Algebra Question - please help!?

if you think of all three as lines, examine their slope.

2x/3 has a slope of 2/3

x has a slope of 1

2x has a slope of 2

when you look at their slopes 2/3 is closest to 1 so, 2x/3 is closest to x.

It's closer to x. 2x/3 is actually saying 2/3 of x. So it's going to be smaller than x. If x is 3 for example, then 2x would be 6. No 2x/3=(2*3)/3=2. Two would be closer to 3 than 6.

It is closer to x than 2x, as long as x does not equal zero (in which case, they are equally close). Here is why:

2x/3 is the same as 2/3 times x, which is close to x (a lot closer than 2 times x would be).

Take x = 1 as an example:

(2x)/3 = (2*1)/3 = 2/3

2x = 2 * 1 = 2

So, which is closer to 1? 2/3 or 2?

This goes for other numbers as well, try it with 30:

2*30/3 = 20

2*30 = 60

So, which is closer to 30? 20 or 60?

It's closer to x.

Here's how you show this:

x - 2x/3 = x/3;

2x - 2x/3 = 2x(1 - 1/3) = 4x/3.

Thus, the difference in the second case is 4 times the difference in the first case.

So 2x/3 is Four times CLOSER to x than it is to 2x.

QED

Live long and prosper.

Plug in some numbers :

x = 3 : (2*3)/3 = 2

Is 2 closer to 3 or 6??

It's closer to x, because 2x/3 is the same as (2/3) x which is less than x. 2x is more than x. So, it has to be closer to x.

You can show this mathematically by subtracting 2x/3 from x and from 2x. The smaller difference is closer to it.

2x/3 - x

=(2x/3) - (3x/3)

=(2x-3x)/3

-x/3

2x/3 -2 x

=(2x/3) - (6x/3)

=(2x-6x)/3

-4x/3

-x/3 is smaller than -4x/3, so x is closer

For expanding cubic binomials the general formula is as follows: (a + b) ^ 3 = a^3 + 3*a^2*b^1 + 3*a^1*b^2 + b^3 In your case, a is x and b is -y^5 So (x - y^5)^3 = x^3 + 3*x^2*(-y^5)^1 + 3*x^1*(-y^5)^2 + (-y^5)^3 Simplified: =x^3 - 3x^2*y^5 + 3x*y^10 - y^15 :D

closer to x

NOTE:

2x/3 = (2/3) * x

x = (1) * x

2x = (2) * x

.......... the question reduces to the simpler question:

"Is 2/3 closer to 1 or to 2."

since 2/3 < 1 < 2 .... the answer is clear

2x/3 is acctually 2/3rds of x. Therefore it is closer to 2x.