Now that you can make equivalent expressions, how do you decide whether two given expressions are equivalent?
Key idea
Two expressions are equivalent when one can be rewritten into the other with properties (distributive, commutative, combining like terms). Then they are equal for every value of the variable. One value where they differ proves they are not equivalent.
Why it works
Properties of numbers are always true, so if properties turn one expression into the other, the two must agree for every number you put in. Testing values works in one direction only. A single mismatch settles it: not equivalent. But a single match proves nothing, because two different expressions can happen to agree at one value.
| $x$ | $x + 3$ | $4x$ |
|---|---|---|
| 1 | 4 | 4 |
| 2 | 5 | 8 |
| 3 | 6 | 12 |
Worked example: Use a property
Is $2(x + 4)$ equivalent to $2x + 8$?
- Distribute: $2(x + 4) = 2x + 2 \times 4 = 2x + 8$.
Answer: Yes: they are equivalent.
Worked example: One matching value is not enough
When $x = 1$, both $x + 3$ and $4x$ equal 4. Are they equivalent?
- Try another value: $x = 2$ gives $5$ and $8$.
- They differ, so they are not equal for every $x$.
Answer: No.
Worked example: Sort the choices
Which are equivalent to $6x + 12$: $6(x + 2)$, $3(2x + 4)$, $6(x + 12)$?
- $6(x + 2) = 6x + 12$. Yes.
- $3(2x + 4) = 6x + 12$. Yes.
- $6(x + 12) = 6x + 72$. No.
Answer: $6(x + 2)$ and $3(2x + 4)$
Common mistake
Deciding by checking a single value. Quick check: to show two expressions are equivalent, rewrite one into the other. To show they are not, find one value where they differ. Picking an unusual value like $x = 2$ or $x = 10$ catches more mistakes than $x = 0$ or $x = 1$.