Equivalent Expressions?

Task

If we multiply $\frac{x}{2} + \frac34$ by 4, we get $2x+3$. Is $2x+3$ an equivalent expression to $\frac{x}{2} + \frac34$?

IM Commentary

The purpose of this task is to directly address a common misconception held by many students who are learning to solve equations. Because a frequent strategy for solving an equation with fractions is to multiply both sides by a common denominator (so all the coefficients are integers), students often forget why this is an "allowable" move in an equation and try to apply the same strategy when they see an expression. Two expressions are equivalent if they have the same value no matter what the value of the variables in them. After learning to transform expressions and equations into equivalent expressions and equations, it is easy to forget the original definition of equivalent expressions and mix up which transformations are allowed for expressions and which are allowed for equations.

Solution

No, $2x + 3$ and $\frac{x}{2} + \frac34$ are not equivalent expressions because they do not yield the same result for most values of $x$. For example, when $x = 1$, we get

$$ 2(1) + 3 = 5 $$

and

$$ \frac{(1)}{2}+\frac34=\frac54 \neq 5 $$

Therefore, they are not equivalent. In fact, the expression $2x + 3$ will be 4 times as big as $\frac{x}{2} + \frac34$ for all values of $x$, since we obtained it by multiplying $\frac{x}{2} + \frac34$ by 4.