document.write( "Question 192510: I should express (x+8)/(x-1)>=0 in interval notation but I always get confused with domain and range.is it (1,infinity)? \n" ); document.write( "

Algebra.Com's Answer #145679 by Edwin McCravy(20086) $\"\"$ $\"About$

You can put this solution on YOUR website!
I should express the solution to
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\n" ); document.write( "\n" ); document.write( "in interval notation but I always
\n" ); document.write( "get confused with domain and range.
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document.write( "No, domain and range finding techniques\r\n" );
document.write( "aren't involved here.\r\n" );
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document.write( "Since 0 is on the right side,\r\n" );
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document.write( "We begin by finding all critical values, by\r\n" );
document.write( "setting the numerator = 0 and solving, then\r\n" );
document.write( "setting the denominator = 0.\r\n" );
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document.write( "Setting the numerator = 0, x+8=0 gives x=-8,\r\n" );
document.write( "so -8 is one critical value.\r\n" );
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document.write( "Setting the denominator = 0, x-1=0 gives x=1,\r\n" );
document.write( "so 1 is the other critical value.\r\n" );
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document.write( "Plot these on a number line.  First make all\r\n" );
document.write( "the circles open, we may have to close some of\r\n" );
document.write( "them, but for now just use open circles:\r\n" );
document.write( "  \r\n" );
document.write( "-------------------o-----------------------------------o-------------\r\n" );
document.write( "-12  -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4 \r\n" );
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document.write( "Now we need to find out what part of the line to shade.\r\n" );
document.write( "First we pick any test value in the region to the left of -8,\r\n" );
document.write( "say -9, and substitute it into the inequality\r\n" );
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document.write( "That is true, so we shade the number line left of -8\r\n" );
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document.write( "<==================o-----------------------------------o-------------\r\n" );
document.write( "-12  -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4\r\n" );
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document.write( "Next we pick any test value in the between -8 and 1,\r\n" );
document.write( "say 0, and substitute it into the inequality\r\n" );
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document.write( "That is false, so we do not shade the number line between\r\n" );
document.write( "-8 and 1\r\n" );
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document.write( "Thirdly we pick any test value in the region to the right of 1,\r\n" );
document.write( "say 2, and substitute it into the inequality\r\n" );
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document.write( "That is true, so we shade the number line right of 1\r\n" );
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document.write( "<==================o-----------------------------------o============>\r\n" );
document.write( "-12  -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4\r\n" );
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document.write( "Now we must test the critical values to see if we can\r\n" );
document.write( "shade them or not.\r\n" );
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document.write( "Test critical value -8 by substituting it\r\n" );
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document.write( "This is true, so we darken the circle at -8\r\n" );
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document.write( "<==================@-----------------------------------o============>\r\n" );
document.write( "-12  -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4\r\n" );
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document.write( "Test critical value 1 by substituting it\r\n" );
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document.write( "Division by zero is always undefined. So we must leave\r\n" );
document.write( "the circle at 1 open, so we still have\r\n" );
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document.write( "<==================@-----------------------------------o============>\r\n" );
document.write( "-12  -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4\r\n" );
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document.write( "Now we want to get the interval notation.  On the far left of the \r\n" );
document.write( "shading we have negative infinity, -oo and the shading stops at\r\n" );
document.write( "-8, so we write the left shaded part as\r\n" );
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document.write( "(-oo,-8]\r\n" );
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document.write( "We use a \"(\" at -oo because infinity is never included.  We use a ] at \r\n" );
document.write( "at -8, because -8 is included and has a darkened circle.\r\n" );
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document.write( "Now we put a \"union\" symbol \"U\"\r\n" );
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document.write( "(-oo,-8] U\r\n" );
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document.write( "Then the shaded part on the right going\r\n" );
document.write( "left to right is from 1 to infinity, or (1,oo)\r\n" );
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document.write( "We use \"(\" at 1 because 1 is not included, so the\r\n" );
document.write( "final answer is:\r\n" );
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document.write( "(-oo,-8] U (1,oo)\r\n" );
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document.write( "Edwin

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