document.write( "Question 154532: I'm having trouble with these. If you could help me with these few I think I will be able to continue.
\n" ); document.write( "4/x < x
\n" ); document.write( "x/x+1 > 3x
\n" ); document.write( "6/x-1 >or= 1
\n" ); document.write( "x/2 >or= 5/x+1 + 4 \n" ); document.write( "
Algebra.Com's Answer #113783 by AnlytcPhil(1806)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I'll do the first and last:\r\n" );
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document.write( "\"4%2Fx+%3C+x\"\r\n" );
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document.write( "Get 0 on one side:\r\n" );
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document.write( "\"4%2Fx-x%3C0\"\r\n" );
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document.write( "\"4%2Fx-x%5E2%2Fx%3C0\"\r\n" );
document.write( "\r\n" );
document.write( "\"%284-x%5E2%29%2Fx%3C0\"\r\n" );
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document.write( "\"%282-x%29%282%2Bx%29%2Fx%3C0\"\r\n" );
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document.write( "To find the critical values set numerator=0,\r\n" );
document.write( "and denominator =0 and solve for x:\r\n" );
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document.write( "(2-x)(2+x)=0 gives 2 and -2 as critical values\r\n" );
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document.write( "x=0 gives 0 as a critical value.\r\n" );
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document.write( "So mark the critical values on a number line:\r\n" );
document.write( "\r\n" );
document.write( "----------o-----o-----o---------\r\n" );
document.write( "-5 -4 -3 -2 -1  0  1  2  3  4  5 \r\n" );
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document.write( "First select a test point in the region left\r\n" );
document.write( "of -2.  Choose x=-3.  Substitute it in\r\n" );
document.write( "\r\n" );
document.write( "\"%282-x%29%282%2Bx%29%2Fx%3C0\"\r\n" );
document.write( "\"%282-%28-3%29%29%282%2B%28-3%29%29%2F%28-3%29%3C0\"\r\n" );
document.write( "\"%282%2B3%29%282-3%29%2F%28-3%29%3C0\"\r\n" );
document.write( "\"%285%29%28-1%29%2F%28-3%29%3C0\"\r\n" );
document.write( "\"%28-5%29%2F%28-3%29%3C0\"\r\n" );
document.write( "\"5%2F3%3C0\"\r\n" );
document.write( "That is false, so we do not shade the region\r\n" );
document.write( "left of -2.  So we still have the same\r\n" );
document.write( "number line:\r\n" );
document.write( "\r\n" );
document.write( "----------o-----o-----o---------\r\n" );
document.write( "-5 -4 -3 -2 -1  0  1  2  3  4  5 \r\n" );
document.write( "\r\n" );
document.write( "Next select a test point in the region between\r\n" );
document.write( "-2 and 0.  Choose x=-1.  Substitute it in\r\n" );
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document.write( "\"%282-x%29%282%2Bx%29%2Fx%3C0\"\r\n" );
document.write( "\"%282-%28-1%29%29%282%2B%28-1%29%29%2F%28-1%29%3C0\"\r\n" );
document.write( "\"%282%2B1%29%282-1%29%2F%28-1%29%3C0\"\r\n" );
document.write( "\"%283%29%281%29%2F%28-3%29%3C0\"\r\n" );
document.write( "\"%283%29%2F%28-3%29%3C0\"\r\n" );
document.write( "\"-1%3C0\"\r\n" );
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document.write( "That is true, so we do shade the region between\r\n" );
document.write( "-2 and 0. So now the number line is\r\n" );
document.write( "\r\n" );
document.write( "----------o=====o-----o---------\r\n" );
document.write( "-5 -4 -3 -2 -1  0  1  2  3  4  5 \r\n" );
document.write( "\r\n" );
document.write( " Next select a test point in the region between\r\n" );
document.write( "0 and 2.  Choose x=1.  Substitute it in\r\n" );
document.write( "\r\n" );
document.write( "\"%282-x%29%282%2Bx%29%2Fx%3C0\"\r\n" );
document.write( "\"%282-%281%29%29%282%2B%281%29%29%2F%281%29%3C0\"\r\n" );
document.write( "\"%282-1%29%282%2B1%29%2F%281%29%3C0\"\r\n" );
document.write( "\"%281%29%283%29%2F%281%29%3C0\"\r\n" );
document.write( "\"%283%29%2F%281%29%3C0\"\r\n" );
document.write( "\"3%3C0\"\r\n" );
document.write( "\r\n" );
document.write( "That is false, so we do not shade the region between\r\n" );
document.write( "0 and 2. So the number line is still\r\n" );
document.write( "\r\n" );
document.write( "----------o=====o-----o---------\r\n" );
document.write( "-5 -4 -3 -2 -1  0  1  2  3  4  5\r\n" );
document.write( "\r\n" );
document.write( "finally select a test point in the region right\r\n" );
document.write( "of 2.  Choose x=3.  Substitute it in\r\n" );
document.write( "\r\n" );
document.write( "\"%282-x%29%282%2Bx%29%2Fx%3C0\"\r\n" );
