Given the system of inequalities below...Help?

iven the system of inequalities below, determine the shape of the feasible region and find the vertices of the feasible region. Report your vertices starting with the one which has the smallest x-value. If more than one vertex has the same, smallest x-value, start with the one that has the smallest y-value. Proceed clockwise from the first vertex. Leave any unnecessary answer spaces blank.

x+y <(or equal to) 7

2x+y >(or equal to) 8

x >(or equal to) 0

y >(or equal to) 0

The feasible region is (a) _______________

The first vertex is ( ____ , ____ )

The second vertex is ( ____ , ____ )

The third vertex is ( ____ , ____ )

The fourth vertex is ( ____ , ____ )

1 Answer

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  • 8 years ago
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    Your use the equations to piece together the shape, I think?

    You'll probably want to graph them to see the shape visually.

    First, I'd solve for y on the first two.

    y <(or equal to) - x +7

    y >(or equal to) -2x + 8

    Now graph these two as if they were equations and note the defined region (the first is defined for the region below its line, the second is defined for the region above its line).

    Also graph the straight lines

    x >(or equal to) 0

    y >(or equal to) 0

    Together, these two lines are saying the defined region is the first quadrant.

    (Because x>0 is defined for all positive x values, and y>0 is defined for all positive y values.)

    Basically, if you've graphed all these, it looks like you have a triangle in the region that's defined. The vertices will be the x-intercept of both the lines, because the "x >(or equal to) 0" line is the one that forms the bottom of the triangle. So, solve for x=0 in the first two inequalities.

    The other vertex will be where the lines formed from first two inequalities intercept each other.

    Here it will be useful to treat them as equations.

    y = - x +7

    y = -2x + 8

    -2y = 2x - 14

    y = -2x + 8

    ---

    -y = -6

    y= 6

    6=-x+7

    -1=-x

    x=1

    (1,6)

    And... I think you will only have three vertices. They probably gave you four blanks so you wouldn't just assume it was a triangle?

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