document.write( "Question 87680: I am no good at graphing. Please help.
\n" ); document.write( "Find and graph the solution set:
\n" ); document.write( "a) x^3 -16x >0
\n" ); document.write( "b) X^2-5x_< -6 \n" ); document.write( "
Algebra.Com's Answer #63606 by Edwin McCravy(20060) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "a) x³ - 16x > 0\r\n" );
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document.write( "Since we already have 0 on the right side, we factor out\r\n" );
document.write( "x on the left side:\r\n" );
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document.write( " x(x² - 16) > 0\r\n" );
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document.write( "Then we factor the expression in parentheses as the\r\n" );
document.write( "difference of two squares:\r\n" );
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document.write( " x(x - 4)(x + 4) > 0\r\n" );
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document.write( "The critical values of the left side are 0, 4 and -4\r\n" );
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document.write( "So we put these on a number line and mark them with\r\n" );
document.write( "an open circle.  (Open circle since it's > and not >\r\n" );
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document.write( "--------o--------o--------o-----------\r\n" );
document.write( "       -4        0        4\r\n" );
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document.write( "Now we substitute any value to the left of -4 to see\r\n" );
document.write( "if we shade that part or not. Suppose we choose -5.\r\n" );
document.write( "Substitute x = -5 into:\r\n" );
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document.write( "   x(x - 4)(x + 4) > 0\r\n" );
document.write( "-5(-5 - 4)(-5 + 4) > 0\r\n" );
document.write( "        -5(-9)(-1) > 0\r\n" );
document.write( "               -45 > 0\r\n" );
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document.write( "This is false so we do not shade the part of the number\r\n" );
document.write( "line left of -4, so the number line is unchanged:\r\n" );
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document.write( "--------o--------o--------o-----------\r\n" );
document.write( "       -4        0        4\r\n" );
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document.write( "Nex we substitute any value between -4 and 0 to see\r\n" );
document.write( "if we shade that part or not. Suppose we choose -1.\r\n" );
document.write( "Substitute x = -1 into:\r\n" );
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document.write( "   x(x - 4)(x + 4) > 0\r\n" );
document.write( "-1(-1 - 4)(-1 + 4) > 0\r\n" );
document.write( "         -1(-5)(3) > 0\r\n" );
document.write( "                15 > 0\r\n" );
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document.write( "This is true so we do shade the part of the number\r\n" );
document.write( "line between -4 and 0, so the number line is now:\r\n" );
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document.write( "--------o========o--------o-----------\r\n" );
document.write( "       -4        0        4\r\n" );
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document.write( "Next we substitute any value between 0 and 4 to see\r\n" );
document.write( "if we shade that part or not. Suppose we choose 1.\r\n" );
document.write( "Substitute x = 1 into:\r\n" );
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document.write( "   x(x - 4)(x + 4) > 0\r\n" );
document.write( "   1(1 - 4)(1 + 4) > 0\r\n" );
document.write( "          1(-3)(5) > 0\r\n" );
document.write( "               -15 > 0\r\n" );
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document.write( "This is false so we do not shade the part of the number\r\n" );
document.write( "line between 0 and 4, so the number line is still:\r\n" );
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document.write( "--------o========o--------o-----------\r\n" );
document.write( "       -4        0        4\r\n" );
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document.write( "Finally we substitute any value to the right of 4 to see\r\n" );
document.write( "if we shade that part or not. Suppose we choose 5.\r\n" );
document.write( "Substitute x = 5 into:\r\n" );
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document.write( "   x(x - 4)(x + 4) > 0\r\n" );
document.write( "   5(5 - 4)(5 + 4) > 0\r\n" );
document.write( "           5(1)(9) > 0\r\n" );
document.write( "                45 > 0\r\n" );
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document.write( "This is true so we do shade the part of the number\r\n" );
document.write( "line right of 4, so the final number line is now:\r\n" );
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document.write( "--------o========o--------o===========>\r\n" );
document.write( "       -4        0        4\r\n" );
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document.write( "The interval notation is an abbreviation of this graph\r\n" );
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document.write( "     (-4, 0) U (4, oo)\r\n" );
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document.write( "==============================================\r\n" );
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document.write( "b) x² - 5x < -6\r\n" );
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document.write( "Get 0 on the right by adding 6 to both sides:\r\n" );
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document.write( "   x² - 5x + 6 < 0\r\n" );
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document.write( "Factor the left side:\r\n" );
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document.write( "(x - 2)(x - 3) < 0\r\n" );
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document.write( "The critical values are 2 and 3\r\n" );
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document.write( "So we put these on a number line, this time using\r\n" );
document.write( "closed circles because the inequality this time is \r\n" );
document.write( "< and not <.\r\n" );
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document.write( "------------------l----l-------\r\n" );
document.write( "  -1    0    1    2    3    4    \r\n" );
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document.write( "Now we substitute any value to the left of 2 to see\r\n" );
document.write( "if we shade that part or not. Suppose we choose 0.\r\n" );
document.write( "Substitute x = 0 into:\r\n" );
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document.write( "    (x - 2)(x - 3) < 0\r\n" );
document.write( "    (0 - 2)(0 - 3) < 0\r\n" );
document.write( "          (-2)(-3) < 0\r\n" );
document.write( "                 6 < 0 \r\n" );
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document.write( "This is false so we do not shade the part of the number\r\n" );
document.write( "line left of 2, so the number line is unchanged:\r\n" );
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document.write( "------------------l----l-------\r\n" );
document.write( "  -1    0    1    2    3    4    \r\n" );
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document.write( "Now we substitute any value between 2 and 3 to see\r\n" );
document.write( "if we shade that part or not. Suppose we choose 2.5.\r\n" );
document.write( "Substitute x = 2.5 into:\r\n" );
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document.write( "    (x - 2)(x - 3) < 0\r\n" );
document.write( "(2.5 - 2)(2.5 - 3) < 0\r\n" );
document.write( "         (.5)(-.5) < 0\r\n" );
document.write( "              -.25 < 0 \r\n" );
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document.write( "This is true so we do shade the part of the number\r\n" );
document.write( "line between 2 and 3, so the number line is now:\r\n" );
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document.write( "------------------l====l-------\r\n" );
document.write( "  -1    0    1    2    3    4    \r\n" );
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document.write( "Finally we substitute any value to the right of 3 \r\n" );
document.write( "to see if we shade that part or not. Suppose we \r\n" );
document.write( "choose 4.  Substitute x = 4 into:\r\n" );
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document.write( "    (x - 2)(x - 3) < 0\r\n" );
document.write( "    (4 - 2)(4 - 3) < 0\r\n" );
document.write( "            (2)(1) < 0\r\n" );
document.write( "                 2 < 0 \r\n" );
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document.write( "This is false so we do not shade the part of the number\r\n" );
document.write( "line to the right of 3, so the final number line is this:\r\n" );
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document.write( "------------------l====l-------\r\n" );
document.write( "  -1    0    1    2    3    4\r\n" );
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document.write( "   In interval notation that is [2, 3]\r\n" );
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document.write( "Edwin
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