graph the  system of inequalities (shade each half-plane solution,not only the overall x<3 y<2x-4 y>-4x  y<2/3x-6 this is one graph how tp graph it 



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I've included all plots necessary.

First attached image is the graph of the lines corresponding to the equalities (e.g. x=3), which serve as the basis for graphing the inequalities.

Second attached image shows the solution to all the inequalities. 

Third attached image is the graph showing the net result, when all inequalities are taken into consideration.


When plotting inequalities, the first thing to be done is to plot the equalities first. This serve as the guideline.

Next, you shade/color/draw lines to indicate the solution to each inequality. This can be done easily by choosing ANY point not on the line. Then, substitute the point to the INEQUALITY. If it gives a true result, the entire domain in which that point is located is the solution to the inequality, otherwise, you choose the other domain (there are only two domains as the line simply divides the cartesian space into two). For example, in the case of x < 3:

First plot x = 3(which is just a vertical line). Then choose any point. I choose (0, 0), for instance. This means x = 0, y = 0. Then I substitute the two values to my inequality. There is no y in the inequality so I just use x: 0 < 3. This statement is true, and hence the solution will be everything to the left of the vertical line.

In the case of y > -4x, Let's say I choose (2, 1) or x = -2, y = 1 (I can't choose (0, 0) in this case, because it is contained by the equality version). Substituting the point to the inequality gives: 1 >  -8 -- which is false! Hence, the entire domain is not included, and I instead shade the opposite side of the graph.

Doing so for the other two cases should give you the correct solution to the four inequalities. The area in which ALL four regions (solutions to each and every inequality) gives the solution to the problem (see attachment 3: result.png)

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