document.write( "Question 512295: I don't get how do we find the solution set for $\"x%5E2-x-2%3C0\"$ . Thank You very much :) \n" ); document.write( "
Algebra.Com's Answer #342598 by Edwin McCravy(20055) $\"\"$ $\"About$
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document.write( "We want values of x such that x² - x - 2 < 0, or\r\n" );
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document.write( "(x - 2)(x + 1) < 0\r\n" );
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document.write( "First we find the zeros of the function f(x) = x² - x - 2 \r\n" );
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document.write( "We write it in factored form\r\n" );
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document.write( "f(x) = (x - 2)(x + 1)\r\n" );
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document.write( "Then we'll test the intervals which they divide the number line into.\r\n" );
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document.write( "To find the zeros of f(x), we set f(x) = 0\r\n" );
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document.write( "(x - 2)(x + 1) = 0\r\n" );
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document.write( "x - 2 = 0;  x + 1 = 0\r\n" );
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document.write( "    x = 2;      x = -1\r\n" );
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document.write( "Nest we draw a number line and mark those points on it with an\r\n" );
document.write( "open circle:\r\n" );
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document.write( "-------------o--------o-------------\r\n" );
document.write( "-5 -4 -3 -2 -1  0  1  2  3  4  5  6\r\n" );
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document.write( "We pick a test value on the infinitely long interval left of\r\n" );
document.write( "-1.  The easiest value to pick is -2, and substitute it into\r\n" );
document.write( "the original inequality\r\n" );
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document.write( "  (x - 2)(x + 1) < 0\r\n" );
document.write( "(-2 - 2)(-2 + 1) < 0\r\n" );
document.write( "        (-4)(-1) < 0\r\n" );
document.write( "               4 < 0   \r\n" );
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document.write( "That is false, so we do not shade the region left of -1, so we\r\n" );
document.write( "still have this:\r\n" );
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document.write( "-------------o--------o-------------\r\n" );
document.write( "-5 -4 -3 -2 -1  0  1  2  3  4  5  6\r\n" );
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document.write( "Next we pick a test value on the interval between -1 and 2.  The \r\n" );
document.write( "easiest value to pick is 0, and substitute it into\r\n" );
document.write( "the original inequality\r\n" );
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document.write( "(x - 2)(x + 1) < 0\r\n" );
document.write( "(0 - 2)(0 + 1) < 0\r\n" );
document.write( "       (-2)(1) < 0\r\n" );
document.write( "            -2 < 0 \r\n" );
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document.write( "That is true, so we DO shade the interval between -1 and 2, so we\r\n" );
document.write( "now have this:\r\n" );
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document.write( "-------------o========o-------------\r\n" );
document.write( "-5 -4 -3 -2 -1  0  1  2  3  4  5  6\r\n" );
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document.write( "We pick a test value on the infinitely long interval right of\r\n" );
document.write( "2.  The easiest value to pick is 3, and substitute it into\r\n" );
document.write( "the original inequality\r\n" );
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document.write( "(x - 2)(x + 1) < 0\r\n" );
document.write( "(3 - 2)(3 + 1) < 0\r\n" );
document.write( "        (1)(4) < 0\r\n" );
document.write( "             4 < 0  \r\n" );
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document.write( "That is false, so we do not shade the region right of 2, so we\r\n" );
document.write( "end up with this:\r\n" );
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document.write( "-------------o========o-------------\r\n" );
document.write( "-5 -4 -3 -2 -1  0  1  2  3  4  5  6\r\n" );
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document.write( "The interval notation is an appreviation for the above number\r\n" );
document.write( "line graph, and this interval notation abbreviation is\r\n" );
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document.write( "              (-1, 2)\r\n" );
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document.write( "[Don't confuse this with the 2D point on the xy-plane which has the\r\n" );
document.write( "same notation.  This is a conflict in mathematical notation.  Sometimes\r\n" );
document.write( "(-1,2) means a point in the xy-plane and other times it means an interval\r\n" );
document.write( "on the number line.  We just have to live with this conflict, and just go\r\n" );
document.write( "by context to discover which it means.)  \r\n" );
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document.write( "Edwin
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