Please explain how to find these solutions: Write a system of linear inequalities that describes the shaded region. x + y ? ≥ ≤ > < 3 9x + 5y ? ≥ ≤ > < 90 3x + 5y ? ≥ ≤ > < 45 x ? ≥ ≤ > < 0 y ? ≥ ≤ > < 0

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Lets assume that you have a graph with a shaded region bounded by the line segments with endpoints (0,3),(0,9),(15/2,9/2),(10,0) and (3,0) (forming a pentagonal shaded region.)(See attachment.))

Using the endpoints to find the slope and then finding the equations of the lines that contain the segments we get:

x+y=3
9x+5y=90
3x+5y=45
x=0
y=0

We need to convert these equations into inequalities to form the given shaded region. First note that if the given line/line segment is dotted then the inequality is strict (either < or >) while if the line/line segment is solid the inequality is inclusive (`<=, >=` ).

To determine if an inequality is less than (or less than or equal to) we note that each such inequality divides the plane into two half-planes. On one side of the line are all solutions to the inequality while the other side has no solutions.

So consider the "line" x+y=3. The point (4,4) is in our shaded region and therefore is a solution to the inequality. Note that if x=4 and y=4 we get 4+4>3 so the inequality must be `x+y>3 "or" x+y >= 3` where the determining factor for whether the inequality is strict or inclusive depends on if the line x+y=3 is solid or dotted.

We continue for each equation:

9(4)+5(4)<90 so we want `9x+5y<90 " or " 9x+5y <= 90`

3(4)+5(4)<45 so we want `3x+5y<45 " or "3x+5y<=45 `

Since 4>0 we want `x>0 " or " x>=0` and `y>0 " or " y>=0`

If the shaded region is not the central pentagon but some other region we proceed as above. Find the equations of the bounding lines and write as inequalities so that a point in the shaded region is a solution to the inequality.

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