document.write( "Question 1058993: Solve the following absolute value equation:\r
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Algebra.Com's Answer #674113 by ikleyn(52791) $\"\"$ $\"About$

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\n" ); document.write( "Solve the following absolute value equation:\r
\n" ); document.write( "\n" ); document.write( "|2x-1|=|x-8|
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\n" ); document.write( "\n" ); document.write( "First look in this plot.\r
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\n" ); document.write( "\n" ); document.write( "Plot y = |2x-1| (red) and y = |x-8| (green)\r
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\n" ); document.write( "\n" ); document.write( "Second, below is the solution and the procedure on how you should treat this equation problem.\r
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\n" ); document.write( "\n" ); document.write( "Divide the entire number line in these subsets:\r
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\n" ); document.write( "\n" ); document.write( "a) $\"-infinity\"$ < x < $\"1%2F2\"$ ; b) $\"1%2F2\"$ <= x <= $\"8\"$ and c) $\"8\"$ < x < $\"infinity\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In each of this \"interval\" the given equation becomes a linear equation (different on different subsets).
\n" ); document.write( "The linear equation you can easily solve.
\n" ); document.write( "But in addition, you must check that the solution you found for the specific equation belongs to the corresponding interval.\r
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document.write( "a)   Interval  < x < .\r\n" );
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document.write( "     In this interval (2x-1) is negative, so |2x-1| = -(2x-1);\r\n" );
document.write( "                      (x-8)  is negative too, so |x-8| = -(x-8).\r\n" );
document.write( "     Therefore the original equation becomes \r\n" );
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document.write( "                      -(2x-1) = -(x-8)\r\n" );
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document.write( "     in this interval. Simplify and solve it:\r\n" );
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document.write( "        -(2x-1) = -(x-8)  --->  2x-1 = x-8  --->  2x-x = -8+1  --->  x = -7.\r\n" );
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document.write( "     The number \"-7\" belongs to the interval (,), so this value really is the solution of the original equation.\r\n" );
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document.write( "b)  Interval  <= x <= .\r\n" );
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document.write( "     In this interval (2x-1) is positive, so |2x-1| = (2x-1);\r\n" );
document.write( "                      (x-8)  is negative, so |x-8| = -(x-8).\r\n" );
document.write( "     Therefore the original equation becomes \r\n" );
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document.write( "                      (2x-1) = -(x-8)\r\n" );
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document.write( "     in this interval. Simplify and solve it:\r\n" );
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document.write( "        (2x-1) = -(x-8)  --->  2x-1 = -x+8  --->  2x+x = 8+1  --->  3x = 9  --->  x = 3.\r\n" );
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document.write( "     The number \"3\" belongs to the interval (,), so this value really is the solution of the original equation.\r\n" );
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document.write( "b)  Interval  < x < .\r\n" );
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document.write( "     In this interval (2x-1) is positive, so |2x-1| = (2x-1);\r\n" );
document.write( "                      (x-8)  is positive too, so |x-8| = (x-8).\r\n" );
document.write( "     Therefore the original equation becomes \r\n" );
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document.write( "                      (2x-1) = (x-8)\r\n" );
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document.write( "     in this interval. Simplify and solve it:\r\n" );
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document.write( "        (2x-1) = (x-8)  --->  2x-1 = x - 8  --->  2x-x = -8+1  --->  x = -7.\r\n" );
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document.write( "     But the number \"-7\" does not belong to the interval (,), so this value really is NOT the solution of the original equation in this interval.\r\n" );
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document.write( "Answer.  The solutions are x = -7 and x = 3.\r\n" );
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document.write( "Compare the answer with the plot.\r\n" );
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\n" ); document.write( "\n" ); document.write( "What I described here is a STANDARD method on solving similar problems.\r
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\n" ); document.write( "Lesson to learn from this solution: \r
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\n" ); document.write( "\n" ); document.write( "A strategy on solving of absolute value equations is to break up the entire set of real numbers into sub-domains (ranges)
\n" ); document.write( "where the absolute value of linear terms is a linear function, and then to solve the corresponding linear equations in each sub-domain (range).
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\n" ); document.write( "\n" ); document.write( "My other lessons on Absolute Value equations in this site are\r
\n" ); document.write( "\n" ); document.write( "    - Absolute Value equations\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 1\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 2\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 3\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO solve equations containing Quadratic Terms under the Absolute Value sign. Lesson 1\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO solve equations containing Quadratic Terms under the Absolute Value sign. Lesson 2 \r
\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of lessons on Absolute Value equations \r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Solving Absolute values equations\". \r
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