document.write( "Question 1093209: |x+24|=-7x\r
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Algebra.Com's Answer #707870 by ikleyn(52788)\"\" \"About 
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\n" ); document.write( "|x + 24| = -7x.         (1)\r
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\n" ); document.write( "\n" ); document.write( "        Let me explain how to solve it and how to present the solution in a way  how it should be done.\r
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document.write( "As you know, the absolute value term may have different expressions.\r\n" );
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document.write( "1.  If x >= - 24,  then  |x+24| = x + 24,\r\n" );
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document.write( "    and the equation  (1)  takes the form\r\n" );
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document.write( "    x + 24 = -7x.       (2)\r\n" );
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document.write( "    Then  24 = -8x,  x = \"24%2F%28-8%29\" = -3,\r\n" );
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document.write( "    and, since  x= -3 >= -24,  it is really the solution of  eq(2) in the domain  x >= - 24.\r\n" );
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document.write( "    Therefore, it is the solution to the original equation (1).\r\n" );
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document.write( "2.  If x < -24,  then  |x+24| = -(x + 24),\r\n" );
document.write( "    \r\n" );
document.write( "    and the equation  (1)  takes the form\r\n" );
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document.write( "    -(x + 24) = -7x.     (3)\r\n" );
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document.write( "    Then  -24 = -6x,  x = \"-24%2F%28-6%29\" = 4.\r\n" );
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document.write( "    BUT  x= 4  is NOT in the domain  x < - 24.\r\n" );
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document.write( "    Therefore,  the value  x= 4  is NOT the solution to the original equation (1).\r\n" );
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document.write( "3.  Thus we found one and only one solution  x = -3.\r\n" );
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document.write( "    The solution  x = 4 was filtered out in the solution process.\r\n" );
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document.write( "4.  The standard way to solve an equation containing the terms under the absolute value sign is to divide the entire number line\r\n" );
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document.write( "    in separate segments, where you can take off the absolute value sign and to write the equation without absolute value sign.\r\n" );
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document.write( "    But then you must remember that the corresponding part of the analysis relates only to this interval of the number line\r\n" );
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document.write( "    which is under consideration.\r\n" );
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document.write( "    If under this local part of the solution you found the root at the considered interval - then it really is the solution \r\n" );
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document.write( "    to the original absolute value  equation.\r\n" );
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document.write( "    If under this local part of the solution you found the root, but it is not at the considered interval - it still is not \r\n" );
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document.write( "    the reason to declare this value as the solution to the original absolute value  equation.\r\n" );
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document.write( "5.  By doing in this way, you will not produce EXTRANEOUS solutions at the final stage.\r\n" );
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document.write( "    You will filter them out at the earlier stages of the (local) analysis.\r\n" );
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\n" ); document.write( "\n" ); document.write( "See the lessons\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( "Read them attentively and become an expert in this area.
\n" ); document.write( "This idea and this methodology were systematically implemented there.
\n" ); document.write( "Actually, it is the only way to solve absolute value equations correctly.\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
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
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