How to add variables

, so .

, so, by definition, , or .

The graph of is a line through the point with coordinates and with slope . The equation of the line can be determined by setting in the slope-intercept form:

.

The equation of the line is , which makes this the definition of . By setting ,

.

Therefore,

If ,

then

.

The absolute value of a negative number is its (positive) opposite, so

Also, if and are both positive, then is positive; the absolute value of a positive number is the number itself, so . Since , it follows that . Therefore,

Since is given to be positive,

.

and

In order to simplify this expression, we will need to add like terms.

There is a lone positive three.

Combine all the terms.

The answer is:

(A) Substitute and evaluate, remembering that by order of operations, squaring precedes addition:

(B) Substitute and evaluate

(B) is greater

Which quantity is the greater depends on the value of .

Case 1:

Then

and

Subsequently, (A) is greater

Case 2:

Then

and

Subsequently, (B) is greater

,

so

and

regardless of the value of .

The expressions and are each other's opposite, so they have the same absolute value. Therefore, regardless of the value of ,

.

Since

,

it follows that

and

regardless of the value of .

By examining two scenarios, we see that we cannot determine which is the greater quantity.

Case 1:

Then

and

This makes (a) the greater quantity.

Case 2:

Then

and

This makes (b) the greater quantity.

, so

and

is the greater quantity, regardless of .