Question

example 3 identify equivalent expressions which of the expressions below are equivalent to $-\frac{2}{3}x - 2$? $-\frac{2}{3}x + (-2)$ $2 - \frac{2}{3}x$ $-x + (\frac{1}{3}x + (-2))$ $= -\frac{2}{3}x - 2$ subtract the additive inverse. $= -\frac{2}{3}x + 2$ use the commutative property. $= (-x + \frac{1}{3}x) + (-2)$ use the associative property. $= -\frac{2}{3}x + (-2)$ the expression is equivalent to $-\frac{2}{3}x - 2$. the expression is not equivalent to $-\frac{2}{3}x - 2$. the expression is equivalent to $-\frac{2}{3}x - 2$. try it! write two expressions that are equivalent to $-\frac{5}{4}x - \frac{3}{4}$.

Explanation:

Step1: Use Additive Inverse

We know that subtracting a number is the same as adding its additive inverse. So, \(-\frac{3}{4}\) can be written as \(+(-\frac{3}{4})\). Thus, \(-\frac{5}{4}x - \frac{3}{4}=-\frac{5}{4}x+(-\frac{3}{4})\).

Step2: Use Associative/Combine Like Terms

We can also try to combine like - terms in a different way. Let's rewrite \(-\frac{5}{4}x\) as \(-x-\frac{1}{4}x\). Then \(-\frac{5}{4}x - \frac{3}{4}=(-x-\frac{1}{4}x)-\frac{3}{4}=(-x)+(-\frac{1}{4}x-\frac{3}{4})\) (using the associative property of addition). Another way: We can factor out \(-\frac{1}{4}\) from the two terms. \(-\frac{5}{4}x-\frac{3}{4}=-\frac{1}{4}(5x + 3)\) (since \(-\frac{5}{4}x=-\frac{1}{4}\times5x\) and \(-\frac{3}{4}=-\frac{1}{4}\times3\)).

Answer:

Two equivalent expressions are \(-\frac{5}{4}x+(-\frac{3}{4})\) and \(-\frac{1}{4}(5x + 3)\) (other valid expressions are also possible, for example \((-\frac{5}{4}x)-\frac{3}{4}\) or \(-x-\frac{1}{4}x-\frac{3}{4}\))