document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=78680>Question 78680</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please solve.  |x - 1.5| <= 3\r<BR>\n" );
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document.write( "I tried and got X <= 4.5   or X >= -1.5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #56550 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=56550\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Your answers are correct.  Good job!<BR>\n" );
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document.write( "In the way of discussion you can solve problems like this by using the plus and minus<BR>\n" );
document.write( "quantities inside the absolute value signs.  Work them like equations, except for one<BR>\n" );
document.write( "exception ... if you multiply or divide both sides by a negative quantity, you reverse <BR>\n" );
document.write( "the direction of the inequality sign.<BR>\n" );
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document.write( "Let's apply these techniques to this problem. <BR>\n" );
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document.write( "First, the quantity inside the absolute value signs is (x - 1.5).<BR>\n" );
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document.write( "Next put a plus sign in front of this quantity and you get the inequality:<BR>\n" );
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document.write( "+(x - 1.5) <= 3<BR>\n" );
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document.write( "Apply the procedures just as you would a equation.  Begin by adding 1.5 to both sides<BR>\n" );
document.write( "to remove the 1.5 from the left side.  This makes the equation become:<BR>\n" );
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document.write( "x <= 3 + 1.5 <BR>\n" );
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document.write( "add the terms on the right side and you get:<BR>\n" );
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document.write( "x <= 4.5 ... the same answer that you got.<BR>\n" );
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document.write( "Now for the second case.  Use the quantity inside the absolute value signs, only precede<BR>\n" );
document.write( "it by a negative sign.  This makes the inequality:<BR>\n" );
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document.write( "-(x - 1.5) <= 3<BR>\n" );
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document.write( "The negative sign in front of the parentheses means that you are multiplying the quantity<BR>\n" );
document.write( "in parentheses by -1.  This results in:<BR>\n" );
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document.write( "-x + 1.5 <= 3<BR>\n" );
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document.write( "Get rid of the 1.5 on the left side by subtracting 1.5 from both sides.  This simplifies<BR>\n" );
document.write( "the equation to:<BR>\n" );
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document.write( "-x <= 3 - 1.5<BR>\n" );
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document.write( "On the right side of the inequality the two terms combine to 1.5 This makes the inequality <BR>\n" );
document.write( "become:<BR>\n" );
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document.write( "-x <= 1.5<BR>\n" );
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document.write( "But we need to solve for +x.  So multiply both sides by -1.  (Don't forget the rule that<BR>\n" );
document.write( "if you multiply or divide both sides by a negative number, you need to reverse the<BR>\n" );
document.write( "direction of the inequality sign.) When you do these to things (multiply by -1 and<BR>\n" );
document.write( "reverse the direction) you get:<BR>\n" );
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document.write( "x >= -1.5<BR>\n" );
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document.write( "This also is the same as your answer.<BR>\n" );
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document.write( "You can write the two answers in a \"one-liner\".  When you say that x must be greater<BR>\n" );
document.write( "than or equal to -1.5 that means it must be at or to the right of 1.5 on the number<BR>\n" );
document.write( "line.  But you also know that x must be equal to or to the left of +4.5 on the number line.<BR>\n" );
document.write( "This can be written as:<BR>\n" );
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document.write( "- 1.5 <= x <= +4.5<BR>\n" );
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document.write( "Just to help you convince yourself that this is correct, select a number between -1.5 and<BR>\n" );
document.write( "+4.5 set x equal to that number. For example, 0 is between those two limits.  Setting<BR>\n" );
document.write( "x equal to zero makes the original inequality:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%280+-+1.5%29+%3C=+3 ALIGN=MIDDLE  ALT=\"abs%280+-+1.5%29+%3C=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This reduces to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28-1.5%29+%3C=+3 ALIGN=MIDDLE  ALT=\"abs%28-1.5%29+%3C=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But the absolute value of -1.5 is +1.5 and so the inequality is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.5+%3C=+3 ALIGN=MIDDLE  ALT=\"1.5+%3C=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is true, so that gives some confidence that we have the right answer.  You can do<BR>\n" );
document.write( "the same sort of analysis by picking a value for x that is less than -1.5 (say for example,<BR>\n" );
document.write( "that x = -5).  Substitute that value for x in the original problem and ensure yourself<BR>\n" );
document.write( "that it does NOT satisfy the inequality.  Do the same sort of analysis by letting x<BR>\n" );
document.write( "be a value greater than +4.5 such as x = +5.  Substitute +5 for x in the original <BR>\n" );
document.write( "inequality and you should see that you get a value on the left side that is greater<BR>\n" );
document.write( "than 3 ... meaning that the inequality does not work for values of x greater than 4.5.<BR>\n" );
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document.write( "Hope this discussion gives you a logical way to work problems of this type and how to<BR>\n" );
document.write( "check to build confidence that your answer is correct.<BR>\n" );
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document.write( "Don't let the above discussion confuse you. You seem to know how to work problems<BR>\n" );
document.write( "of this type.  Keep up the good work. </SPAN>\n" );
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