MathRevolution wrote:

If x>0, y<0, and a>b>0, then which of the following is (are) positive?

Ⅰ. ax+by Ⅱ. ax-by Ⅲ. by-ax

A. Ⅰonly

B. Ⅱ only

C. Ⅲ only

D.Ⅰ& Ⅱ only

E. Ⅱ & Ⅲ only

x, a, and b are positive

y is negative

a > b

Let P be the final product of two terms

Let Q be the final product of the other two terms

I. ax+by[(pos)(pos)] + [(pos)(neg)] =

[pos] + [neg] =

[P] + [-Q] =

P - Q: could be positive or negative

We don't have enough information about values.

If |P| > |Q|, it's positive.

If |P| < |Q|, it's negative.

a = 2, x = 2, b = 1, y = -1 -->

(4) + (-1) => 4 - 1 = 3. Positive.

a = 2, x = 2, b = 1, y = -10 -->

(4) + (-10) => 4 - 10 = -6. Negative. REJECT

II. ax-by[(pos)(pos)] - [(pos)(neg)] =

[pos] - [neg] => (pos) + (pos)

[P] - [-Q] =>

P + Q ---> ALWAYS positive

Subtracting a negative means addition, which always increases the value of the number to which it is being added (and the first number here is already positive).

a = 2, x = 2, b = 1, y = -10

(4) - (-10) => 4 + 10 = 14. KEEP

III. by-ax [(pos)(neg)] - [(pos)(pos)] =

[neg] - [pos] =

(-Q) - (P) =>

-Q - P = ALWAYS negative

Subtracting a positive number from ANY other number always decreases the other number's value (and here, original number is already negative) .

a = 2, x = 1, b = 1, y = -1

(-1) - (2) => -1 - 2 = -3 Negative. REJECT

Only II must be positive. Answer B