Zero Product Property or inquality regarding finding the domain

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I'm confused and need an explanation. Sometimes when I find the domain of an algebraic expression I mostly use the zero product property rule but then I look at the back my textbook to get the answer and I see they used an inequality equation to find the domain. So the question is, how do I know when to use the zero product property rule or inequality to find the domain? I think the answer is (and correct me if I'm wrong) use the product property for quadratic equations and use the inequality equation for polynomial that have 2 terms or less (ex: 2x+3) so I'm i right or wrong? Thanks in advance.:D
 
I'm confused and need an explanation. Sometimes when I find the domain of an algebraic expression I mostly use the zero product property rule but then I look at the back my textbook to get the answer and I see they used an inequality equation to find the domain. So the question is, how do I know when to use the zero product property rule or inequality to find the domain? I think the answer is (and correct me if I'm wrong) use the product property for quadratic equations and use the inequality equation for polynomial that have 2 terms or less (ex: 2x+3) so I'm i right or wrong? Thanks in advance.:D
I really have no idea what you are asking or what you mean?
The best way to get help is to present an actual question? Write out a question that you have done but your answer is different from that given in the text. Also tell us what answer is given.
 
I think I just answered my own problem. I think I can use any method for expressions with two terms or less. I just got to be aware of what the domain needs to be with regards to the inequality symbol, but correct me if I'm wrong.

I took a picture of my work:

https://postimg.org/image/dl1adbbi3/
 
Erm... okay. I'm still a bit confused, but I think I understand what's going on. There are actually two completely separate concepts you're asking about, and they're really not related at all. The domain of a function is the set of all values the variable can take on. The zero product property has no role in determining this whatsoever. Using your example function:

\(\displaystyle f(x)=\dfrac{14x}{3x+5}\)

The domain is the set of all inputs (i.e. x-values) for the function. Because there's an infinite amount of numbers, it's often easier to determine which inputs are invalid, rather than determine which are valid. You first used the inequality \(\displaystyle 3x+5 \ge 0\) but I'm not sure what this is for. That inequality establishes what x values produce a negative denominator, but it doesn't matter if the denominator is negative. As an example, if x=1, the function is \(\displaystyle f(x)=\dfrac{14(1)}{3(1)-5}=\dfrac{14}{-2}=-7\). The denominator is negative, but the function is still defined. So what can you conclude about if x=1 is part of the domain? What value(s) of x make the function undefined? Are those x-values part of the domain?

Next, we have the zero product principle. This is used for determining roots (sometimes called solutions, or zeroes of a function, not the domain. In your example:

\(\displaystyle f(x)=x^2+7x+10\)

The roots of this function are when the function equals 0. So you would set it equal, and factor, as you did. Essentially, all the zero product principle says is this:

If "something" times "something else" equals 0, then we know that either "something" is 0, or "something else" is 0. Do you see why that must be the case?
 
Yeah you cleared it up. And it makes sense after you explained it. I failed to realize that it really depends on the problem that's given to me...if it was lets say a fraction and I know that the denominator can't be zero then I would proceed with the zero product property to find the domain and if it was a problem about with a square root then i would setup inequality to see which values would trigger a negative number. but yeah thanks. :D
 
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