# 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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### 1 Answer

Solve the first inequality:

(-5)(2)=-10 and -5+2=-3

x^2-3x-10=(x-5)(x+2)<0

Therefore: -2<x<5

A) For the necessary condition we with to find the range of a for which |x-2| must be true in order for x^2-3x-10 to be true:

In order for -2 < x < 5 to map to 2-a < x < 2+a, a cannot be more than 4 (a<=4)

Consequently, the necessary condition is: a<=4

B) For the sufficient condition we wish to find the range of a for which if |x-2|<a is true then x^2-3x-10<0 is also true:

-2-2<|x-2|<=5-2

-4<|x-2|<=3

The magnitude of any value must be positive, therefore:

0<|x-2|<=3

Consequently, the sufficient condition is: 0<a<=3

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