document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207733>Question 1207733</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the absolute value equation. \r<BR>\n" );
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document.write( "Solve |x + |3x-2|| = 2\r<BR>\n" );
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document.write( "Note: Here we have an absolute value inside another absolute value.  </SPAN>\n" );
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document.write( "Answers: <font color=red>x = 0, x = 1</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "Rule: If |x| = k then x = -k or x = k <BR>\n" );
document.write( "where k is nonnegative\r<BR>\n" );
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document.write( "Using that rule we go from this<BR>\n" );
document.write( "|x + |3x-2|| = 2<BR>\n" );
document.write( "to this<BR>\n" );
document.write( "x + |3x-2| = 2 or x + |3x-2| = -2\r<BR>\n" );
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document.write( "Let's solve each separately.<BR>\n" );
document.write( "x + |3x-2| = 2<BR>\n" );
document.write( "|3x-2| = 2-x<BR>\n" );
document.write( "3x-2 = 2-x or 3x-2 = -(2-x)<BR>\n" );
document.write( "3x-2 = 2-x or 3x-2 = -2+x<BR>\n" );
document.write( "3x+x = 2+2 or 3x-x = -2+2<BR>\n" );
document.write( "4x = 4 or 2x = 0<BR>\n" );
document.write( "x = 4/4 or x = 0/2<BR>\n" );
document.write( "x = 1 or x = 0 are two potential solutions.\r<BR>\n" );
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document.write( "Now solve the other equation<BR>\n" );
document.write( "x + |3x-2| = -2<BR>\n" );
document.write( "|3x-2| = -2-x<BR>\n" );
document.write( "3x-2 = -2-x or 3x-2 = -(-2-x)<BR>\n" );
document.write( "3x-2 = -2-x or 3x-2 = 2+x<BR>\n" );
document.write( "3x+x = -2+2 or 3x-x = 2+2<BR>\n" );
document.write( "4x = 0 or 2x = 4<BR>\n" );
document.write( "x = 0/4 or x = 4/2<BR>\n" );
document.write( "x = 0 or x = 2 \r<BR>\n" );
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document.write( "We have x = 0 show up again, but x = 2 is a new potential solution.\r<BR>\n" );
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document.write( "Summary so far:<BR>\n" );
document.write( "The 3 potential solutions are<BR>\n" );
document.write( "x = 0, x = 1, or x = 2\r<BR>\n" );
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document.write( "Like with any equation, we need to check said solutions.<BR>\n" );
document.write( "This step is very important.<BR>\n" );
document.write( "Let's start with x = 0.<BR>\n" );
document.write( "Replace each x with 0 and then apply PEMDAS.<BR>\n" );
document.write( "|x + |3x-2|| = 2<BR>\n" );
document.write( "|0 + |3*0-2|| = 2<BR>\n" );
document.write( "|0 + |0-2|| = 2<BR>\n" );
document.write( "|0 + |-2|| = 2<BR>\n" );
document.write( "|0 + 2| = 2<BR>\n" );
document.write( "|2| = 2<BR>\n" );
document.write( "2 = 2<BR>\n" );
document.write( "That checks out. <BR>\n" );
document.write( "<font color=red>x = 0 is confirmed as a solution</font>.\r<BR>\n" );
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document.write( "Now let's try x = 1<BR>\n" );
document.write( "|x + |3x-2|| = 2<BR>\n" );
document.write( "|1 + |3*1-2|| = 2<BR>\n" );
document.write( "|1 + |3-2|| = 2<BR>\n" );
document.write( "|1 + |1|| = 2<BR>\n" );
document.write( "|1 + 1| = 2<BR>\n" );
document.write( "|2| = 2<BR>\n" );
document.write( "2 = 2<BR>\n" );
document.write( "That works as well. <BR>\n" );
document.write( "<font color=red>x = 1 is confirmed as a solution</font>.\r<BR>\n" );
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document.write( "Lastly x = 2<BR>\n" );
document.write( "|x + |3x-2|| = 2<BR>\n" );
document.write( "|2 + |3*2-2|| = 2<BR>\n" );
document.write( "|2 + |6-2|| = 2<BR>\n" );
document.write( "|2 + |4|| = 2<BR>\n" );
document.write( "|2 + 4| = 2<BR>\n" );
document.write( "|6| = 2<BR>\n" );
document.write( "6 = 2<BR>\n" );
document.write( "The last statement is false since the numbers don't match up.<BR>\n" );
document.write( "A false equation at the end means the first equation of this sub-block is false when x = 2.<BR>\n" );
document.write( "It proves that x = 2 is <font color=red>NOT</font> a solution. It's considered extraneous. <BR>\n" );
document.write( "You can see why we needed to check each potential solution.\r<BR>\n" );
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document.write( "Ultimately there are 2 solutions and they are <font color=red>x = 0, x = 1</font>\r<BR>\n" );
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document.write( "If we use Desmos to graph things out, then we get a V shape curve that represents y = |x + |3x-2||<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/aavxbn9myo\">https://www.desmos.com/calculator/aavxbn9myo</a><BR>\n" );
document.write( "I used the \"abs\" function to indicate absolute value in Desmos.<BR>\n" );
document.write( "Example: abs(3x-2) = |3x-2|<BR>\n" );
document.write( "The V shape and the horizontal line y = 2 intersect at the two points (0,2) and (1,2) shown in green and purple respectively.<BR>\n" );
document.write( "The x coordinates of these points are the solutions mentioned above.<BR>\n" );
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