Algebra.Com's Answer #67624 by Edwin McCravy(20056)![\"\"](\"/images/stars/stars-13.gif\")  
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For the following equation, find the interval(s) where  .
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document.write( "First let's learn what we want to find. Let's look at \r\n" );
document.write( "the graph of\r\n" );
document.write( " which can be found by plotting\r\n" );
document.write( "points and drawing dot-to-dot:\r\n" );
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document.write( "The two green lines are not part of the graph of f(x). Only \r\n" );
document.write( "the blue curves make up the graph of . The blue \r\n" );
document.write( "curves approach but never reach the green lines. The green \r\n" );
document.write( "lines are called \"asymptotes\". But that's beside the point.\r\n" );
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document.write( "The question we are asking is this:\r\n" );
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document.write( "What part of the x-axis is the blue graph BELOW the x-axis?\r\n" );
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document.write( "Why BELOW the x-axis? Because the problem is to find\r\n" );
document.write( "where and a graph is less than zero when it is\r\n" );
document.write( "BELOW the x-axis.\r\n" );
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document.write( "We can tell that the blue graph is BELOW the x-axis on the\r\n" );
document.write( "part of the x-axis which is between -2 and 4, which we\r\n" );
document.write( "can write as -2 < x < 4 or in interval notation as (-2,4).\r\n" );
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document.write( "That is the correct answer, but we did it graphically. But\r\n" );
document.write( "you are supposed to do it algebraically. Here is how:\r\n" );
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document.write( " and we want , so\r\n" );
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document.write( "we want:\r\n" );
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document.write( "We factor the denominator\r\n" );
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document.write( "The critical values are values that make the\r\n" );
document.write( "expression either 0 or undefined. So we set\r\n" );
document.write( "each factor = 0.\r\n" );
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document.write( "x-4=0 gives critical value x=4\r\n" );
document.write( "x+2=0 gives critical value x=-2\r\n" );
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document.write( "Now we draw a number line and mark these two points:\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( "(This number line is actually the x-axis of the\r\n" );
document.write( "graph above without the graph or the y-axis.)\r\n" );
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document.write( "Anyway these two points divide the number line into \r\n" );
document.write( "three parts. Let's pick a point in each part.\r\n" );
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document.write( "Pick a point less that (left of) -2, say -3.\r\n" );
document.write( "Substitute -3 for x in\r\n" );
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document.write( "This is FALSE! So we DO NOT shade the part of \r\n" );
document.write( "the number line to the left of -2. So our number \r\n" );
document.write( "line is still as it was:\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 any point in the middle part,\r\n" );
document.write( "between -2 and 4. Suppose we choose x=0\r\n" );
document.write( "Substitute 0 for x in\r\n" );
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document.write( "This is TRUE! So we DO shade the part of the \r\n" );
document.write( "number line between -2 and 4. So our number \r\n" );
document.write( "line now looks like 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( "Finally we pick a point to the right of 4.\r\n" );
document.write( "Suppose we choose x=5.\r\n" );
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document.write( "Substitute 5 for x in\r\n" );
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document.write( "This is FALSE! So we do not shade the part to the\r\n" );
document.write( "right of x=4. So the answer is this number line:\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( "which is represented by the interval notation (-2,4),\r\n" );
document.write( "which is what we observed graphically.\r\n" );
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document.write( "Edwin \r
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