Is z>0? (1) (z+1)(z)(z-1)<0; (2) |z|<1

[OliviaP316]

The question below is from MGMAT Guide 2 Algebra, 6th edition, page 149, question 3:

Is z > 0?

(1) (z + 1)(z)(z - 1) < 0
(2) |z| < 1

My question is, can I simplify (1) as follows:

(z^2 - 1)(z) < 0
Divide both sides by z (this is the part that I'm unsure about... is this "legal"?)
(z^2 - 1) < 0
z^2 < 1

Thank you!

[RonPurewal]

remember the basics, from high-school algebra:

• if you divide both sides of an inequality by a POSITIVE quantity, then, the inequality sign stays the same.

• if you divide both sides of an inequality by a NEGATIVE quantity, then, the inequality sign is reversed.

here, we don't know whether "z" is positive or negative... so, you can't perform this division, because we don't know what needs to happen to the inequality sign (it might need to be reversed... but it might not).

this is definitely something there's no excuse for overlooking, by the way—since the goal of the problem is "Is z > 0?"
in other words, the unknown sign of "z" is the whole point of the problem. so, you definitely should have been thinking about that going in.

[OliviaP316]

Thanks, Ron!

[RonPurewal]

you're welcome.

[BergurR797]

Can I however simplify it to:

z^3 - z < 0 ?


(z+1) (z) (z-1) < 0
z (z^2 - 1) < 0
z^3 - z < 0

[RonPurewal]

^^ i wouldn't use the word "simplify" there, since you aren't actually simplifying anything -- you're just expanding a product of factors.
(your result has fewer terms, but, it's still a third-degree expression -- in other words, it's no simpler than what you started with.)

in fact -- while that version is algebraically correct, it's actually a lot LESS useful than the ORIGINAL version of statement #1.
viz.:
• in the original version, it's immediately clear that the left-hand side is zero when z is –1, 0, or 1... and thus that we need to investigate the "chunks" into which those values break up the number line.
• in your transformed version, it's actually much harder to see these things.

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[RonPurewal]

please do note -- i'm not just picking at words, in that post above.

when we talk about SIMPLIFYING, we're talking about something that:
• GETS RID of some of the parts/complexity of the original problem;
• generally isn't done in reverse—EXCEPT as part of a very specific process (e.g., making a common denominator);
• should basically ALWAYS be done, as soon as possible;

e.g.,

if you SIMPLIFY 84/2, you get 42.
• this is actually SIMPLER than the original (it's not a fraction anymore).
• it would be really strange to do this in reverse (= to take 41 and turn it INTO 84/2), unless you were doing that specifically as part of making a common denominator.
• if you have the expression 84/2 basically ANYWHERE in a problem, you should pretty much turn it into 41 as soon as you possibly can.

__

this distinction is important:

• SIMPLIFICATION should be DONE RIGHT AWAY... unless you have a specific reason NOT to do it.

• reversible processes -- such as expanding (z – 1)(z)(z + 1) into z^3 – 1, or factoring z^3 – 1 back into (z – 1)(z)(z + 1) -- you generally shouldn't do unless you have a REASON to do them.