Answer: Choice C. (3,-2)
A diagram is shown below.
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Explanation:
There are two approaches we can take. Since we're given a list of choices, we can plug each of the coordinates one at a time into the inequality. If we get a true statement, then it is a solution.
For choice A, we have x = 2 and y = 3 paired up together. Plugging them into the given inequality leads to...
y < (-1/2)*x + 2
3 < (-1/2)*2 + 2
3 < -1 + 2
3 < 1
This is false because 3 is not smaller than 1. So (2,3) is not a solution. You should find that choice B leads to a similar situation because
y < (-1/2)*x + 2
1 < (-1/2)*2 + 2
1 < -1+2
1 < 1
and 1 cannot be smaller than itself. So (2,1) isn't a solution either.
However, choice C is the answer since,
y < (-1/2)*x + 2
-2 < (-1/2)*3 + 2
-2 < -1.5 + 2
-2 < 0.5
This is true because -2 is to the left of 0.5 on the number line. So that's why (3,-2) is a solution.
For choice D, you should find that (-1,3) doesn't make the inequality true.
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An alternative approach:
As a visual way to solve this problem, we can graph the given inequality by following these steps
- Draw a dashed line through the points (0,2) and (4,0). This forms the graph of y = (-1/2)x + 2
- Shade below the dashed line
We make the boundary line dashed because points on the boundary do not count as solution points. This is due to the lack of "or equal to" in the inequality sign. We shade below the boundary line because of the "less than".
As the diagram below indicates, the blue region consists of every (x,y) solution point that makes y < (-1/2)*x + 2 true.
The diagram also shows that only point C is in that blue region, making it a solution point. Point B on the boundary does not count as a solution. You can think of the dashed line as an electric fence you can only get closer to, but can't touch, in terms of determining a solution point.
So effectively, this is a quick and easy way to find solutions if you have a graphing calculator or online graphing tools (such as Desmos or GeoGebra).
