Question

Is the following true for all positive values of a and b: If a>b then a^2>b^2

Answer 1

Yes

Explanation:

If
`a>b`, then
`a-b>0`.

If
`a,b` are positive, then
`a+b>0`.

So if I multiply both sides of
`a-b>0` by
`(a+b)` it will not effect the direction of the inequality since it is of positive value.

This actions gives us
`(a+b)(a-b)>0`.

Using foil we can multiply the left hand side out resulting in:

`a^2-ba+ab-b^2>0`

`a^2-b^2>0`

Add
`b^2` on both sides:

`a^2>b^2`.

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Also consider the function
`f(x)=x^2`. This function increases after
`x` is 0. This means as you increase the value for
`x` the value for
`x^2` increases for positive value inputs of
`x`.