answer:-8<x<8
step by step: (x^2) <64 => (x^2) -64 < 64-64 => (x^2) - 64 < 0
64= 8^2 so (x^2) - (8^2) < 0
To solve the inequality we first find the roots (values of x that make (x^2) - (8^2) = 0 )
Note that if we can express (x^2) - (y^2) as (x-y)* (x+y) You can work backwards and verify this is true.
so let's set (x^2) - (8^2) equal to zero to find the roots:
(x^2) - (8^2) = 0 => (x-8)*(x+8) = 0
if x-8 = 0 => x=8 and if x+8 = 0 => x=-8
So x= +/-8 are the roots of x^2) - (8^2)
Now you need to pick any x values less than -8 (the smaller root) , one x value between -8 and +8 (the two roots), and one x value greater than 8 (the greater root) and see if the sign is positive or negative.
1) Let's pick -10 (which is smaller than -8). If x=-10, then (x^2) - (8^2) = 100-64 = 36>0 so it is positive
2) Let's pick 0 (which is greater than -8, larger than 8). If x=0, then (x^2) - (8^2) = 0-64 = -64 <0 so it is negative
3) Let's pick +10 (which is greater than 10). If x=-10, then (x^2) - (8^2) = 100-64 = 36>0 so it is positive
Since we are interested in (x^2) - 64 < 0, then x should be between -8 and positive 8.
So -8<x<8
