document.write( "Question 1139420: 2|4-3x|-3|2x+1| < 7\r
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Algebra.Com's Answer #759894 by ikleyn(52847)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            This inequality has two linear functions under the absolute value sign each.\r
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\n" ); document.write( "\n" ); document.write( "            This \"absolute value\" sign transform a linear function into non-linear, which is not so simple to solve.\r
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\n" ); document.write( "\n" ); document.write( "            Therefore, the solution strategy is to divide the entire number line into separate intervals / segments
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\n" ); document.write( "\n" ); document.write( "            Then the solution is doable and simple.\r
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\n" ); document.write( "\n" ); document.write( "            Below is how I implement this idea.\r
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document.write( "In this case we have two critical points, x= \"-1%2F2\"  and x= \"4%2F3\", where the functions change their linear behavior.\r\n" );
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document.write( "These points divide the entire number line in 3 non-intersecting intervals/segments\r\n" );
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document.write( "    1)  x < \"-1%2F2\";    2)  \"-1%2F2\" <= x <= \"4%2F3\";   and   3)  x > \"4%2F3\".\r\n" );
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document.write( "Let's analyze each interval separately.\r\n" );
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document.write( "1)  If x < \"-1%2F2\",  then  | 2x+1 | = -(2x+1)  and  | 4-3x | = 4-3x.\r\n" );
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document.write( "                therefore, the original inequality takes the form\r\n" );
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document.write( "                    (-3)*(-(2x+1)) + 2*(4-3x) < 7.\r\n" );
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document.write( "                Simplify and solve it step by step\r\n" );
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document.write( "                     6x + 3 + 8 - 6x < 7\r\n" );
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document.write( "                     11 < 7.\r\n" );
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document.write( "               This inequality is FALSE.\r\n" );
document.write( "               It means that the interval  x < \"-1%2F2\" is NOT the solution to the original inequality.\r\n" );
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document.write( "2)  If \"-1%2F2\" <= x <= \"4%2F3\",  then  | 2x+1 | = 2x+1  and  | 4-3x | = 4-3x.\r\n" );
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document.write( "                therefore, the original inequality takes the form\r\n" );
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document.write( "                    (-3)*(2x+1) + 2*(4-3x) < 7.\r\n" );
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document.write( "                Simplify and solve it step by step\r\n" );
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document.write( "                     -6x - 3 + 8 - 6x < 7\r\n" );
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document.write( "                     -12x < 2.\r\n" );
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document.write( "                      x   > \"2%2F%28-12%29\" = \"-1%2F6\"\r\n" );
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document.write( "               It means that in the interval \"-1%2F6\" <= x <= \"4%2F3\" is the partial solution to the original inequality.\r\n" );
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document.write( "2)  If  x > \"4%2F3\",  then  | 2x+1 | = 2x+1  and  | 4-3x | = -(4-3x).\r\n" );
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document.write( "                therefore, the original inequality takes the form\r\n" );
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document.write( "                    (-3)*(2x+1) + 2*(-(4-3x)) < 7.\r\n" );
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document.write( "                Simplify and solve it step by step\r\n" );
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document.write( "                     -6x - 3 - 8 + 6x < 7\r\n" );
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document.write( "                     -11 < 7.\r\n" );
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document.write( "               This inequality is TRUE.\r\n" );
document.write( "               It means that in the interval x >= \"4%2F3\" is the partial solution to the original inequality.\r\n" );
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document.write( "Thus, after completing analyses of all 3 cases/intervals we come to this conclusion\r\n" );
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document.write( "ANSWER.  The given inequality has the solution set  x >= \"-1%2F6\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "See the plot below, which visually confirms the found solution.\r
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document.write( "    \r\n" );
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document.write( "    Plot y = 2*|4-3x| - 3*|2x+1| (red)  and  y = 7  (green).\r\n" );
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\n" ); document.write( "\n" ); document.write( "--------------\r
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\n" ); document.write( "\n" ); document.write( "To see many other similar solved problems, 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
\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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