document.write( "Question 1094264: |5 x -4| = |x + 17| the 2 soulutions for it \n" ); document.write( "
Algebra.Com's Answer #708861 by ikleyn(52777)\"\" \"About 
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\n" ); document.write( "|5x -4| = |x + 17|         (1)\r
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\n" ); document.write( "\n" ); document.write( "        As you know, the absolute value functions have different expressions in different parts of the number line.\r
\n" ); document.write( "\n" ); document.write( "        The strategy of solving such problems is to divide the number line in segments/intervals, \r
\n" ); document.write( "\n" ); document.write( "        where the absolute value functions can be written in more simple way, and then solve the equations over each such interval.\r
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\n" ); document.write( "\n" ); document.write( "        Below find an implementation of this strategy.\r
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document.write( "There are 2 critical points: x = \"4%2F5\" and x = -17,  and 3 intervals:\r\n" );
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document.write( "(-infinity,-17],  (-17,\"4%2F5\")  and  [\"4%2F5\",infinity).\r\n" );
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document.write( "1)  Interval (-infinity,-17].\r\n" );
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document.write( "    In this interval,  5x-4 < 0;    therefore, |5x-4| = -(5x-4).\r\n" );
document.write( "                       x + 17 <=0;  therefore, |x+17| = -(x+17).\r\n" );
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document.write( "    Therefore, the original equation (1) takes the form\r\n" );
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document.write( "        -(5x-4) = -(x+17)   over the domain  x <= -17.    (2)\r\n" );
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document.write( "     Simplify:\r\n" );
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document.write( "        -5x + 4 = -x - 17  ====>  4 + 17 = -x + 5x  ====>  4x = 21  ====>  x = \"21%2F4\". \r\n" );
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document.write( "      Now notice that this value \"21%2F4\"  does not belong  to the domain x <= -17.  So, the equation (2) HAS NO solution/solutions over this domain.\r\n" );
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document.write( "      Thus this case is completed.\r\n" );
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document.write( "2)  Interval (-17,\"4%2F5\").\r\n" );
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document.write( "    In this interval,  5x-4 < 0;    therefore, |5x-4| = -(5x-4).\r\n" );
document.write( "                       x + 17 > 0;  therefore, |x+17| =   x+17.\r\n" );
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document.write( "    Therefore, the original equation (1) takes the form\r\n" );
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document.write( "        -(5x-4) = x+17   over the domain   -17 < x < \"4%2F5\".    (3)\r\n" );
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document.write( "     Simplify:\r\n" );
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document.write( "        -5x + 4 = x + 17  ====>  4 - 17 = x + 5x  ====>  6x = -13  ====>  x = \"-13%2F6\". \r\n" );
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document.write( "      This value \"-13%2F6\"  does belong  to the domain -17 < x < \"4%2F5\".  \r\n" );
document.write( "      So, the equation (3) HAS the solution {{-13/6}}} in this domain.\r\n" );
document.write( "      Hence, it is the solution to the original equation (1), too.\r\n" );
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document.write( "      Thus this case is completed.\r\n" );
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document.write( "3)  Interval [\"4%2F5\",infinity).\r\n" );
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document.write( "    In this interval,  5x-4 > 0;    therefore, |5x-4| =   5x-4.\r\n" );
document.write( "                       x + 17 > 0;  therefore, |x+17| =   x+17.\r\n" );
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document.write( "    Therefore, the original equation (1) takes the form\r\n" );
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document.write( "         5x-4 = x+17   over the domain  x > \"4%2F5\".    (4)\r\n" );
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document.write( "     Simplify:\r\n" );
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document.write( "         5x-4 = x+17  ====>  5x - x = 17+4 = 21  ====>  4x = 21  ====>  x = \"21%2F4\". \r\n" );
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document.write( "      This value \"21%2F4\"  does belong  to the domain  x > \"4%2F5\".  \r\n" );
document.write( "      So, the equation (4) HAS the solution {{21/4}}} in this domain.\r\n" );
document.write( "      Hence, it is the solution to the original equation (1), too.\r\n" );
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document.write( "      Thus this case is completed, too.\r\n" );
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document.write( "Answer.  The original equation (1) has two solutions:  x = \"-13%2F6\"   and  x = \"21%2F4\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "Plot y = |5x-4| (red) and y = |x+17| (green)\r
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\n" ); document.write( "\n" ); document.write( "---------------
\n" ); document.write( "On solving  Absolute Value equations  see the lessons in this site\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.\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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\n" ); document.write( "WRITING  BY  @josgarithmetic  IS  ABSOLUTELY  INCORRECT  APPROACH  TO  THE  SOLUTION  OF  THIS  PROBLEM.\r
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\n" ); document.write( "\n" ); document.write( "So,   I G N O R E   and   D I S R E G A R D   it.\r
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\n" ); document.write( "\n" ); document.write( "This person is absolutely unfamiliar with the subject.\r
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