Question 116556Start with the given system of inequalities {{{x-2y<4}}} {{{x>1}}} In order to graph this system of inequalities, we need to graph each inequality one at a time. First lets graph the first inequality {{{x-2y<4}}} In order to graph {{{x-2y<4}}}, we need to graph the <b>equation</b> {{{x-2y=4}}} (just replace the inequality sign with an equal sign). So lets graph the line {{{x-2y=4}}} (note: if you need help with graphing, check out this <a href=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>) {{{ graph( 500, 500, -20, 20, -20, 20, (1/2)x-2) }}} graph of {{{x-2y=4}}} Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality {{{x-2y<4}}} with the test point Substitute (0,0) into the inequality {{{(0)-2(0)<4}}} Plug in {{{x=0}}} and {{{y=0}}} {{{0<4}}} Simplify (note: for some reason, some of the following images do not display correctly in Internet Explorer. So I recommend the use of <a href=http://www.mozilla.com/en-US/firefox/>Firefox</a> to see these images.) Since this inequality is true, we simply shade the entire region that contains (0,0) {{{drawing( 500, 500, -20, 20, -20, 20, graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+3), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+6), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+9), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+12), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+15), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+18), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+21), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+24), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+27), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+30), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+33), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+36), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+39), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+42), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+45), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+48))}}} Graph of {{{x-2y<4}}} with the boundary (which is the line {{{x-2y=4}}} in red) and the shaded region (in green) (note: since the inequality contains a less-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line) --------------------------------------------------------------- Now lets graph the second inequality {{{x>1}}} In order to graph {{{x>1}}}, we need to graph the <b>equation</b> {{{x=1}}} (just replace the inequality sign with an equal sign). So lets graph the line {{{x=1}}} (simply draw a vertical line through {{{x=1}}}) {{{ graph( 500, 500, -20, 20, -20, 20, 1000(x-1)) }}} graph of {{{x=1}}} Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality {{{x>1}}} with the test point Substitute (0,0) into the inequality {{{(0)>1}}} Plug in {{{x=0}}} and {{{y=0}}} {{{0>1}}} Simplify Since this inequality is <font size=4><b>not</b></font> true, we do <font size=4><b>not</b></font> shade the entire region that contains (0,0). So this means we shade the region that is on the opposite side of the line {{{drawing( 500, 500, -20, 20, -20, 20, graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-2)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-4)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-6)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-8)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-10)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-12)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-14)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-16)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-18)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-20)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-22)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-24)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-26)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-28)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-30)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-32)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-34)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-36)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-38)))}}} Graph of {{{x>1}}} with the boundary (which is the line {{{x=1}}} in red) and the shaded region (in green) (note: since the inequality contains a greater-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line) --------------------------------------------------------------- So we essentially have these 2 regions: Region #1 {{{drawing( 500, 500, -20, 20, -20, 20, graph( 500, 500, -20, 20, -20, 20,(1/2)x-2), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+3), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+6), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+9), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+12), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+15), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+18), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+21), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+24), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+27), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+30), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+33), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+36), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+39), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+42), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+45), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+48))}}} Graph of {{{x-2y<4}}} Region #2 {{{drawing( 500, 500, -20, 20, -20, 20, graph( 500, 500, -20, 20, -20, 20,1000(x-1)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-2)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-4)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-6)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-8)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-10)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-12)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-14)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-16)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-18)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-20)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-22)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-24)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-26)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-28)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-30)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-32)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-34)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-36)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-38)))}}} Graph of {{{x>1}}} When these inequalities are graphed on the same coordinate system, the regions overlap to produce this region. It's a little hard to see, but after evenly shading each region, the intersecting region will be the most shaded in. {{{drawing( 500, 500, -20, 20, -20, 20, graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+3), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+6), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+9), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+12), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+15), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+18), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+21), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+24), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+27), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+30), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+33), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+36), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+39), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+42), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+45), graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,(1/2)x-2+48), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-2)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-4)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-6)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-8)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-10)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-12)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-14)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-16)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-18)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-20)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-22)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-24)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-26)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-28)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-30)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-32)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-34)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-36)), graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-38)))}}} Here is a cleaner look at the intersection of regions {{{drawing( 500, 500, -20, 20, -20, 20, graph( 500, 500, -20, 20, -20, 20,(1/2)x-2,1000(x-1)),circle(4,1,0.2), circle(4,4,0.2), circle(4,7,0.2), circle(4,10,0.2), circle(4,13,0.2), circle(4,16,0.2), circle(4,19,0.2), circle(7,4,0.2), circle(7,7,0.2), circle(7,10,0.2), circle(7,13,0.2), circle(7,16,0.2), circle(7,19,0.2), circle(10,4,0.2), circle(10,7,0.2), circle(10,10,0.2), circle(10,13,0.2), circle(10,16,0.2), circle(10,19,0.2), circle(13,7,0.2), circle(13,10,0.2), circle(13,13,0.2), circle(13,16,0.2), circle(13,19,0.2), circle(16,7,0.2), circle(16,10,0.2), circle(16,13,0.2), circle(16,16,0.2), circle(16,19,0.2), circle(19,10,0.2), circle(19,13,0.2), circle(19,16,0.2), circle(19,19,0.2), circle(4,1,0.2), circle(4,4,0.2), circle(4,7,0.2), circle(4,10,0.2), circle(4,13,0.2), circle(4,16,0.2), circle(4,19,0.2), circle(7,4,0.2), circle(7,7,0.2), circle(7,10,0.2), circle(7,13,0.2), circle(7,16,0.2), circle(7,19,0.2), circle(10,4,0.2), circle(10,7,0.2), circle(10,10,0.2), circle(10,13,0.2), circle(10,16,0.2), circle(10,19,0.2), circle(13,7,0.2), circle(13,10,0.2), circle(13,13,0.2), circle(13,16,0.2), circle(13,19,0.2), circle(16,7,0.2), circle(16,10,0.2), circle(16,13,0.2), circle(16,16,0.2), circle(16,19,0.2), circle(19,10,0.2), circle(19,13,0.2), circle(19,16,0.2), circle(19,19,0.2))}}} Here is the intersection of the 2 regions represented by the series of dots