document.write( "\"%282-%283%29%29%282%2B%283%29%29%2F%283%29%3C0\"\r\n" );
document.write( "\"%282-3%29%282%2B3%29%2F%283%29%3C0\"\r\n" );
document.write( "\"%28-1%29%285%29%2F%283%29%3C0\"\r\n" );
document.write( "\"%28-5%29%2F%283%29%3C0\"\r\n" );
document.write( "\"-5%2F3%3C0\"\r\n" );
document.write( "That is true, so we do shade the region\r\n" );
document.write( "right of 2.  So the final number line is:\r\n" );
document.write( "\r\n" );
document.write( "----------o=====o-----o=========>\r\n" );
document.write( "-5 -4 -3 -2 -1  0  1  2  3  4  5\r\n" );
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document.write( "Now we have to test the critical values\r\n" );
document.write( "to see if they are solutions. But that's \r\n" );
document.write( "easy because with strict inequality, none \r\n" );
document.write( "of the critical values can be solutions.\r\n" );
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document.write( "In interval notation:\r\n" );
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document.write( " (-2,0) U (2,\"infinity\")\r\n" );
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document.write( "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n" );
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document.write( "\"x%2F2+%3E=+5%2F%28x%2B1%29+%2B+4\"\r\n" );
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document.write( "Get 0 on one side\r\n" );
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document.write( "\"x%2F2+-+5%2F%28x%2B1%29+-+4%3E=0\"\r\n" );
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document.write( "Get LCD of \"2%28x%2B1%29\"\r\n" );
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document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"%28x%5E2%2Bx+-+10+-+8x-8%29%2F%282%28x%2B1%29%29%3E=0\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28x%5E2-7x-18%29%2F%282%28x%2B1%29%29%3E=0\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28%28x-9%29%28x%2B2%29%29%2F%282%28x%2B1%29%29%3E=0\"\r\n" );
document.write( "\r\n" );
document.write( "To find the critical values set numerator=0,\r\n" );
document.write( "and denominator =0 and solve for x:\r\n" );
document.write( "\r\n" );
document.write( "(x-9)(x+2)=0 gives 9 and -2 as critical values\r\n" );
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document.write( "2(x+1)=0 gives -1 as a critical value.\r\n" );
document.write( "\r\n" );
document.write( "So mark the critical values on a number line:\r\n" );
document.write( "\r\n" );
document.write( "-------o--o-----------------------------o---------\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12 \r\n" );
document.write( "\r\n" );
document.write( "---\r\n" );
document.write( "First select a test point in the region left\r\n" );
document.write( "of -2.  Choose x=-3.  Substitute it in\r\n" );
document.write( "\r\n" );
document.write( "\"%28%28x-9%29%28x%2B2%29%29%2F%282%28x%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28%28-3%29-9%29%28%28-3%29%2B2%29%29%2F%282%28%28-3%29%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28-3-9%29%28-3%2B2%29%29%2F%282%28-3%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28-12%29%28-1%29%29%2F%282%28-2%29%29%3E=0\"\r\n" );
document.write( "\"%2812%29%2F%28-4%29%3E=0\"\r\n" );
document.write( "\"-3%3E=0\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "That is false, so we do not shade the region\r\n" );
document.write( "left of -2.  So we still have the same\r\n" );
document.write( "number line:\r\n" );
document.write( "\r\n" );
document.write( "-------o--o-----------------------------o---------\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12 \r\n" );
document.write( "\r\n" );
document.write( "Next select a test point in the region between\r\n" );
document.write( "-2 and -1.  Choose x=-1.5.  Substitute it in\r\n" );
document.write( "\r\n" );
document.write( "\"%28%28x-9%29%28x%2B2%29%29%2F%282%28x%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28%28-1.5%29-9%29%28%28-1.5%29%2B2%29%29%2F%282%28%28-1.5%29%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28-1.5-9%29%28-1.5%2B2%29%29%2F%282%28-1.5%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28-10.5%29%28.5%29%29%2F%282%28-.5%29%29%3E=0\"\r\n" );
document.write( "\"%28-5.25%29%2F%28-1%29%3E=0\"\r\n" );
document.write( "\"5.25%3E=0\"\r\n" );
document.write( "That is true, so we do shade the region between\r\n" );
document.write( "-2 and -1. So now the number line is\r\n" );
document.write( "\r\n" );
