Polynomial inequalities

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The Velociraptors

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6AA8A5BC-1531-4AAB-AADC-35236535BBDF.jpegI know that the inequality has to be less than or equal to 0 because that’s what the answer said. But I’m having trouble graphing this. Why is three and 1 included if the inequality is less than or equal to 0? They’re both greater than 0. Is it just the set of y values that must be less than 0, or both x and y values less than or equal to 0? If it was only y-values though, then it would be odd how the regions of 3 and 1 aren’t shaded making the domain (-infinity,3). I really don’t understand what is happening, if anyone has any resources or know of any videos that touch on this, that would be extremely helpful.
 
Very strange problem. Is it complete?!

I will assume that the number line is showing the points where y (NOT x) is less than zero.

Your answer should be in the format of y<=(x+2)^2*(x-1)^2*(x-3) [Not x<=(x+2)^2*(x-1)^2*(x-3)].
The problem is this inequality is <=0 when x<=3. You only want y<=0 when x<=-2.
Try fixing your answer.
 
Well, it is complete, but has many solutions.
If the y values were less than 0, would it look like this:
CE3C3276-30AC-4680-A02E-F3A4F7873371.jpegBut I guess my problem with this is why the interval (-2,3) isn’t shaded. Why is it [1,1] and [3,3]?
 
The Velociraptors said:
Well, it is complete, but has many solutions.
If the y values were less than 0, would it look like this:
View attachment 29433But I guess my problem with this is why the interval (-2,3) isn’t shaded. Why is it [1,1] and [3,3]?
Click to expand...
You first diagram was perfectly fine, why did you change it? The solution set includes _points_ 1 and 3 - this is where y=0. If you make the graph deep below the x axis to the right of x=-2 you'll include more than 1 and 3.
Can you explain why you decided to square some factors and not the others? How does squaring affect the behavior of the graph?
 
lev888 said:
You first diagram was perfectly fine, why did you change it? The solution set includes _points_ 1 and 3 - this is where y=0. If you make the graph deep below the x axis to the right of x=-2 you'll include more than 1 and 3.
Can you explain why you decided to square some factors and not the others? How does squaring affect the behavior of the graph?
Click to expand...
Oh right! I totally forgot that when x=3, y=0, or x=1, y=0; because of the greater than or equal to sign, these coordinates are included. This actually makes a lot of sense, thank you! I decided to square some quantities to try and install a tangent at those points. I was just experimenting with what I could do to satisfy the requirements. It would affect the behavior because it prevents the line from passing through the x-axis.
 
The Velociraptors said:
Well, it is complete, but has many solutions.
If the y values were less than 0, would it look like this:
View attachment 29433But I guess my problem with this is why the interval (-2,3) isn’t shaded. Why is it [1,1] and [3,3]?
Click to expand...
You wrote on your own a + sign right above 4 on the number line! You put that because the graph is positive there BUT you want the graph to always be negative or 0.
 
To OP and helpers, what on earth does the bold on the number line to the left of -2 mean? To me, it would mean y=0, if x<=-2.
 
Jomo said:
To OP and helpers, what on earth does the bold on the number line to the left of -2 mean? To me, it would mean y=0, if x<=-2.
Click to expand...
Yes, that is what it means. It also means that the interval is (-infinity,-2], and I’m just supposed to find whether or not that region is positive or negative. There are multiple solutions apparently, I just have to come up with a possible one.
 
Jomo said:
You wrote on your own a + sign right above 4 on the number line! You put that because the graph is positive there BUT you want the graph to always be negative or 0.
Click to expand...
Well yeah, I knew that the leading coefficient was positive which is why that specific region would be positive. I just forgot about the part where y=0 when x= 1 and 3 which is what got me confused.
 
Jomo said:
To OP and helpers, what on earth does the bold on the number line to the left of -2 mean? To me, it would mean y=0, if x<=-2.
Click to expand...

I think the bold line (and points) on the number line just indicate graphically the domain, all x values, where the following inequality is satisfied

P(x) ≤ 0

There is no y. However, if the graph did show y=P(x) then it would look like OP's dotted line in post #1 (NOT post #3)
 
Cubist said:
OP, your suggested polynomial

[imath](x+2)^2 (x-1)^2 (x-3) \le 0[/imath]

is almost correct. You just need to rethink which factors need to be squared.
Click to expand...
I think I realized a trick. Because there are many possible solutions, if I take the shaded region to be below the graph, then the interval [-2,3] would go above the graph. If I take the shaded region to be above the graph, then the opposite would be true for [-2,3]. I’ve found two possible solutions following this:
 

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The Velociraptors said:
I think I realized a trick. Because there are many possible solutions, if I take the shaded region to be below the graph, then the interval [-2,3] would go above the graph. If I take the shaded region to be above the graph, then the opposite would be true for [-2,3]. I’ve found two possible solutions following this:
Click to expand...
Well actually, the first graph would have a negative sign in front of it because the end behavior is negative. Forgot to add that.
 
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