document.write( "Question 1116383: Find the set of values of x which satisfies each of the following inequalities.
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Algebra.Com's Answer #731272 by ikleyn(52814)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            Nothing is as far from the correct solution as this writing by @josgarithmetic.\r
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document.write( "It is an absolute value inequality.\r\n" );
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document.write( "The method of solving such inequalities is to divide the number line in intervals where each term of the equation is a linear function,\r\n" );
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document.write( "and then solve the resulting inequalities for linear functions for each separate interval.\r\n" );
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document.write( "In this case the critical values, where functions change their behavior, are x= 0 and x= 2.  They divide the number line\r\n" );
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document.write( "in three intervals:  1) x < 0;  2) 0 < x < 2,  and  3) x > 2.\r\n" );
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document.write( "Plot y = \"abs%28x-2%29\" (red) and y = x + abs(x) (green)\r\n" );
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document.write( "In the FIRST interval,  x < 0,  we have (x-2) < 0;  therefore  |x-2| = -(x-2).\r\n" );
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document.write( "Also, in this interval x < 0, therefore  |x| = -x  and then  x + |x| = x + (-x) = 0.\r\n" );
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document.write( "Thus in the first interval the inequality takes the form \r\n" );
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document.write( "     -(x-2) > 0 ,  or, equivalently,\r\n" );
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document.write( "      x - 2 < 0,   or\r\n" );
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document.write( "      x < 2,   which is always TRUE in the interval x < 0.\r\n" );
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document.write( "Thus the first interval,  x < 0, is the part of the solution to the original inequality.  \r\n" );
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document.write( "    At this point I completed with the first interval and now start working with the second interval.\r\n" );
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document.write( "In the SECOND interval,  0 < x < 2,  we have (x-2) < 0;  therefore  |x-2| = -(x-2).\r\n" );
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document.write( "Also, in this interval x > 0, therefore  |x| = x,  and then  x + |x| = x + x = 2x.\r\n" );
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document.write( "Thus in the second interval the inequality takes the form \r\n" );
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document.write( "     -(x-2) > 2x ,  or, equivalently,\r\n" );
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document.write( "      2x + (x - 2) < 0,   or\r\n" );
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document.write( "      3x < 2,   which is  TRUE  if  x < \"2%2F3\".\r\n" );
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document.write( "Thus for the second interval,  0 < x < 2,  ONLY PART OF IT  x < \"2%2F3\" is the solution to the original inequality.  \r\n" );
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document.write( "In the THIRD interval,  x > 2,  we have (x-2) > 0;  therefore  |x-2| = (x-2).\r\n" );
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document.write( "Also, in this interval x > 0, therefore  |x| = x,  and then  x + |x| = x + x = 2x.\r\n" );
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document.write( "Thus in the third interval the inequality takes the form \r\n" );
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document.write( "     (x-2) > 2x ,  or, equivalently,\r\n" );
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document.write( "      2x - (x - 2) < 0,   or\r\n" );
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document.write( "      x < 2,   which is always FALSE  in the interval x > 2.\r\n" );
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document.write( "Thus for the third interval, x > 2.  NO PART OF IT  is the solution to the original inequality.  \r\n" );
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document.write( "After this analysis you can conclude that the solution set to the original inequality is  x < \"3%2F2\",  or, in interval form,  (\"-infinity\",\"3%2F2\").\r\n" );
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document.write( "Answer.  The solution set to the original inequality is  x < \"3%2F2\",  or, in interval form,  (\"-infinity\",\"3%2F2\").\r\n" );
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\n" ); document.write( "If you want to see other similar solved problems on absolute value inequalities, look into 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.\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
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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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document.write( "     I think that these sources are the only place in the american school math literature, where you can learn the subject adequately.\r\n" );
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