Solving Polynomial Inequalities

[Veronica_Oles]

Homework Statement

Solve the following. Express answers in set notation.
-2(x-2)(x-4)(x+3)<0

Homework Equations

The Attempt at a Solution

I know my four intervals are x<-3 , -3<x<2 , 2<x<4 , x>4.

I thought the answer would be x<-3 and 2<x<4 however the answers are opposite of what I thought, they are -3<x<2 and x>4. I am having trouble understanding why this is. Would it be because there is a -2 that needs to leave so it is divided under both sides thus meaning instead of it being "less than zero" the sign changes and it becomes "greater than zero"? Would that be correct? Thanks just need claification.

 

[fresh_42]

Homework Statement

Solve the following. Express answers in set notation.
-2(x-2)(x-4)(x+3)<0

Homework Equations

The Attempt at a Solution

I know my four intervals are x<-3 , -3<x<2 , 2<x<4 , x>4.

I thought the answer would be x<-3 and 2<x<4 however the answers are opposite of what I thought, they are -3<x<2 and x>4. I am having trouble understanding why this is. Would it be because there is a -2 that needs to leave so it is divided under both sides thus meaning instead of it being "less than zero" the sign changes and it becomes "greater than zero"? Would that be correct? Thanks just need claification.

Yes.

If $-2z < 0$ then $2z > 0$ as you said.
To check the entire expression you could simply take a number in one of your intervals and calculate the product.
Do you know how to write the answer in set notation? How would you do that?

 

[Ray Vickson]

Homework Statement

Solve the following. Express answers in set notation.
-2(x-2)(x-4)(x+3)<0

Homework Equations

The Attempt at a Solution

I know my four intervals are x<-3 , -3<x<2 , 2<x<4 , x>4.

I thought the answer would be x<-3 and 2<x<4 however the answers are opposite of what I thought, they are -3<x<2 and x>4. I am having trouble understanding why this is. Would it be because there is a -2 that needs to leave so it is divided under both sides thus meaning instead of it being "less than zero" the sign changes and it becomes "greater than zero"? Would that be correct? Thanks just need claification.

If you set $A = x-2$, $B = x-4$ and $C = x+3$ you must have $A B C > 0$ (changing the "-" to "+" and reversing the inequality, as you have already noted).

If a product of three numbers is positive, what are all the possibilities for their signs?

 

[haruspex]

I suggest the easiest way is to figure out the sign at $+\infty$ then consider varying x and what happens to the sign as x passes through one of the roots (given that they are none of them repeated roots).