I was tutoring a student today and was given a question that stated the following:
Find $P(|z| > -0.29)$.
Seems simple enough. But this doesn't really make sense in concept. Firstly, absolute values always yield a positive result, so it wouldn't really make sense for this question. Would the answer simply be $1$?
And secondly, for standard questions where $P(|z|>a)$, is the formula simplfied just $P(z>a) + P(z<-a)$ for values of $a>0$? For this one, it wouldn't make sense to just say $P(z<0.29) + P(z>-0.29)$ because you can't just flip the sign on absolute values. It just doesn't make sense for an absolute value equation to have a negative on the other side of the inequality.
Tried finding a YouTube video or anything just searching "probability of absolute value" and found nothing. It's shocking such a simple concept found in the first statistics course you take doesn't have anything to click on with that prompt so that I can show my student a simpler example that makes more concrete sense.
The question "Find $P(|z| > -0.29)$" has the obviously correct answer of $1$. It is likely either that it was a test of obviousness, or that it was asked in error.
As to your other question related to $P(|z|>a) = P(z>a)+P(z<-a)$, that expression is only correct when the events $z>a$ and $z<-a$ are mutually exclusive, as they are in this case when $a\ge 0$ but not necessarily when $a<0$.
More generally $P(A \cup B)=P(A) +P(B)-P(A \cap B)$ using inclusion-exclusion. So here you would have for general $a$:
$$P(|z|>a)=P((z>a) \cup (z<-a)) = P(z>a)+P(z<-a) - P(a < z < -a)$$
and thus for $a \ge 0$ you have $P(a < z < -a)=0$ giving $$P(|z|>a) = P(z>a)+P(z<-a),$$
while for $a \lt 0$ you have $P(a < z < -a)$ $=1 - P(z \le a) - P(z \ge -a) $ $= 1 - (1- P(z>a))+(1-P(z<-a)) = P(z>a)+P(z<-a)-1$ giving $$P(|z|>a) = 1.$$