Inverse Properties

Mentor: We have talked about the basics of solving linear equations by getting the "x" by itself on one side of the equals sign, but we need to learn some terminology related to solving equations. How would you begin this problem?

Student 1: Since we want the x to be alone, I would divide both sides by 8.

Mentor: Very good. Can anyone think of a number that you could multiply both sides by and still have the x be alone?

Student 1: What? You want to multiply something by 8 and then get 1? Doesn't multiplying always make a number bigger?

Student 2: Does it have to be a whole number or integer?

Mentor: No, it doesn't. What are you thinking about?

Student 2: Well, isn't multiplying by 1/8 the same thing as dividing by 8?

Mentor: Very good. 1/8 has a special name in this problem, it is the multiplicative inverse. What do you think of when you hear the word "inverse"?

Student 1: Opposite.

Mentor: Very good. The multiplicative inverse is a special kind of "opposite." A number multiplied by its multiplicative inverse will always result in a product of one.

Student 1: And 8 multiplied by 1/8 is one!

Mentor: Right. So when we divide by 8 or multiply by 1/8 in this problem, we are using the multiplicative inverse property.

Mentor: Remember that each number will have a unique multiplicative inverse and different numbers will have different multiplicative inverses. What is the multiplicative inverse of -2?

Student 1: A number that when multiplied by -2 equals 1.

Student 2: Yeah, but isn't that number -1/2?

Mentor: Very good.

Student 2: What about the problems where one side is "x plus a number"? Are you using the multiplicative inverse?

Mentor: We aren't using the multiplicative inverse, but we are using the additive inverse in that type of problem. Can you make a hypothesis about the definition of an "additive inverse"?

Student 1: Since a number times its multiplicative inverse is always one, maybe it is a number that when added to another number equals one.

Mentor: Do we want our sums to equal one? Do you want to reduce a problem to 8 = x + 1?

Student 1: Oh no. We want it to be zero. So it is the number that you add to get 0.

Mentor: Exactly, the sum of a number and its additive inverse is always 0. Remember that an additive inverse for one number is unique to that number, so different numbers will have different additive inverses. What is the additive inverse in this problem?

Student 2: Negative five because 5 plus -5 equals 0.

Mentor: Very good. What is the additive inverse of -3?

Student 1: Well, I think you just change the sign, so it must be 3.

Mentor: Excellent. When you add or subtract a number to get zero you are using the additive inverse. Always remember that the additive and multiplicative inverses are specific to the problem that you are working on.