Comparing Bipolarity: How to Determine the Greater Value of x in an Equation?

[lalapnt]

how do i work this out?

 

[arildno]

Remember that for any two numbers, a and b, we have: (a-b)*(a+b)=a^2-b^2

 

[tahayassen]

Think of the square root function graphed.

 

[Mark44]

Think of the square root function graphed.

I like this approach. You can see at a glance which difference is larger.

 

[arildno]

Think of the square root function graphed.

Very good!

 

[lalapnt]

Remember that for any two numbers, a and b, we have: (a-b)*(a+b)=a^2-b^2

i knot this. but how does it help? the terms are different

Think of the square root function graphed.

i don't want to solve this graphically. (not like i even know how. I need some help here)

 

[arildno]

sqrt(12)-sqrt(11)=(sqrt(12)-sqrt(11))*(sqrt(12)+sqrt(11))/(sqrt(12)+sqrt(11))

 

[Mark44]

i don't want to solve this graphically. (not like i even know how. I need some help here)

Are you saying you don't know what the graph of y = $\sqrt{x}$ looks like?

 

[lalapnt]

Are you saying you don't know what the graph of y = $\sqrt{x}$ looks like?

oh no! i know that! but first, i don't want to solve this graphically even if i did, how does the graph of y = √x help out?

EDIT: if i knew everything in math, i wouldn't be here.

 

[Mark44]

oh no! i know that! but first, i don't want to solve this graphically even if i did, how does the graph of y = √x help out?

EDIT: if i knew everything in math, i wouldn't be here.

The graph of y = √x is one of the first ones you learn when you learn to graph functions. If you are asking questions about square roots, it's one you should know.

Look at the graph of this function. Does the y value on the graph change more between 11 and 12 than it does between 12 and 13, or does it change less between 11 and 12 than it does between 12 and 13?

 

[lalapnt]

The graph of y = √x is one of the first ones you learn when you learn to graph functions. If you are asking questions about square roots, it's one you should know.

Look at the graph of this function. Does the y value on the graph change more between 11 and 12 than it does between 12 and 13, or does it change less between 11 and 12 than it does between 12 and 13?

i get it now. but the problem is. i don't want to solve it geobetrically at all. is there a non-graphical way? please?

 

[dextercioby]

Purely algebra ?

Place ? i/o =, > or <.

sqrt(12)-sqrt(11) ? sqrt(13)-sqrt(12)
2sqrt(12) ? sqrt(11)+sqrt(13)
48 ? ...

Can you continue ?

 

[lalapnt]

Purely algebra ?

Place ? i/o =, > or <.

sqrt(12)-sqrt(11) ? sqrt(13)-sqrt(12)
2sqrt(12) ? sqrt(11)+sqrt(13)
48 ? ...

Can you continue ?

but how is 2sqrt(12) = 48? :/

 

[lalapnt]

i think i get it. please check this image:

 

[lalapnt]

am i correct working it this way?

 

[Mark44]

I doubt that any instructor would accept a proof in which most of the symbols are ?.

 

[Dalek1099]

Think about it like this what is √4-√1=1 obviously what is √7-√4, √9-√4=1 so less than 1, which means that the difference between higher roots is less than lower roots-you could also think that there are more roots to share between 2 numbers higher up eg.√100 to √81, for 9-10 and √4-√1, for 2-1 which shows there is a greater difference between smaller roots.

 

[Bipolarity]

If you want a purely analytical method:

Note that $\sqrt{n}$ is a twice-differentiable function, yielding a second-derivative of $-.25n^{-1.5}$ which is negative for all positive n. Thus the first-derivative of the function decreases monotonically for positive n.

Apply the mean value theorem to the intervals [11,12] and [12,13]. You will get an interesting result which wil give you your answer.

BiP

 

[pierce15]

I doubt that any instructor would accept a proof in which most of the symbols are ?.

You could make it an equation and use the same method... i.e. let root(13)-root(12)+x=root(12)-root(11). If x>0, then the left side is greater. If x<0, the right side is greater

 

[micromass]

You could make it an equation and use the same method... i.e. let root(13)-root(12)+x=root(12)-root(11). If x>0, then the left side is greater. If x<0, the right side is greater

If the left side is greater, then you can't write an = between the sides.

Am I understanding you wrong?

 

[pierce15]

If the left side is greater, then you can't write an = between the sides.

Am I understanding you wrong?

let x be the difference between the two. if, in the above scenario, x>0, then the left side must be less than the right, and vice versa.

 

[Mark44]

You could make it an equation and use the same method... i.e. let root(13)-root(12)+x=root(12)-root(11). If x>0, then the left side is greater. If x<0, the right side is greater

And then the problem becomes determining the sign of x.

 

[tahayassen]

And then the problem becomes determining the sign of x.

Why's that?

Edit: Never-mind. I see why. It's because you square both sides, so when you solve for x, you get |x|=some number.

I would go with Bipolarity's method if you need a proof.