Solve for x: x^2 < 64

Question

x² < 64

 

values for x to graph on number line

Answer 1

(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

 

Note: If you choose any number outside this range for x, and square it it will be greater than 64 and so it is not valid.