SOLUTION: Solve each of a linear inequalities graphically. x+2y<4 x-y>1

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Question 107771: Solve each of a linear inequalities graphically.
x+2y<4
x-y>1
Found 2 solutions by MathLover1, jim_thompson5910:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
1.
x%2B2%3C4...move 2 to the right
x+%3C+4-2
x+%3C+2}
here is the graph, and you need to draw dashed line because solution is every x from -infinity to 2, excluding 2.
(-infinity, 2]

Solved by pluggable solver: Graphing Linear Equations

1%2Ax+=+2


x+=+2%2F%281%29


x+=+2


This is the vertical line that cuts through 2 on the x-axis and the graph is shown below





The slope of this line is undefined. All vertical lines have an undefined slope since rise/run = x/0 and division by zero is undefined.



2.

x-y%3E1

graph:
thisi is a function x-1=y, and you need to shade the area where y%3Cx-1.
That will be area from (-infinity,1]
Solved by pluggable solver: Graphing Linear Equations


1%2Ax-1%2Ay=1Start with the given equation



-1%2Ay=1-1%2Ax Subtract 1%2Ax from both sides

y=%28-1%29%281-1%2Ax%29 Multiply both sides by -1

y=%28-1%29%281%29%2B%281%29%281%29x%29 Distribute -1

y=-1%2B%281%29x Multiply

y=1%2Ax-1 Rearrange the terms

y=1%2Ax-1 Reduce any fractions

So the equation is now in slope-intercept form (y=mx%2Bb) where m=1 (the slope) and b=-1 (the y-intercept)

So to graph this equation lets plug in some points

Plug in x=-8

y=1%2A%28-8%29-1

y=-8-1 Multiply

y=-9 Add

So here's one point (-8,-9)





Now lets find another point

Plug in x=-7

y=1%2A%28-7%29-1

y=-7-1 Multiply

y=-8 Add

So here's another point (-7,-8). Add this to our graph





Now draw a line through these points

So this is the graph of y=1%2Ax-1 through the points (-8,-9) and (-7,-8)


So from the graph we can see that the slope is 1%2F1 (which tells us that in order to go from point to point we have to start at one point and go up 1 units and to the right 1 units to get to the next point) the y-intercept is (0,-1)and the x-intercept is (1,0) . So all of this information verifies our graph.


We could graph this equation another way. Since b=-1 this tells us that the y-intercept (the point where the graph intersects with the y-axis) is (0,-1).


So we have one point (0,-1)






Now since the slope is 1%2F1, this means that in order to go from point to point we can use the slope to do so. So starting at (0,-1), we can go up 1 units


and to the right 1 units to get to our next point



Now draw a line through those points to graph y=1%2Ax-1


So this is the graph of y=1%2Ax-1 through the points (0,-1) and (1,0)

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

Start with the given system of inequalities
x%2B2y%3C4
x-y%3E1

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%2B2y%3C4
In order to graph x%2B2y%3C4, we need to graph the equation x%2B2y=4 (just replace the inequality sign with an equal sign).
So lets graph the line x%2B2y=4 (note: if you need help with graphing, check out this solver)
+graph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C+-%281%2F2%29x%2B2%29+ graph of x%2B2y=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%2B2y%3C4 with the test point

Substitute (0,0) into the inequality
%280%29%2B2%280%29%3C4 Plug in x=0 and y=0
0%3C4 Simplify



(note: for some reason, some of the following images do not display correctly in Internet Explorer. So I would recommend the use of Firefox to see these images.)


Since this inequality is true, we simply shade the entire region that contains (0,0)
Graph of x%2B2y%3C4 with the boundary (which is the line x%2B2y=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-y%3E1
In order to graph x-y%3E1, we need to graph the equation x-y=1 (just replace the inequality sign with an equal sign).
So lets graph the line x-y=1 (note: if you need help with graphing, check out this solver)
+graph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C+x-1%29+ graph of x-y=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-y%3E1 with the test point

Substitute (0,0) into the inequality
%280%29-%280%29%3E1 Plug in x=0 and y=0
0%3E1 Simplify



Since this inequality is not true, we do not shade the entire region that contains (0,0). So this means we shade the region that is on the opposite side of the line
Graph of x-y%3E1 with the boundary (which is the line x-y=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
Graph of x%2B2y%3C4


Region #2
Graph of x-y%3E1




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.







Here is a cleaner look at the intersection of regions




Here is the intersection of the 2 regions represented by the series of dots