∣ 9 − x^2∣ <8x^2. Find possible solutions of x.?

Think of this as its graphical meaning: The distance from 9 to a number is less than eight times the number. If the number would be 1, the distances are equal. So the number must be greater than 1

so x² >1 which means that x²-1>0, (x-1)(x+1) > 0

So x is either less than -1 or greater than 1

if x € [ - 3 , 3 ] then x^2 <= 9

so the equation gives

9 - x^2 < 8x^2

9x^2 > 9

x^2 > 1

so the set of solutions is [ - 3 , - 1) U ( 1 , 3 ]

if |x|>= 3 then x^2 >= 9

so the equation gives

- 9 + x^2 < 8x^2

7x^2 > - 9

so any x in the "outside intervals R \ [ - 3 , 3 ] is convenient

finally

the set S of solutions is

S = ( -oo ; -1 ) U ( 1 , +oo )

hope it' ll help !!