BR>\n" );
document.write( "sign.<BR>\n" );
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document.write( "The method will become a little more clear when we work through the problem.<BR>\n" );
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document.write( "Begin with the given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3E%281%2F2%29%2Aabs%286-x%29+-+%283%2F4%29 ALIGN=MIDDLE  ALT=\"1%3E%281%2F2%29%2Aabs%286-x%29+-+%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Let's get rid of the denominators by multiplying both sides (all terms) by +4. This results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%3E%284%2F2%29%2Aabs%286-x%29+-%284%2A3%29%2F4 ALIGN=MIDDLE  ALT=\"4%3E%284%2F2%29%2Aabs%286-x%29+-%284%2A3%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and by dividing the denominators into the numerators of each term on the right side, we end<BR>\n" );
document.write( "up with:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%3E2%2Aabs%286-x%29+-+3 ALIGN=MIDDLE  ALT=\"4%3E2%2Aabs%286-x%29+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the -3 on the right side by adding 3 to both sides:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%3E2%2Aabs%286-x%29 ALIGN=MIDDLE  ALT=\"7%3E2%2Aabs%286-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Divide both sides by 2 so that the right side is just the absolute value term:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F2%3Eabs%286-x%29 ALIGN=MIDDLE  ALT=\"7%2F2%3Eabs%286-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So far we've worked this pretty much as we would have done an equation.  Now let's split<BR>\n" );
document.write( "this into two separate problems. In one problem we replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%286-x%29 ALIGN=MIDDLE  ALT=\"abs%286-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the quantity<BR>\n" );
document.write( "+(6 - x) and then solve for x. In the other problem we replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%286-x%29 ALIGN=MIDDLE  ALT=\"abs%286-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the quantity<BR>\n" );
document.write( "-(6 - x) and then solve for x again. [That's the typical way to solve absolute value<BR>\n" );
document.write( "equations.]<BR>\n" );
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document.write( "The first problem then is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F2%3E6-x ALIGN=MIDDLE  ALT=\"7%2F2%3E6-x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "we can get rid of the denominator 2 on the left side by multiplying both sides by +2 to<BR>\n" );
document.write( "get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%3E12+-+2x ALIGN=MIDDLE  ALT=\"7%3E12+-+2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract 12 from both sides to get rid of the 12 on the right side and the inequality <BR>\n" );
document.write( "becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5+%3E+-2x ALIGN=MIDDLE  ALT=\"-5+%3E+-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve for +x by dividing both sides by -2 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F2+%3E+x ALIGN=MIDDLE  ALT=\"5%2F2+%3E+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But recall the rule that if we multiply or divide both sides by a negative quantity<BR>\n" );
document.write( "we must reverse the direction of the inequality sign. In this case we divided both sides <BR>\n" );
document.write( "by -2 so the inequality sign changes direction and the answer is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F2+%3C+x ALIGN=MIDDLE  ALT=\"5%2F2+%3C+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > <=== remember this solution for +x<BR>\n" );
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document.write( "Now we need to work a second problem. This time we precede the quantity that was in the <BR>\n" );
document.write( "absolute value signs by a minus sign and solve for +x. So the setup for this problem is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F2%3E+-%286-x%29 ALIGN=MIDDLE  ALT=\"7%2F2%3E+-%286-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Since the parentheses on the right side are preceded by a minus sign, we can remove the<BR>\n" );
document.write( "parentheses (and the minus sign) if we change the signs of all the terms in the parentheses.<BR>\n" );
document.write( "When we do that the problem becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F2+%3E+-6+%2B+x ALIGN=MIDDLE  ALT=\"7%2F2+%3E+-6+%2B+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the denominator on the right side by multiplying everything on both sides by +2 <BR>\n" );
document.write( "to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7+%3E+-12+%2B+2x ALIGN=MIDDLE  ALT=\"7+%3E+-12+%2B+2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the -12 on the right side by adding 12 to both sides:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19+%3E+2x ALIGN=MIDDLE  ALT=\"19+%3E+2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve for +x by dividing both sides by +2:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19%2F2+%3Ex ALIGN=MIDDLE  ALT=\"19%2F2+%3Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This answer can be interpreted as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19%2F2 ALIGN=MIDDLE  ALT=\"19%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is greater than x and this is equivalent<BR>\n" );
document.write( "to saying x is less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19%2F2 ALIGN=MIDDLE  ALT=\"19%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So an equivalent form of this answer is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C+19%2F2 ALIGN=MIDDLE  ALT=\"x+%3C+19%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <=== remember this.<BR>\n" );
