Solving the inequalitites in term of x, applying rules

[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

Homework Statement

I have typed it all as I find it easier to use equations in words

Homework Equations

Its attached in PDF file

The Attempt at a Solution

is also attached

 

[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.

 

[rohan03]

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

 

[Mark44]

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

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.

 

[Ray Vickson]

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

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

 

[Mark44]

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.

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.

(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

 

[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!

 

[rohan03]

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

 

[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

 

[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.

 

[rohan03]

Now I am lost!

 

[rohan03]

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

 

[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.

 

[rohan03]

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

 

[rohan03]

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.

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

 

[rohan03]

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

 

[Joffan]

May be I am on mobile- time to switch pc on

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.

 

[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

 

[rohan03]

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

 

[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.

 

[rohan03]

do you mean I write out reasoing for all my steps? Yes I would and thank you so much.