Solving Tricky Inequalities: Help with a Non-Factorable Numerator

[J-Girl]

Inequalities- tricky question!

hiii,
i was wondering if anybody knew how to help me with this one tricky homework question. i can do most of the inequaliies I've come across, but how do you solve an inequality if you can't factor it?
the question is: (x^2-4x+7)/(x^2+x-6)
i know that the denominator is easily factorable ((x+3)(x-2)) but the numerator isnt? help pleasezzz I've been stuck on this for ages!:(:(

 

[disregardthat]

Rewrite (x^2-4x+7) to (x-2)^2+3. What can you say about this?

 

[Mentallic]

Ok so I'm guessing by inequalities you mean something like $$\frac{x^2-4x+7}{x^2+x-6}>1$$ for example?

Try multiplying both sides by $$(x^2+x-6)^2$$ since you know this has to be a non-negative number, and don't expand! Use your knowledge of factorizing to solve it.

 

[J-Girl]

ohh sorry i didnt even put the rest of the question in! it was (x^2-4x+7)/(x^2+x-6)≤ 0
i just wasnt sure if i was allowed to automatically multiply by the denominator because i didnt know if it was positive or negative. if i rephrase it as (x-2)^2 + 3, i still can't cancel out any brackets because its not completely factored. sorry:( just haven't done maths in ages and i suck at it..

 

[disregardthat]

Can you tell when (x-2)^2+3 is positive and negative though? Notice the squared term.

 

[J-Girl]

oohhh i got it now!:) yay took ages tho lol but thanks every1:)