document.write( "Question 1208939: Solve |3x - |2x + 1|| = 4\r
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Algebra.Com's Answer #847488 by ikleyn(52794) $\"\"$ $\"About$

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Solve |3x - |2x + 1|| = 4
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\n" ); document.write( "\n" ); document.write( "Looking into the solutions by other tutors, you may ask yourself, if there exist a method,
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\n" ); document.write( "\n" ); document.write( "Yes, such a method does exist. It is shown below.\r
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document.write( "Starting equation is\r\n" );
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document.write( "    |3x - |2x + 1|| = 4.    (1)\r\n" );
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document.write( "It means that \r\n" );
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document.write( "    either  3x - |2x + 1| =  4    (2)\r\n" );
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document.write( "    or      3x - |2x + 1| = -4.   (3)\r\n" );
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document.write( "Next consider equations (2) and (3) separately.\r\n" );
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document.write( "       Equation (2)\r\n" );
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document.write( "Equation (2)  is equivalent to\r\n" );
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document.write( "    |2x+1| = 3x-4.    (4)\r\n" );
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document.write( "In the domain 2x+1 >= 0,  equation (4) is equivalent to\r\n" );
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document.write( "    2x+1 = 3x-4,  1+4 = 3x - 2x,  5 = x,  x= 5.\r\n" );
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document.write( "    For this value of x,  the expression 2x+1 = 2*5+1 = 11 is positive,\r\n" );
document.write( "    so, the premise 3x-2 >= 0  is valid;  hence,  x= 5 is a valid solution to equation (4).\r\n" );
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document.write( "In the domain 2x+1 < 0,  equation (4) is equivalent to\r\n" );
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document.write( "    2x+1 = -(3x-4),  2x+1 = -3x+4,  2x+3x = 4 - 1,  5x= 3,  x= 3/5.\r\n" );
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document.write( "    For this value of x,  the expression 2x+1 = 3*(3/5)+1 = 9/5+1 is positive,\r\n" );
document.write( "    so, the premise 2x+1 < 0  is NOT valid;  hence,  x= 3/5 is NOT a valid solution to equation (4).\r\n" );
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document.write( "       Equation (3)\r\n" );
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document.write( "Equation (3)  is equivalent to\r\n" );
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document.write( "    |2x+1| = 3x+4.    (5)\r\n" );
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document.write( "In the domain 2x+1 >= 0,  equation (5) is equivalent to\r\n" );
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document.write( "    2x+1 = 3x+4,  1-4 = 3x-2x,  -3 = x,  x= -3.\r\n" );
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document.write( "    For this value of x,  the expression 2x+1 = 2*(-3)+1 = -6+1 = -5 is negative,\r\n" );
document.write( "    so, the premise 2x+1 >= 0  is NOT valid;  hence,  x= -3 is NOT a valid solution to equation (5).\r\n" );
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document.write( "In the domain 2x+1 < 0,  equation (5) is equivalent to\r\n" );
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document.write( "    2x+1 = -(3x+4),  2x+1 = -3x-4,  2x + 3x = -4 - 1,  5x= -5,  x= -5/5 = -1.\r\n" );
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document.write( "    For this value of x,  the expression 2x+1 = 2*(-1)+1 = -2+1 = -1 is negative,\r\n" );
document.write( "    so, the premise 2x+1 < 0  is valid;  hence,  x= -1 is a valid solution to equation (5).\r\n" );
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document.write( "ANSWER.  After this analysis, we see that the only solutions for the given equation (1)  \r\n" );
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document.write( "         are x= -1  and  x= 5.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Notice that this solution follows the strict logic at every step, with the analysis of possible
\n" ); document.write( "domains, so it provides the true roots of the original equation without the necessity to check them at the end.\r
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\n" ); document.write( "\n" ); document.write( "This method of solution and this logic do not create excessive erroneous solutions,
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\n" ); document.write( "\n" ); document.write( "The possible excessive erroneous solutions are rejected (are excluded) in the course of analysis.\r
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\n" ); document.write( "\n" ); document.write( "The plot using plotting tool www.desmos/calculator (free of charge for common use) does confirm the solution visually\r
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