document.write( "Question 1139419: |(5-2x)/(x+2)| ≤ 1 \n" ); document.write( "

Algebra.Com's Answer #759891 by ikleyn(52835) $\"\"$ $\"About$

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document.write( "An inequality\r\n" );
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document.write( "     ≤ 1\r\n" );
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document.write( "is equivalent to\r\n" );
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document.write( "    -1 <=  <= 1.\r\n" );
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document.write( "It means that we should solve TWO inequalities \r\n" );
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document.write( "    (a)  -1 <=    and   (b)   <= 1\r\n" );
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document.write( "and then take the intersection set of their solutions.\r\n" );
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document.write( "The domain (the set of real numbers where the rational function is defined) is the set of all real numbers except of x = -2.\r\n" );
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document.write( "I will solve the inequality (a)   -1 <=     first.\r\n" );
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document.write( "Let us consider the domain x > -2;  where (x+2) > 0 is positive.\r\n" );
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document.write( "     Then the inequality (a) is equivalent to\r\n" );
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document.write( "        -(x+2) <= 5-2x,   -x - 2 <= 5 - 2x,   2x - x < = 5 +2,   x <= 7.\r\n" );
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document.write( "     So, the interval (-2,7] is the part of the solution to the inequality (a) in the domain x > -2.\r\n" );
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document.write( "We should continue and complete the solution to the inequality (a), now in the domain x < -2.\r\n" );
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document.write( "     Then (x+2) < 0 is negative, and the inequality (a) is equivalent to \r\n" );
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document.write( "          -(x+2) >= 5-2x,   -x - 2 >= 5-2x,   2x - x >= 5 + 2,   x >= 7.\r\n" );
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document.write( "     So, in the domain x < -2  the inequality (a) has no solution.\r\n" );
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document.write( "Thus we completed solving inequality (a) and found the set of its solutions as the interval (-2,7].\r\n" );
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document.write( "Now we should solve the inequality  (b)    <= 1.\r\n" );
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document.write( "Again let's start with the domain  x > -2.\r\n" );
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document.write( "     In this case, the inequality (b) is equivalent to\r\n" );
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document.write( "        5-2x <= x+2,   5-2 <= x + 2x,   3 < = 3x,   x >= 1.\r\n" );
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document.write( "     So, the interval x >= 1 is the part of the solution to the inequality (b) in the domain x > -2.\r\n" );
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document.write( "We should continue and complete the solution to the inequality (b), now in the domain x < -2.\r\n" );
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document.write( "     Then (x+2) < 0 is negative, and the inequality (b) is equivalent to \r\n" );
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document.write( "          5-2x >= x+2,   5-2 >= x + 2x,   3 >= 3x,   x <= 1.\r\n" );
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document.write( "     So, the interval x < -2  is the solution to inequality (b) in this case.\r\n" );
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document.write( "Thus we completed solving inequality (b) and found the set of its solutions as the union of intervals (,) U [,).\r\n" );
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document.write( "The intersection of solutions to (a) and (b)  is the intersection of the sets\r\n" );
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document.write( "    (-2,7]   (for (a))   and   (,) U [,)   (for (b)).\r\n" );
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document.write( "This intersection is the set [1,7].\r\n" );
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document.write( "ANSWER.  The solution to the original inequality is the set  [1,7].\r\n" );
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\n" ); document.write( "\n" ); document.write( "Below is the plot providing visual check of the solution.\r
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document.write( "    \r\n" );
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document.write( "    Plot y =  (red) and  y = 1 (green)\r\n" );
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\n" ); document.write( "\n" ); document.write( "Concluding remark. The analysis (the solution) by @stanbon was incomplete; it is why I wrote my solution.\r
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\n" ); document.write( "\n" ); document.write( "Fortunately, both answers are the same.\r
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