please help me im stuck .................

Answer: Choice A) Empty set

This is another way of saying "no solutions"

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Explanation:

Your teacher made a typo. The given inequalities will have boundary lines that both have positive slopes, not negative slopes. So the boundary lines should go uphill as you move from left to right.

See the diagram below to see what the graph should look like.

The upper dashed line is y = (1/2)x+3. The lower dashed line is y = (1/2)x-2

The upper dashed line represents the boundary for y > (1/2)x+3. Region 1 is the set of solution points for y > (1/2)x+3

While the solution set for y < (1/2)x-2 is represented with region 3

There is no overlap between regions 1 and 3 due to the parallel boundary lines.

Put another way: there are no (x,y) points that are in both region 1 and region 3 at the same time.

So that's why there are no solutions to this system of inequalities.

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Alternative method:

Convert the first inequality into Ax+By > C form

y > (1/2)x+3

2y > x + 6 .... multiply both sides by 2

-x+2y > 6 ... subtract x from both sides

z > 6 .... let z = -x+2y

Convert the second inequality into Ax+By < C form

y < (1/2)x-2

2y < x - 4

-x+2y < -4

z < -4 .... let z = -x+2y

We have some number z that is both z > 6 and z < -4 at the same time, but no such number exists. So this is another way to see we have an inconsistent system with no solutions.

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