document.write( "Question 710958: Hi I am trying to understand how to graph y>3x. I am trying to graph inequalities in two variables and was sick from school today and only need help in 2 or 3 problems so I can do the rest on my own. If you could help me it would be greatly appreciated and thank you very much. \n" ); document.write( "
Algebra.Com's Answer #437252 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
Hi I am trying to understand how to graph y > 3x. I am trying to graph inequalities in two variables and was sick from school today and only need help in 2 or 3 problems so I can do the rest on my own. If you could help me it would be greatly appreciated and thank you very much.
\n" ); document.write( "
\r\n" );
document.write( "This is a shaded region graph.\r\n" );
document.write( "\r\n" );
document.write( "Start out by drawing the graph of the boundary line, which is\r\n" );
document.write( "just like the inequality y > 3x except that you replace the \r\n" );
document.write( "inequality symbol > by an equal sign =.\r\n" );
document.write( "\r\n" );
document.write( "That is we draw the line whose equation is \r\n" );
document.write( "\r\n" );
document.write( "    y = 3x  Which contains the points (-1,3), (0,0), and (1,3) \r\n" );
document.write( "\r\n" );
document.write( "Since the inequality symbol is > and not ≥, the graph will \r\n" );
document.write( "not include its boundary line so we draw the graph of the boundary \r\n" );
document.write( "line as a dotted line like this:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Since the graph of the line y = 3x is the boundary of the graph \r\n" );
document.write( "of y > 3x, all the solutions are points on one side of the line.\r\n" );
document.write( "\r\n" );
document.write( "So we arbitrarily choose a test point, which can be on either side \r\n" );
document.write( "of the line, but not on the line. Let's arbitrarily choose the test\r\n" );
document.write( "point (6,5).\r\n" );
document.write( "\r\n" );
document.write( "Now if (6,5) is a solution to y > 3x, then all the other points on the\r\n" );
document.write( "same side of the line as (6,5) will also be solutions.  So we will\r\n" );
document.write( "shade that side of the line.  On the other hand, if it (6,5) is not\r\n" );
document.write( "a solution to y > 3x, then none of the other points on that side of\r\n" );
document.write( "the line will be solutions, either, and so we will then know that\r\n" );
document.write( "all the solutions are on the other side.  So we would shade the other\r\n" );
document.write( "side from the side the test point is on.\r\n" );
document.write( "\r\n" );
document.write( "So let's test the arbitrarily chosen point (6,5), marked on the graph below:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We test it by substituting x=6 and y=5 into the inequality:\r\n" );
document.write( "\r\n" );
document.write( "y > 3x\r\n" );
document.write( "5 > 3(6)\r\n" );
document.write( "5 > 18\r\n" );
document.write( "\r\n" );
document.write( "That is false so we know that none of the points on that\r\n" );
document.write( "side of the dotted line are solutions, so we know that all\r\n" );
document.write( "the solutions are on the other side than the side (6,5) is\r\n" );
document.write( "on.\r\n" );
document.write( "\r\n" );
document.write( "Therefore we shade the side of the line that the test point \r\n" );
document.write( "(6,5) was NOT on, which is the upper left region, so we\r\n" );
document.write( "shade it like this green region:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Notice that if we had arbitrarily picked a test point on the other\r\n" );
document.write( "side of the line say (-4,3),\r\n" );
document.write( "\r\n" );
document.write( "We would have tested it by substituting x=-4 and y=3 into the inequality:\r\n" );
document.write( "\r\n" );
document.write( "y > 3x\r\n" );
document.write( "3 > 3(-4)\r\n" );
document.write( "3 > -12\r\n" );
document.write( "\r\n" );
document.write( "and that is true, so we would have known that all the solutions\r\n" );
document.write( "are on the same side as that test point.  So we would have ended\r\n" );
document.write( "up with the same graph, and shaded the same side of the line.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );