document.write( "Question 657547: How would I graph the system of inequalities y<2x-1
\n" ); document.write( " x+y>=2 \n" ); document.write( "

Algebra.Com's Answer #409988 by KMST(5328) $\"\"$ $\"About$

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Since there is an $\"x\"$ and a $\"y\"$ in your system, you need an x-axis and a y-axis in your graph.
\n" ); document.write( "Each inequality will be represented as the half of the x-y plane to one side of a boundary line.
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\n" ); document.write( "The equation $\"x%2By=2\"$ represents one of those boundary lines, and is graphed as a solid line, because all the points on that line are part of the solution to
\n" ); document.write( " $\"x%2By%3E=2\"$ . Any point (x,y) in that line satisfies $\"x%2By=2\"$ and also satisfies $\"x%2By%3E=2\"$ .
\n" ); document.write( " $\"graph%28300%2C300%2C-2%2C8%2C-2%2C8%2C2-x%29\"$ is the graph (line) for $\"x%2By=2\"$ , and the inequality $\"x%2By%3E=2\"$ graphs as $\"graph%28300%2C300%2C-2%2C8%2C-2%2C8%2C2-x%2Cx%2By%3E=2%29\"$
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\n" ); document.write( "The equation $\"y=2x-1\"$ represents the boundary line for $\"y%3C2x-1\"$ , and is graphed as a dashed line, because the points (x,y) with $\"y=2x-1\"$ do not satisfy $\"y%3C2x-1\"$
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\n" ); document.write( "To graph each line you need 2 points.
\n" ); document.write( "To get a point you chose a value for $\"x\"$ and find the corresponding $\"y\"$ or the other way around. A value of zero is often a good choice.
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\n" ); document.write( "FOR $\"x%2By=2\"$ :
\n" ); document.write( " $\"x=0\"$ gives you $\"0%2By=2\"$ --> $\"y=2\"$ for the point (0,2).
\n" ); document.write( " $\"y=0\"$ gives you $\"x%2B0=2\"$ --> $\"x=2\"$ for the point (2,0).
\n" ); document.write( "You plot the two points and connect them with a line to graph $\"x%2By=2\"$
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\n" ); document.write( "FOR $\"x%2By%3E=2\"$ :
\n" ); document.write( "You graph the line as shown above. It is a solid line because there is a \"=\" in the inequality, and that causes the solutions to $\"x%2By=2\"$ to be solutions of $\"x%2By%3E=2\"$ .
\n" ); document.write( "Next, you figure out which side of the line to shade or color.
\n" ); document.write( "An easy way to do it is to pick a point on one side of the line and check if it is part of the solution.
\n" ); document.write( "The easiest choice (if not on the line) is the point (0,0), with $\"x=y=0\"$
\n" ); document.write( "For that point $\"x%2By=0%2B0%3C2\"$ , so that point is not part of the solution.
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\n" ); document.write( "FOR $\"y%3C2x-1\"$ :
\n" ); document.write( "You plot the line for $\"y=2x-1\"$ as a dashed line, because it is not part of the solution.
\n" ); document.write( " $\"x=0\"$ gives you $\"y=2%2A0-1\"$ --> $\"y=-1\"$ for point (0,-1).
\n" ); document.write( "Choosing $\"y=0\"$ would not make the calculations or the graphing easy, so I pick $\"x=3\"$ for the second point.
\n" ); document.write( " $\"x=3\"$ gives me $\"y=2%2A3-1\"$ --> $\"y=5\"$ for point (3,5).
\n" ); document.write( "I plot the points and connect them with a dashed line:
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Finally, I decide which side of the line is the solution to $\"y%3C2x-1\"$ .
\n" ); document.write( "Using (0,0), the origin, as a test point, I figure that the solution is the side that does not contain the origin,
\n" ); document.write( "because substituting $\"x=y=0\"$ into $\"y%3C2x-1\"$ , I get
\n" ); document.write( " $\"0%3C2%2A0-1=-1\"$ , which is not true.
\n" ); document.write( "The colored graph of the inequality will look like this:
\n" ); document.write( " $\"graph%28300%2C300%2C-2%2C8%2C-2%2C8%2Cy%3C2x-1%29\"$
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\n" ); document.write( "The solution to the system is the part of the x-y plane that is a solution to both inequalities. It graphs as the colored or shaded pie wedge bounded by the lines, as shown below.
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