document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=902237>Question 902237</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the absolute value inequality. Express the answer using interval notation.<BR>\n" );
document.write( "7 &#8722; |2x + 3| &#8804; 6\r<BR>\n" );
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document.write( "Must graph the solution set. Please explain how to solve. 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 #547205 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=547205\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> start with 7 - abs(2x+3) <= 6\r<BR>\n" );
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document.write( "subtract 7 from both sides of the equation to get:\r<BR>\n" );
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document.write( "- abs(2x+3) <= -1\r<BR>\n" );
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document.write( "multiply both sides of this equation by -1 to get:\r<BR>\n" );
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document.write( "abs(2x+3) >= 1\r<BR>\n" );
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document.write( "when 2x+3 >= 0, this equation becomes:\r<BR>\n" );
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document.write( "2x+3 >= 1\r<BR>\n" );
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document.write( "solve for x to get x >= -1\r<BR>\n" );
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document.write( "when 2x+3 < 0, this equation becomes:\r<BR>\n" );
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document.write( "2x+3 <= -1\r<BR>\n" );
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document.write( "solve for x to get x <= -2\r<BR>\n" );
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document.write( "your solutions are:\r<BR>\n" );
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document.write( "x >= -1 or x <= -2\r<BR>\n" );
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document.write( "in interval notation this solution is:\r<BR>\n" );
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document.write( "(-infinity,-2] or [-1,infinity)\r<BR>\n" );
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document.write( "you would graph this equation as follows:\r<BR>\n" );
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document.write( "start with abs(2x+3) >= 1\r<BR>\n" );
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document.write( "graph y = abs(2x+3) and graph y = 1 on the same graph.\r<BR>\n" );
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document.write( "your solution lies in the region where the graph of y = abs(2x+3) is greater than or equal to the graph of y = 1.\r<BR>\n" );
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document.write( "that occurs when x <= -2 and when x >= -1.\r<BR>\n" );
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document.write( "that graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/091603.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "you can confirm the solution is good by going back to the original equation and replacing x with selected values to see when the equation is true and when the equation is false.\r<BR>\n" );
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document.write( "the original equation is 7 - abs(2x+3) <= 6\r<BR>\n" );
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document.write( "when x is -3, you get 7 - abs(-6+3) = 7 - abs(-3) = 7 - 3 = 4 which is smaller than or equal to 6 so all is good because x is smaller than or equal to -2.\r<BR>\n" );
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document.write( "when x is -2, you get 7 - abs(-4+3) = 7 - abs(-1) = 7 - 1 = 6 which is smaller than or equal to 6 so all is still good because x is smaller than or equal to -2.\r<BR>\n" );
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document.write( "when x ix 0, you get 7 - abs(0+3) = 7 - abs(3) = 7 - 3 = 4 which is smaller than or equal to 6 so all is still good because x is greater than or equal to -1.\r<BR>\n" );
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document.write( "when x is -1, you get 7 - abs(-2+3) = 7 - abs(1) = 7 - 1 = 6 which is smaller than or equal to 6 so all is still good because x is greater than or equal to -1.\r<BR>\n" );
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document.write( "when x is -1.5, you get 7 - abs(-3+3) = 7 - abs(0) = 7 - 0 = 7 which is greater than 6 so all is still good because x is not greater than or equal to -1 and x is not smaller than or equal to -2 and you should not get a result smaller than or equal to 6 which you don't.\r<BR>\n" );
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document.write( "everything looks good, so your solution is:\r<BR>\n" );
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document.write( "x <= -2 or x >- -1\r<BR>\n" );
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document.write( "interval notation is as shown above and below:\r<BR>\n" );
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document.write( "(-infinity,-2] or [-1,infinity)\r<BR>\n" );
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document.write( " </SPAN>\n" );
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