Nonlinear Inequalities and unions?

• Aug 27th 2008, 11:49 AM
ithompsonline
Nonlinear Inequalities and unions?
if you can help me with this, well, i'd be very happy.

x(2x + 7) ≥ 0

first off, i understand most of the problem, i just don't know how to tell when an inequality has a union, if i need to use or in the answer, etc

my intervals:

-∞, -7/2
-7/2, 0
0, ∞

sign of x -, -, +
sign of 2x+7 -, +, +

sign of x(2x+7) +, -, +

in the back of the book it shows {x|x ≤ -7/2 or 0 ≤ x}
Interval: ( -∞, -7/2] U [0, ∞)

how the heck am i supposed to know when to use a union?
i have an alternative problem that i can solve on my own to show what i mean...a problem without a union...:
(x+2) (x-3) < 0

answer = {x|-2 < x < 3}. Interval: (-2, 3)

so the real question is, why does one problem have a union, and the other one does not???
• Aug 27th 2008, 11:54 AM
Moo
Hello,

The first one is a union, because wherever x is in the 2 intervals, the inequality is OK.
It may be in either the first or the third interval, it'd work. The union is quite equivalent to "or".

For the second one, there's no union, because it's a single interval, there's no need to make a union with another interval...
For the inequality to be verified, x has to be in this single interval, and not outside it.

I don't know if it's clear enough (Wondering), if it helps a little, ok, if not, I'm sure someone else will tell you something else :D
• Aug 27th 2008, 11:59 AM
ithompsonline
Quote:

Originally Posted by Moo
Hello,

The first one is a union, because wherever x is in the 2 intervals, the inequality is OK.
It may be in either the first or the third interval, it'd work. The union is quite equivalent to "or".

For the second one, there's no union, because it's a single interval, there's no need to make a union with another interval...
For the inequality to be verified, x has to be in this single interval, and not outside it.

I don't know if it's clear enough (Wondering), if it helps a little, ok, if not, I'm sure someone else will tell you something else :D

ok, this is gonna sound really stupid on my end....how do i know if an inequality is *ok* and what happened to the 3rd interval? what is a single interval? i'm so lost, lol, i last took this math class 9 years ago if you can't tell.
• Aug 27th 2008, 12:08 PM
Moo
Quote:

Originally Posted by ithompsonline
ok, this is gonna sound really stupid on my end....how do i know if an inequality is *ok* and what happened to the 3rd interval? what is a single interval? i'm so lost, lol, i last took this math class 9 years ago if you can't tell.

An inequality is *ok* (sorry, that was a misuse from me), or "satisfied" when you have x(2x + 7) ≥ 0 for a given x.

Single interval is like (-2, 3), whereas when I was talking to the first and third intervals, I was talking about ( -∞, -7/2] U [0, ∞)

Sorry for my speaking (Doh)

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You can't talk about ( -∞, -7/2] U [0, ∞) without using the union sign, because there's no connection between the two parts.

Whereas if the solution is (-2, 3), there's a continuous connection between any element of it, and every element of it satisfies the inequality.