﻿ Renaming Fractional Expressions
RENAMING FRACTIONAL EXPRESSIONS

It's often necessary to take a somewhat complicated-looking fraction,
like (say) $\,-\frac{5x}{-3}\,$, and rename it.

One popular name is the form $\,kx\,$:   i.e., a number first, and the variable $\,x\,$ last.
In general, it is efficient to make two ‘passes’ through the expression:
figure out the sign (plus or minus) on the first pass, and the size on the second pass: $$-\frac{5x}{-3}\ \ \overset{\text{first pass, determine plus/minus sign:}}{ \overset{\text{even # of negative factors, so positive}}{\overbrace{\strut\ \ \ =\ \ \ }}} \ \ \frac{5x}{3}\ \ \overset{\text{‘peel off’ the coefficient}}{ \overset{\text{and write it in front}}{\overbrace{\strut\ \ \ =\ \ \ }}} \ \ \underset{k}{\underbrace{\ \frac53\ }} x$$ This exercise gives you practice with this type of renaming.

EXAMPLES:
Question: Rename in the form $\,kx\,$:   $\displaystyle\frac{5x}{-2}$
Solution: $\displaystyle \frac{5x}{-2} = -\frac{5}{2}x$
Question: Rename in the form $\,kx\,$:   $\displaystyle-\frac{-x}{-4}$
Solution: $\displaystyle -\frac{-x}{-4} = -\frac{1}{4}x$
Master the ideas from this section

When you're done practicing, move on to:
Practice with Multiples

CONCEPT QUESTIONS EXERCISE:
 Rename in the form $\,kx\,$: