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Question 172799: Find the solution of the system of inequalities

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Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Graphing Inequality #1
Graphing Inequality #2
Graphing Inequality #3
Graphing all three inequalities together
Solution






Inequality #1


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2y%2Bx%3E=-2 Start with the 1st inequality.


2y%3E=-2-x Subtract x from both sides.


y%3E=%28-2-x%29%2F%282%29 Divide both sides by 2 to isolate y.


y%3E=-%282%29%2F%282%29-%28x%29%2F%282%29 Break up the fraction.


y%3E=-1-%281%2F2%29x Reduce


y%3E=-%281%2F2%29x-1 Rearrange the terms.


So in order to graph y%3E=-%281%2F2%29x-1, we need to graph the equation y=-%281%2F2%29x-1


Graph y=-%281%2F2%29x-1


Now since the inequality sign of y%3E=-%281%2F2%29x-1 is a "greater-than or equal to" sign, this tells us to shade the region above the graph (since points in that region have larger y values)


So the shading looks like this:


Graph of y%3E=-%281%2F2%29x-1 with the shaded region in green


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Inequality #2


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y%2B2x%3C4 Start with the 2nd inequality.


y%3C4-2x Subtract 2x from both sides.


y%3C-2x%2B4 Rearrange the terms.


So in order to graph y%3C-2x%2B4, we need to graph the equation y=-2x%2B4


Graph y=-2x%2B4


Now since the inequality sign of y%3C-2x%2B4 is a "less-than" sign, this tells us to shade the region below the graph (since points in that region have smaller y values)


So the shading looks like this:


Graph of y%3C-2x%2B4 with the shaded region in green


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Inequality #3


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Finally, to graph y%3E3, we need to graph y=3 (which is a horizontal line at y=3) like this

Graph y=3



Now shade the region above the line (since the sign is a "greater-than" sign) to get


Graph of y%3E3 with the shaded region in green

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Graphing the inequalities together on the same coordinate systems


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Now simply graph the inequalities together. To do this, simply graph each line together and their corresponding region to get (note: use different colors to represent the shaded regions)







Here is a cleaner look at the intersection of regions




Here is the intersection of the 3 regions represented by the series of dots



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Solution:

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So the solution of the system of inequalities is