document.write( "-------o==o-----------------------------o---------\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12 \r\n" );
document.write( "\r\n" );
document.write( "Next select a test point in the region between\r\n" );
document.write( "-1 and 9.  Choose x=0.  Substitute it in\r\n" );
document.write( "\r\n" );
document.write( "\"%28%28x-9%29%28x%2B2%29%29%2F%282%28x%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28%280%29-9%29%28%280%29%2B2%29%29%2F%282%28%280%29%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28-9%29%282%29%29%2F%282%281%29%29%3E=0\"\r\n" );
document.write( "\"%28%28-18%29%282%29%29%2F%282%29%3E=0\"\r\n" );
document.write( "\"%28-36%29%2F%282%29%3E=0\"\r\n" );
document.write( "\"-18%3E=0\"\r\n" );
document.write( "\r\n" );
document.write( "That is false, so we do not shade the region\r\n" );
document.write( "between -1 and 9.  So we still have the same\r\n" );
document.write( "number line:\r\n" );
document.write( "\r\n" );
document.write( "-------o==o-----------------------------o---------\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12 \r\n" );
document.write( "\r\n" );
document.write( "Finally select a test point in the region right\r\n" );
document.write( "of 9.  Choose x=10.  Substitute it in\r\n" );
document.write( "\r\n" );
document.write( "\"%28%28x-9%29%28x%2B2%29%29%2F%282%28x%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28%2810%29-9%29%28%2810%29%2B2%29%29%2F%282%28%2810%29%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%281%29%2812%29%29%2F%282%2811%29%29%3E=0\"\r\n" );
document.write( "\"%28%2812%29%29%2F%2822%29%3E=0\"\r\n" );
document.write( "\"6%29%2F11%3E=0\"\r\n" );
document.write( "\r\n" );
document.write( "That is true, so we do shade the region\r\n" );
document.write( "right of 9.  So the number line is now:\r\n" );
document.write( "\r\n" );
document.write( "-------o==o-----------------------------o=========>\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12 \r\n" );
document.write( "\r\n" );
document.write( "Now we have to test the critical values\r\n" );
document.write( "to see if they are solutions. \r\n" );
document.write( "\r\n" );
document.write( "We test critical value -2\r\n" );
document.write( "\r\n" );
document.write( "\"%28%28x-9%29%28x%2B2%29%29%2F%282%28x%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28%28-2%29-9%29%28%28-2%29%2B2%29%29%2F%282%28%28-2%29%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28-2-9%29%28-2%2B2%29%29%2F%282%28-2%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28-11%29%280%29%29%2F%282%28-1%29%29%3E=0\"\r\n" );
document.write( "\"%280%29%2F%28-2%29%3E=0\"\r\n" );
document.write( "\"0%3E=0\"\r\n" );
document.write( "\r\n" );
document.write( "That is true so we darken the circle at -2, like this:\r\n" );
document.write( "\r\n" );
document.write( "-------@==o-----------------------------o=========>\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12\r\n" );
document.write( "\r\n" );
document.write( "We test critical value -1\r\n" );
document.write( "\r\n" );
document.write( "\"%28%28x-9%29%28x%2B2%29%29%2F%282%28x%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28%28-1%29-9%29%28%28-1%29%2B2%29%29%2F%282%28%28-1%29%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28-1-9%29%28-1%2B2%29%29%2F%282%28-1%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28-10%29%281%29%29%2F%282%280%29%29%3E=0\"\r\n" );
document.write( "\"%28-10%29%2F0%3E=0\"\r\n" );
document.write( "This undefined!  So we do not darken the circle at -1,\r\n" );
document.write( "but leave it open\r\n" );
document.write( "\r\n" );
document.write( "Now we test critical point x=9\r\n" );
document.write( "\r\n" );
document.write( "\"%28%28x-9%29%28x%2B2%29%29%2F%282%28x%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%28%289%29-9%29%28%289%29%2B2%29%29%2F%282%28%289%29%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%28%280%29%2811%29%29%2F%282%289%2B1%29%29%3E=0\"\r\n" );
document.write( "\"%280%29%2F%282%2810%29%29%3E=0\"\r\n" );
document.write( "\"0%2F20%3E=0\"\r\n" );
document.write( "\"0%3E=0\"\r\n" );
document.write( "\r\n" );
document.write( "That is true so we darken the circle at 9, like this:\r\n" );
document.write( "\r\n" );
document.write( "-------@==o-----------------------------@=========>\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12\r\n" );
document.write( "\r\n" );
document.write( "The interval notation is:\r\n" );
document.write( "\r\n" );
document.write( "     [-2,-1) U [9,\"infinity\")\r\n" );
document.write( "\r\n" );
document.write( "Edwin
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