9. sherry is playing the integer game and is given a chance to discard a set of matching cards. sherry determines that if she discards one set of cards, her score will increase by 12. if she discards another set, then her score will decrease by 8. if her matching cards make up all six cards in her hand, what cards are in sherry's hand?

Sherry might have 4 (-3) cards and 2 (+4) cards or 4(+2) cards and 2(-6) cards.

SOLUTION:

Given that, Sherry determines that if she discards one set of cards, her score will increase by 12.  If she discards another set, then her score will decrease by 8.  Her matching cards make up all six cards in her hand,  

We have to find what cards are there in Sherry's hand. Now, in total sherry has 6 cards. And her points increase by 12 if she discards negative point cards.

Now, let us see set of possibilities for 12 with negative cards

12 (-1) cards, 6 (-2) cards, 4 (-3) cards, 3 (-4) cards, 2 (-6) cards

We can discard the two possibilities, as she has only 6 cards and in that 6 cards there are two types, if we choose 6 (-2) cards then other type will become 0 so, we have to discard them.

Now, let us see, set of possibilities for 8 with positive cards

8 (+1) cards, 4 (+2) cards, 2 (+4) cards, 1 (+8) card

Here again we have to discard possibility because of the same reason we discarded in above case.

Now, remaining possibilities are

Negative cards = 4 (-3) cards, 3 (-4) cards, 2 ( -6) cards

Positive cards = 4 (+2) cards, 2 (+4) cards,  1 (+8) card

Now, we have to select one set from negative cards and one set from the positive cards such that number of cards equals to 6.

Then, we again get two possibilities,

4 (-3) cards and 2 (+4) cards or 4(+2) cards and 2(-6) cards.