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document.write( "We now have two answers. Our first answer said that x was greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F2 ALIGN=MIDDLE  ALT=\"5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and our second<BR>\n" );
document.write( "answer said that x was less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19%2F2 ALIGN=MIDDLE  ALT=\"19%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We can combine these to say that:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F2%3Cx%3C19%2F2 ALIGN=MIDDLE  ALT=\"5%2F2%3Cx%3C19%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "And this tells us that on a number line the value of x must be between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F2 ALIGN=MIDDLE  ALT=\"5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > [in decimal<BR>\n" );
document.write( "form 2.5] and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19%2F2 ALIGN=MIDDLE  ALT=\"19%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >[in decimal form 9.5]. So x can be any value between those two <BR>\n" );
document.write( "limits. <BR>\n" );
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document.write( "We can do a few quick checks to help convince ourselves that this answer is likely to<BR>\n" );
document.write( "be correct.<BR>\n" );
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document.write( "Let's pick a value for x greater than 9.5.  This is outside the allowable domain of x and<BR>\n" );
document.write( "so the original inequality should not be true. The original inequality is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3E%281%2F2%29%2Aabs%286-x%29+-+%283%2F4%29 ALIGN=MIDDLE  ALT=\"1%3E%281%2F2%29%2Aabs%286-x%29+-+%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substitute +10 for x and it becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3E%281%2F2%29%2Aabs%286-10%29+-+%283%2F4%29 ALIGN=MIDDLE  ALT=\"1%3E%281%2F2%29%2Aabs%286-10%29+-+%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "In the absolute value signs the quantity 6 - 10 becomes -4, and the absolute value signs<BR>\n" );
document.write( "change that to +4. So the inequality becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3E%281%2F2%29%2A4+-+%283%2F4%29 ALIGN=MIDDLE  ALT=\"1%3E%281%2F2%29%2A4+-+%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and this reduces to<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3E2+-+3%2F4 ALIGN=MIDDLE  ALT=\"1%3E2+-+3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The terms on the right side combine. In decimal form they are 2 - 0.75 and this is 1.25. So<BR>\n" );
document.write( "the inequality becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3E1.25 ALIGN=MIDDLE  ALT=\"1%3E1.25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is obviously wrong, just as we suspected it would be because letting x = 10 was beyond<BR>\n" );
document.write( "the acceptable range we found.<BR>\n" );
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document.write( "Now suppose we let x = +2. This is below the acceptable range and so the inequality should<BR>\n" );
document.write( "not be true if x has this value. Substituting +2 for x we get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3E%281%2F2%29%2Aabs%286-2%29+-+%283%2F4%29 ALIGN=MIDDLE  ALT=\"1%3E%281%2F2%29%2Aabs%286-2%29+-+%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "In the absolute value signs the 6 - 2 becomes +4 and since this is positive, the absolute value<BR>\n" );
document.write( "signs can just be removed to make the inequality become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3E%281%2F2%29%2A4+-3%2F4 ALIGN=MIDDLE  ALT=\"1%3E%281%2F2%29%2A4+-3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Doing the multiplication on the right side:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3E2+-+3%2F4 ALIGN=MIDDLE  ALT=\"1%3E2+-+3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and in decimal form (easier to type) the 2 - 3/4 of the right side becomes 2 - 0.75 and<BR>\n" );
document.write( "this is 1.25. So the inequality is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%3E+1.25 ALIGN=MIDDLE  ALT=\"1+%3E+1.25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and just as we suspected, this is not true because it is outside the limits we found. <BR>\n" );
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document.write( "Finally, suppose we let x = 6. That is within the limits that we found, so the inequality<BR>\n" );
document.write( "should work.  When we substitute 6 for x the inequality becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3E%281%2F2%29%2Aabs%286-6%29+-+%283%2F4%29 ALIGN=MIDDLE  ALT=\"1%3E%281%2F2%29%2Aabs%286-6%29+-+%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This makes the absolute value quantity go to zero and this means that the first term on the<BR>\n" );
document.write( "right side equals zero. Therefore, the inequality is reduced to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%3E+-3%2F4 ALIGN=MIDDLE  ALT=\"1+%3E+-3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is a true statement (a positive number is greater than a negative number, that's for sure)<BR>\n" );
document.write( "and so a number within our acceptable range for x makes the inequality true. <BR>\n" );
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document.write( "We can, therefore, have some degree of confidence that our solution is correct.<BR>\n" );
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document.write( "Hope this helps you to see the basics of working with absolute values and with inequalities<BR>\n" );
document.write( "... two lessons in one problem.<BR>\n" );
document.write( ". </SPAN>\n" );
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