Solving the inequalitites in term of x, applying rules

1. Mar 30, 2012

rohan03

I am working through this chapter and trying out exercises - I am stuck on this one- this is what I have done so far but since this is my first experience with this topic I am just not sure what to do next
i have typed it and attached it here with

Thanks
1. The problem statement, all variables and given/known data
I have typed it all as I find it easier to use equations in words

2. Relevant equations
Its attached in PDF file

3. The attempt at a solution is also attached

Last edited: Mar 31, 2012
2. Mar 30, 2012

Joffan

It is certainly always the case that, given $x>0$, it is true that $-x<0$. More generally, $x>y \implies -x<-y$.

Note that this means that you cannot always multiply inequalities by unknowns, because you don't know whether the unknowns are positive or negative.

I'm assuming that the factorization is not an issue for you.

3. Mar 30, 2012

rohan03

No factorising is fine but that makes all positive fraction smaller than zero and that's what I was worried about.

4. Mar 30, 2012

Staff: Mentor

Starting from
$$\frac{-x^2-3x-2}{x^2+7x+12} > 0$$
If you multiply both sides by -1, the inequality sign must change direction.
$$\frac{x^2+3x+2}{x^2+7x+12} < 0$$

I think you are assuming that both fractions are positive, since all the coefficients of the terms are positive. That isn't necessarily true, since x could be negative in value.

5. Mar 30, 2012

Ray Vickson

If $$\frac{-x^2 - 3x - 2}{x^2 + 7x + 12} > 0,$$ then either both the numerator and denominator are > 0 or both are < 0.

(1) Assuming the numerator and denominator are both > 0 we have $-x^2 - 3x - 2 > 0 \Longrightarrow x^2 + 3x < - 2, \text{ and } x^2 + 7x + 12 > 0 \Longrightarrow x^2 + 7x > -12.$ You can figure out what the values of x must be to have that.

(2) Assuming the numerator and denominator are both < 0 we have $-x^2 - 3x - 2 < 0 \Longrightarrow x^2 + 3x > -2, \text{ and } x^2 + 7x + 12 < 0 \Longrightarrow x^2 + 7x < -12.$ Again, you can figure out what (if any) x are allowed.

RGV

6. Mar 30, 2012

Staff: Mentor

I think it's better to leave the three terms on the left side of the inequality, and then factor it, rather than move the constant to the other side.

7. Mar 30, 2012

Saitama

Maybe you would like to write the equation like this:-
$$\frac{(x+1)(x+2)}{(x+4)(x+3)}<0$$

Plot the roots of the quadratics on the number line. For example, what sign you get when you substitute a number greater than -3 and less than -2 in x. Make sure you first plot the numbers on the number line. When you substitute -2.5, you get a positive sign. Similarly check what happens when you substitute a number less than -3 and greater than -4. Check for all the possible ranges. If you get a positive sign in a range, put a plus sign in that range on the number line. The range having the negative sign will be your answer.

Hope that helped!

8. Mar 31, 2012

rohan03

Thank you all. Finding the range is my next step. I will post my finding today and see what I get

9. Mar 31, 2012

rohan03

after workign out the quadratic roots I get solution set as
{x:(x^(2 )+11x+22)/(x^(2 )+7x+12)>2} = (- ∞,-2) u (-1,∞) is this correct

10. Mar 31, 2012

Joffan

No.

One free check you can do is look at the value at x=zero. Another is to think about trends to both infinities.

But basically once you had the equation recast into an inequality against zero, you can simply find ranges where the numerator and denominator have the appropriate sign.

11. Mar 31, 2012

rohan03

Now I am lost!

12. Mar 31, 2012

rohan03

I think that this should be {x:-2<x<-1}

13. Mar 31, 2012

Joffan

No. Can you show the inequality you are considering to derive these?

Useful tip: Press "Quote" next to one of the existing messages on the thread if you want to see how to format expressions on this board.

14. Mar 31, 2012

rohan03

I am typing out my sign table and will reattached my attachment. But any help in between be great!

15. Mar 31, 2012

rohan03

I pressed quote but can't see any input format!!

16. Mar 31, 2012

rohan03

May be I am no mobile- time to switch pc on

17. Mar 31, 2012

Joffan

Yes, that would probably help; easier to review the material already discussed too.

You should have your expression as a fraction consisting of a polynomial in the numerator (top) and another polynomial in the denominator limited by inequality to zero.

Finding the roots of those two polynomials shows you where they change sign.

18. Mar 31, 2012

rohan03

x (-∞,-4) -4 (-4,-3) -3 (-3,-2) -2 (-2,-1) -1 (-1,∞
(x+2) -ve -ve -ve -ve -ve 0 + + +
(x+1) -ve -ve -ve -ve -ve - 0 +
(x+3) -ve -ve -ve 0 +ve + +
(x+4) -ve 0 +ve -ve +ve + +

gives + 0 - * + 0 - 0 +

Now what? this I think gives me (-∞,-4)U(-2,-1) hope I am right now

19. Mar 31, 2012

rohan03

sorry I think its (-4,-3)U(-2,-1)

20. Mar 31, 2012

Joffan

Yes, that's the right range. I recommend that you try to make a start-to-finish argument for how you produced the answer; otherwise in any significant exam you would be losing a lot of marks for method.