\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "Get 0 on the right.\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "In order for
to be a real number we require\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

which is the same as \r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "So 5 is a critical number, since no value of x\r\n" );
document.write( "can be greater than 5\r\n" );
document.write( "\r\n" );
document.write( "Next we look for critical numbers of\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "By solving 

\r\n" );
document.write( " \r\n" );
document.write( " 

\r\n" );
document.write( "\r\n" );
document.write( " 

\r\n" );
document.write( "\r\n" );
document.write( " 

\r\n" );
document.write( " \r\n" );
document.write( " 

\r\n" );
document.write( "\r\n" );
document.write( " 

\r\n" );
document.write( "\r\n" );
document.write( " x+4=0; x-1=0\r\n" );
document.write( "\r\n" );
document.write( " x=-4; x=1\r\n" );
document.write( "\r\n" );
document.write( "So there are two more critical numbers\r\n" );
document.write( "\r\n" );
document.write( "We plot the three critical numbers on a number line:\r\n" );
document.write( "\r\n" );
document.write( "----------o--------------o-----------o\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5\r\n" );
document.write( "\r\n" );
document.write( "We test a value in each of the three regions\r\n" );
document.write( "\r\n" );
document.write( "For the leftmost region, we test x=-5 in \r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "The left side is a positive number, so that is false.\r\n" );
document.write( "\r\n" );
document.write( "So the left region is not part of the solution:\r\n" );
document.write( "\r\n" );
document.write( "----------o--------------o-----------o\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5\r\n" );
document.write( "\r\n" );
document.write( "For the middle region, we test x=0 in \r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "The left side is a positive number, so that is false.\r\n" );
document.write( "\r\n" );
document.write( "So the middle region is also not part of the solution:\r\n" );
document.write( "\r\n" );
document.write( "----------o--------------o-----------o\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5\r\n" );
document.write( "\r\n" );
document.write( "For the righmost region, we test x=2 in \r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "The left side is a negative number, so that is true.\r\n" );
document.write( "\r\n" );
document.write( "So the rightmost region is part of the solution,\r\n" );
document.write( "so we shade that interval:\r\n" );
document.write( "\r\n" );
document.write( "----------o--------------o===========o\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5\r\n" );
document.write( "\r\n" );
document.write( "Finally we test the critical numbers themselves to\r\n" );
document.write( "see if they are solutions;\r\n" );
document.write( "\r\n" );
document.write( "We test critical number -4 in \r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( " \r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "That's false so critical number -4 is not a solution.\r\n" );
document.write( "\r\n" );
document.write( "So we erase the open circle at -4:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "-------------------------o===========o\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We test critical number 1 in \r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( " \r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "That's false so critical number 1 is not a solution.\r\n" );
document.write( "\r\n" );
document.write( "-------------------------o===========o\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5\r\n" );
document.write( "\r\n" );
document.write( "We test critical number 5 in \r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( " \r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "That's true so 5 is a solution.\r\n" );
document.write( "\r\n" );
document.write( "So we darken it:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "-------------------------o===========●\r\n" );
document.write( "-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5\r\n" );
document.write( "\r\n" );
document.write( "So the solution is \r\n" );
document.write( "\r\n" );
document.write( "1 < x ≤ 5\r\n" );
document.write( "\r\n" );
document.write( "That can be written in interval notation as (1,5]\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "