document.write( "Question 1159458: Solve the inequality|x-2| > 2x-5. \n" ); document.write( "

Algebra.Com's Answer #852222 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Solve the inequality |x-2| > 2x-5.
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\n" ); document.write( "\n" ); document.write( " The solution by @MowMow in his post is FATALLY wrong.\r
\n" ); document.write( "\n" ); document.write( " I came to bring a correct solution.\r
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document.write( "We should consider two cases.\r\n" );
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document.write( "            Case 1.\r\n" );
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document.write( "If x >= 2, then the difference  (x-2)  is positive, and |x-2| = x-2.\r\n" );
document.write( "So, we should find the solutions to this inequality\r\n" );
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document.write( "    x-2 > 2x - 5     (1)\r\n" );
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document.write( "in the domain  x >= 2.\r\n" );
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document.write( "Inequality  (1)  is equivalent to\r\n" );
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document.write( "    5 - 2 > 2x - x,\r\n" );
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document.write( "or\r\n" );
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document.write( "      3   > x.\r\n" );
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document.write( "Thus, in the domain  x >= 2,  the solution set is  x < 3.\r\n" );
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document.write( "In other words, in the domain  x >= 2  the solution set is  2 <= x < 3.\r\n" );
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document.write( "            Case 2.\r\n" );
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document.write( "If x < 2, then the difference (x-2)  is negative, and |x-2| = -(x-2).\r\n" );
document.write( "So, we should find the solutions to this inequality\r\n" );
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document.write( "    -(x-2) > 2x - 5    (2)\r\n" );
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document.write( "in the domain x < 2. \r\n" );
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document.write( "Inequality (2) is equivalent to\r\n" );
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document.write( "    -x + 2 > 2x - 5,\r\n" );
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document.write( "    5  + 2 > 2x + x\r\n" );
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document.write( "      7   >    3x\r\n" );
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document.write( "      x   <    7/3.\r\n" );
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document.write( "Thus in the domain  x < 2, all the values satisfy inequality (2).\r\n" );
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document.write( "Combining cases 1 and 2, we see that the solution set for the original equation is  x < 3.\r\n" );
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document.write( "ANSWER.  The solution set is  x < 3,  or, in interval notation,  ( , ).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "In the online picture\r
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\n" ); document.write( "\n" ); document.write( "https://www.desmos.com/calculator/nzwdgckzfi \r
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\n" ); document.write( "\n" ); document.write( "I prepared a plot of functions y = |x-2| and y = 2x-5.\r
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\n" ); document.write( "\n" ); document.write( "Looking at this plot, you may check visually that my answer is correct.\r
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