Question

For nitrogen to be a liquid, its temperature must be within 12.78 °F of –333.22 °F. The equation |x + 333.22 | = 12.78 can be used to find x, which represents the maximum and minimum temperatures at which nitrogen is a liquid.

im not quite understanding this on how to explain this to my daugher

Answer 1

The concept seems to make sense. Higher than 12.78 degrees, Nitrogen is a gas; lower than -333.22 degrees, Nitrogen is a solid; between 12.78 and -333.22 degrees, Nitrogen is a liquid. That part shouldn't be too hard.

Is it the equation that is confusing? Or what is the source of the confusion?

Answer 2

the equation

Answer 3

So we have
|x + 333.22 | = 12.78
The (x + 333.22) part is between absolute value symbols ||, which means that (x + 333.22) could be 12.78, or it could be -12.78. We could rewrite this as:
x + 333.22 = 12.78 OR x + 333.22 = -12.78
If we follow this equation, this means that x could be (12.78 - 333.22) or x could be (-12.78 - 333.22). Simplifying this, we get two possible values for x:
x = -320.44 OR x = -346
Only one of these actually falls within the range of temperatures for which Nitrogen is a liquid, so x can only be 320.44, not -345.

Is that what the equation means?

Answer 4

Perhaps I misunderstood the purpose of the equation, but if we are meant to solve for x, which represents a temperature at which Nitrogen is a liquid, I think that is how we would do it.

Answer 5

so this was the actual question: im just trying to figure out how they came up with the equation how would you know its x+333.22 vs x-333.22

question: For nitrogen to be a liquid, its temperature must be within 12.78 °F of –333.22 °F. Which equation can be used to find the maximum and minimum temperatures at which nitrogen is a liquid, x?

Answer 6

sorry for the confusion

Answer 7

Oh! I see, the question is about figuring out the correct equation to use. That makes sense.

In that case, it would have to be an equation that is true whenever the value of x is between -333.22 and 12.78.
One way to test the answer choices would be to plug in values of x within that range to see whether or not they satisfy the equations. Alternately, if any value of x outside of that range satisfies the equation, you would know it is not an appropriate equation for this situation.

Answer 8

Are there choices of equations for this question, or does it ask that we write our own equation to represent the relationship?

Answer 9

unfortunately it asks you for the equation.. no choices

Answer 10

Ok, I should apologize. I made a pretty bad misunderstanding earlier. I misread "temperature must be within 12.78 °F of –333.22 °F" as "temperature must be between 12.78 and -333.22". I think I got it now.
Actually, Nitrogen is a liquid between -346 degrees and -320.44 degrees, which are both -333.22 - 12.78 and -333.22 + 12.78, respectively.

So, the original equation, |x + 333.22 | = 12.78 would actually be correct to find the upper and lower bounds of the range for the temperature!
The results we got for x (-346, -320.44) are the lowest and highest temperatures at which Nitrogen can still be a liquid.

Sorry for misunderstanding and making things more complicated than they had to be. It seems like you had the right solution from the beginning!

Answer 11

nono that was my fault for not being clear...just trying to figure out how to explain how did i come up with the equation so it makes sense to my daughter

Answer 12

In order to explain why the equation is that way, we know first of all that "x", the temperature, has to be within 12.78 of -333.22. That's the condition we start with.
So, we can set 12.78 on one side of the equation, because that is the furthest distance we can go from -333.22. On the other side of the equation, we can subtract (-333.22) from x, to represent a distance being taken. And we put all of that between the absolute value signs || to show that it is a distance, which can go in either direction.
The result |x + 333.22| = 12.78, accurately represents the range of temperatures that Nitrogen is a liquid.

Answer 13

I guess another way to demonstrate how the equation makes sense would be to use algebra to find the values of x that work with the equation?
Maybe seeing the numbers at work would help with understanding

Answer 14

she is asking why it wouldn't be x-333.22=12.78

Answer 15

its being a little more clear now so maybe i can wiggle my way and explain it

Answer 16

Ah I think I understand.
It's not "x-333.22" because you are subtracting (-333.22). So, when you subtract a negative, it becomes positive. Is that what she means?

Answer 17

yes that part i get i know what might make my confusion more clear. so if it has to be within 12.78 how can you make an equation no know if its below or above -333.22

Answer 18

sorry i feel so dumb even asking this lol

Answer 19

No worries, it's all a learning process

Answer 20

I think that's where the absolute value comes in ||.
With that as part of the equation, you can account for the values that are both above and below -333.22

Answer 21

|x + 333.22| = 12.78
is just another way of saying that
x + 333.22
could be
-12.78
or
+12.78

Answer 22

\[\left| x+a \right|=b\]
\[x+a=\pm b\]
\[x=-a \pm b\]
so
x=-a+b
or x=-a-b

Answer 23

oooohhhhh yessssss

Answer 24

^^ That's a pretty clear demonstration. Thank you!

Answer 25

so it basically incorporates either way up or down

Answer 26

thank you guys so much!

Answer 27

You're very welcome!
And thank you @surjithayer for putting the equation in a clear and easy to understand format!