# consider two condition x^2-3x-10<0 and |x-2|<α on a real number x, where a is a positive real numberi. the range of values of α such that |x-2|<α is a necessary condition...

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`

`x^2-3x-10=(x-5)(x+2)lt0`

Therefore: `-2ltxlt5`

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:

`-2-2lt|x-2|lt=5-2`

`-4lt|x-2|lt=3`

The magnitude of any value must be positive, therefore:

`0lt|x-2|lt=3`

**Consequently, the sufficient condition is: `0ltalt=3` **

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