Juan used the expression 16 – 9 – 12 + 22 to find his profit for days 2 and 3. He rewrote the expression as 16 + (–9) + (–12) + 22.

Question

Juan can use the associative and commutative properties to rewrite the expression again. Explain why he had to use the additive inverse before he could use these properties.

Answer 1

If Juan uses the associative property (improperly) without using additive inverses to replace the minus signs, he may end up with, for example

16 - 9 - (12 + 22),

which is not equal to the original expression, because 16 - 9 - 12 + 22 = 17, whereas 16 - 9 - (12 + 22) = -27

In fact, the associative property is never meant to be used unless any and all minus signs present are first turned into additive inverses.

If Juan uses the commutative property (improperly) without using additive inverses to replace the minus signs, he may end up with, for example

16 - 22 - 12 + 9,

which is not equal to the original expression, because 16 - 9 - 12 + 22 = 17, whereas 16 - 22 - 12 + 9 = -9

In fact, the commutative property is never meant to be used unless any and all minus signs present are first turned into additive inverses.