1. I need help in solving the following compound\r\n" );
document.write( "inequality.\r\n" );
document.write( "I also need to write solution set in interval notation \r\n" );
document.write( "and graph. \r\n" );
document.write( "\r\n" );
document.write( " -1 < 3 - 2x < 11\r\n" );
document.write( "\r\n" );
document.write( "The idea is to isolate x in the middle.\r\n" );
document.write( "Get rid of the 3 by adding -3 to all three sides:\r\n" );
document.write( "\r\n" );
document.write( " -1 < 3 - 2x < 11\r\n" );
document.write( " -3 -3 -3\r\n" );
document.write( "-------------------\r\n" );
document.write( " -4 < -2x < 8\r\n" );
document.write( "\r\n" );
document.write( "So we have\r\n" );
document.write( " \r\n" );
document.write( " -4 < -2x < 8\r\n" );
document.write( "\r\n" );
document.write( "Now we divide all three siade by the coefficient of x,\r\n" );
document.write( "which is -2. But since we are dividing by a negative\r\n" );
document.write( "number that reverses the inequalities:\r\n" );
document.write( "\r\n" );
document.write( " -4 -2x 8\r\n" );
document.write( " ———— > ————— > ————\r\n" );
document.write( " -2 -2 -2\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( " 2 > x > -4\r\n" );
document.write( "\r\n" );
document.write( "Which is the same as\r\n" );
document.write( "\r\n" );
document.write( " -4 < x < 2\r\n" );
document.write( "\r\n" );
document.write( "On a number line this is graphed as\r\n" );
document.write( "\r\n" );
document.write( "--------o=================·-------\r\n" );
document.write( " -6 -5 -4 -3 -2 -1 0 1 2 3 4 \r\n" );
document.write( "\r\n" );
document.write( "It is an open circle at -4, because\r\n" );
document.write( "it is < and not <, and closed (dark\r\n" );
document.write( "circle) at 2, because it is < and not <.\r\n" );
document.write( "\r\n" );
document.write( "The interval notation is a shorthand\r\n" );
document.write( "for this graph:\r\n" );
document.write( "\r\n" );
document.write( " (-4, 2]\r\n" );
document.write( "\r\n" );
document.write( "The \"(\" on the left represents the\r\n" );
document.write( "open circle, the -4 is the left endpoint\r\n" );
document.write( "of the shaded part, the 2 is the right endpoint\r\n" );
document.write( "of the shaded part, and the \"]\" represents\r\n" );
document.write( "the closed circle.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@\r\n" );
document.write( "\r\n" );
document.write( "2. I need to graph the following absolute value \r\n" );
document.write( "inequality:\r\n" );
document.write( "\r\n" );
document.write( "|x+2y|(greater than or equal to) 6\r\n" );
document.write( "\r\n" );
document.write( "First we draw the boundary graph, which is found by\r\n" );
document.write( "replacing the > sign by =\r\n" );
document.write( "\r\n" );
document.write( "|x + 2y| = 6 \r\n" );
document.write( "\r\n" );
document.write( "x + 2y = 6 or x + 2y = -6\r\n" );
document.write( "\r\n" );
document.write( "So we graph both these lines, which are parallel:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " 
\r\n" );
document.write( "\r\n" );
document.write( "This gives us three regions, the region above the red \r\n" );
document.write( "line, the region between the two lines, and the region \r\n" );
document.write( "below the green line. We select an arbitrary test point \r\n" );
document.write( "in each of the three regions to see which we shade as \r\n" );
document.write( "the solution set, and which we don't.\r\n" );
document.write( "\r\n" );
document.write( "I arbitrarily select the point (0,5) as a test point \r\n" );
document.write( "for the region above the red line and substitute it into \r\n" );
document.write( "the original inequality:\r\n" );
document.write( "\r\n" );
document.write( " |x + 2y| > 6\r\n" );
document.write( " |0 + 2(5)| > 6\r\n" );
document.write( " 10 > 6\r\n" );
document.write( "\r\n" );
document.write( "This is true so we shade the part above the red line.\r\n" );
document.write( "\r\n" );
document.write( "I arbitrarily select the point (0,0) as a test point for \r\n" );
document.write( "the region between the two lines and substitute it into \r\n" );
document.write( "the original inequality:\r\n" );
document.write( "\r\n" );
document.write( " |x + 2y| > 6\r\n" );
document.write( " |0 + 2(0)| > 6\r\n" );
document.write( " 0 > 6\r\n" );
document.write( "\r\n" );
document.write( "This is false so we do not shade the region between the \r\n" );
document.write( "lines:\r\n" );
document.write( "\r\n" );
document.write( "I arbitrarily select the point (0,-5) as a test point \r\n" );
document.write( "for the region below the green line and substitute it \r\n" );
document.write( "into the original inequality:\r\n" );
document.write( "\r\n" );
document.write( " |x + 2y| > 6\r\n" );
document.write( " |0 + 2(-5)| > 6\r\n" );
document.write( " 10 > 6\r\n" );
document.write( "\r\n" );
document.write( "This is true so we shade the part below the green line.\r\n" );
document.write( "\r\n" );
document.write( "Now I can't shade on here but you can on your paper. \r\n" );
document.write( "Shade the big area above the red line, and the big area \r\n" );
document.write( "below the green line. But do not shade the region \r\n" );
document.write( "between the two lines. Also draw both lines solid \r\n" );
document.write( "because the inequality symbol is > and not >.\r\n" );
document.write( "If it had been > instead you would draw the line\r\n" );
document.write( "dotted.\r\n" );
document.write( "\r\n" );
document.write( "@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@\r\n" );
document.write( "\r\n" );
document.write( "3. I need to graph the following compound inequality:\r\n" );
document.write( " 3-xy+5 \r\n" );
document.write( "\r\n" );
document.write( "Sorry, you mistyped that. There's no >, <, > or <\r\n" );
document.write( "\r\n" );
document.write( "@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@\r\n" );
document.write( "\r\n" );
document.write( "4. I need to determine whether -4 satisfies the \r\n" );
document.write( "following compound equality: \r\n" );
document.write( "\r\n" );
document.write( " -3x > 0 AND 3x - 4 < 11\r\n" );
document.write( "\r\n" );
document.write( "Plug it in and see if it's true or false:\r\n" );
document.write( "\r\n" );
document.write( " -3x > 0 AND 3x - 4 < 11\r\n" );
document.write( "\r\n" );
document.write( "-3(-4) > 0 AND 3(-4) - 4 < 11\r\n" );
document.write( "\r\n" );
document.write( " 4 > 0 AND -12 - 4 < 11\r\n" );
document.write( "\r\n" );
document.write( " 4 > 0 AND -16 < 11\r\n" );
document.write( "\r\n" );
document.write( "They are BOTH true so yes, it satisfies the\r\n" );
document.write( "compound inequality. If either had been false\r\n" );
document.write( "then it would not have satisfied it. If it had\r\n" );
document.write( "been OR instead of AND only one side would have\r\n" );
document.write( "needed to be true for it to have been a solution.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
