I want to verify my solution for $x(x+\bar{y}+\bar{z})\\=xx+\bar{y}x+\bar{z}x\ \text{by Distributive laws}\\ =x+\bar{y}x+\bar{z}x\ \text{by Idempotent laws}\\ =x+\bar{z}x\ \text{by Absorption laws}\\ =x\ \text{by Absorption laws}$
(N.B. I donte $\neg x := \bar{x}$) Is this solution correct?
Looks good.
For your information, the other absorption law is $$x(x+w) = x,$$ so your simplification result is right by letting $$w = \bar y + \bar z.$$