Could someone help me out with this ineqaulity property

My calculus book gives the factors: (x - 1 + sqrt2/2 )(x - 1 - sqrt2/2) > 0

It then says that since the inequality is greater than zero either both factors are positive or negative.
Therefore, we require that either x < 1 - sqrt2/2 or x > 1 + sqrt2/2.

My question is why?

I tried justifying it different ways, but my logic ends up not making any sense.

I really appreciate any help you guys can give me.

In the case that both quantities are positive, the inequality

x - 1 - sqrt2/2 > 0

implies that

x - 1 + sqrt2/2 > 0,

so you only have to say the first one.

In the case that both factors are negative, the inequality

x - 1 + sqrt2/2 < 0

implies that

x - 1 - sqrt2/2 < 0,

so you only have to say the first one. So, to summarize the solution, it is sufficient to say that either

x > 1 + sqrt2/2

or

x < 1 - sqrt2/2.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The product of two negative numbers is positive. The product of two positive numbers is positive. But the product of a negative number and a positive number is negative. So if you have a product ab > 0, then either both a and b are negative or they are both positive.

In this particular case, (x - 1 + sqrt2/2) > (x - 1 - sqrt2/2). If they are both negative then
x - 1 + sqrt2/2 < 0, so x < 1 - sqrt2/2. Similarly when they are both positive, then x > 1 + sqrt2/2.