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Problem 12 (Do We Need The Associative Law)

In this problem, your job is to explain each step in the process of solving $2x=3$ for $x$. Try to break each part of your computation down into the most basic processes. The only two operations we have on the real numbers are addition and multiplication. Subtraction and division are inverse operations, so rather than saying "divide by 2", you would multiply by the multiplicative inverse of 2. As always, write your solution using complete sentences.

  • If you did not use the associative law of multiplication in your work, then go back through your work. It should be there somewhere.
  • Did you use the commutative law of multiplication in your work? If you did, make sure you pinpoint where you did. It is possible to solve this problem without the commutative law.
  • You might try to justify this as follows:
    First divide both sides of $2x=3$ by 2. Then cancel the 2's on the left side of the equation and we have $x=3/2$.
This correctly uses sentences to explain the solution. However, there are several questions left. First, what does divide mean? The key operations defined on the real numbers are multiplication and addition. What does divide mean? Second, what does cancel the 2's mean? What steps go into cancelling the 2's? Carefully explain every step.

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