Fraction inequality denominator 0 rule?

-x+8

----- (greater than or equal) 0

x-7

Do the problem I get x=8 and x=7

I graph it, and test it, and am wondering what exactly is the rule for when to put () or []

My understanding is that since it is a (greater than or equal) sign it should be []

But, the answer states it is (7,8] and Im not sure why.

I read something that if a number in the solution makes the denominator 0 then you must include it with () and I would have gotten it wrong as I was going to enter it as [7,8]

Is it basically this? if a number that is the solution on my graph makes the original equations denominator 0 then that number will have a () by it?

Maybe someone can dumb it down for me (Headbang)

Re: Fraction inequality denominator 0 rule?

The general fact is that $\displaystyle \frac{a}{b}\ge0$ iff (a ≥ 0 and b > 0) or (a ≤ 0 and b < 0). Yes, the inequality for b is strict because the denominator cannot become zero.

Applying this fact to this problem, we get (1) -x + 8 ≥ 0 and x - 7 > 0 or (2) -x + 8 ≤ 0 and x - 7 < 0. Variant (1) is equivalent to 7 < x ≤ 8, i.e., x ∈ (7, 8]. Variant (2) is equivalent to 8 ≤ x < 7, which is impossible.

Re: Fraction inequality denominator 0 rule?

So in dummer terms, for me, I need it simple, is it true then that any time I get an answer with numbers that make the denominator 0, use () with that number when writing the answer?

Re: Fraction inequality denominator 0 rule?

Quote:

Originally Posted by

**itgl72** So in dummer terms, for me, I need it simple, is it true then that any time I get an answer with numbers that make the denominator 0, use () with that number when writing the answer?

Yes, if a number that makes a denominator 0 appears as the end of an interval in the final answer, then it must be excluded by using a parenthesis instead of a square bracket.

Re: Fraction inequality denominator 0 rule?

(a, b) means the points a and b are NOT included in the set.

(a, b] means the point a is not included but b is.

[a, b) means the point a is included but b is not.

[a, b] means the points a and b are both included in the set.

Put the endpoints into the original inequality to see if they satify it or not.