Answer: 3 < y-x < 6
The set of all possible y-x values is between 3 and 6, excluding both endpoints.
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Step-by-step explanation:
The y-x value is smallest when:
- The y value is as small as possible
- The x value is as large as possible
I recommend to draw out a number line. The idea is to place x and y as close as possible to each other.
Based on those two conditions, we use y = 10 and x = 7 respectively. That gives y-x = 10-7 = 3 as the smallest possible difference. Keep in mind we can't actually reach this value so we would write y-x > 3 which is the same as 3 < y-x.
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Similarly, y-x is largest when:
- The y value is largest
- The x value is smallest
Like earlier, use a number line to see why this would be the case. We're trying to space x and y out as far as possible.
The largest y value and smallest x value are y = 12 and x = 6 in that order. So the upper bound is y-x = 12-6 = 6. Like before, we can't actually reach this number which means we write y-x < 6.
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Recap:
- In the 1st section, we found that 3 < y-x
- In the 2nd section, we found that y-x < 6
Put those two results together in a tidy compound inequality to get the final answer 3 < y-x < 6
It says: The expression y-x is between 3 and 6, excluding both endpoints.