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consider two condition x^2-3x-10<0 and |x-2|<α on a real number x, where a...
consider two condition x^2-3x-10<0 and |x-2|<α on a real number x, where a is a positive real number
i. the range of values of α such that |x-2|<α is a necessary condition for x^2-3x-10<0 is (A)
2. the range of values α such that |x-2|<α is a sufficient condition for x^2-3x-10<0 is (B)
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Solve the first inequality:
`(-5)(2)=-10` and `-5+2=-3`
A) For the necessary condition we with to find the range of a for which `|x-2|<a` must be true in order for` x^2-3x-10` to be true:
In order for `-2 lt x lt 5` to map to `2-a lt x lt 2+a` , a cannot be more than 4 `(alt=4)`
Consequently, the necessary condition is: `alt=4`
B) For the sufficient condition we wish to find the range of a for which if `|x-2|lta` is true then `x^2-3x-10lt0` is also true:
The magnitude of any value must be positive, therefore:
Consequently, the sufficient condition is: `0ltalt=3`
Posted by crmhaske on May 24, 2013 at 3:02 PM (Answer #1)
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