Equivalent Expressions

Task

Which of the following expressions are equivalent? Why? If an expression has no match, write 2 equivalent expressions to match it.

1. $2(x+4)$
2. $8+2x$
3. $2x+4$
4. $3(x+4)-(4+x)$
5. $x+4$

IM Commentary

In this problem we have to transform expressions using the commutative, associative, and distributive properties to decide which expressions are equivalent. Common mistakes are addressed, such as not distributing the 2 correctly. This task also addresses 6.EE.3.

Solution

First, we notice that the expressions in (a) and (d) can be rewritten so that they do not contain parentheses.

• To rewrite (a), we can use the distributive property: $2(x + 4) = 2x + 8$

• To rewrite (d):

  $$ \begin{alignat}{2} 3(x+4)-(4+x) &= 3(x+4) - (x+4) &\qquad & \text{using the commutative property of addition}\\ &= (3-1)(x+4) &\qquad&\text{using the distributive property}\\ &= 2(x+4) &\qquad & \\ &=2x+8 &\qquad &\text{using the distributive property again} \end{alignat} $$

So the five expressions are equivalent to:

1. $2x + 8$
2. $8+2x$
3. $2x+4$
4. $2x + 8$
5. $x+4$

Right away we see that expressions (a), (b), and (d) are equivalent, since addition is commutative: $2x + 8 = 8 + 2x$. We now only have to check (c) and (e). If $x=0$, (c) and (e) have the value 4, whereas (a), (b), and (d) have the value 8. So (c) and (e) are not equivalent to (a), (b), and (d). Moreover, if $x=1$, then (c) has the value 6 and (e) has the value 5, so those two expressions are not equivalent either.

For (c), we can use the distributive property and decompose one of the numbers to write two equivalent expressions:

Many other equivalent expressions are possible.

For (e), we can add and subtract the same number or the same term to write two equivalent expressions:

Many other equivalent expressions are possible.