audio read-through Renaming Fractional Expressions

Need some basic practice renaming fractional expressions first?   Practice with the Form $\,a\cdot\frac{b}{c}$ and More Practice with the Form $\,a\cdot\frac{b}{c}$

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:

$$ \begin{align} \cssId{s10}{-\frac{5x}{-3}}\ \ &\cssId{s11}{\overset{\text{first pass, determine plus/minus sign:}}{ \overset{\text{even # of negative factors, so positive}}{\overbrace{\strut\ \ \ =\ \ \ }}}} \cssId{s12}{\ \ \frac{5x}{3}\ \ }\cr\cr &\qquad\ \cssId{s13}{\overset{\text{‘peel off’ the coefficient}}{ \overset{\text{and write it in front}}{\overbrace{\strut\ \ \ =\ \ \ }}}} \ \ \cssId{s14}{ \underset{k}{\underbrace{\ \frac53\ }} x} \end{align} $$

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: $\displaystyle -\frac{-x}{-4} = -\frac{1}{4}x$

Concept Practice

Rename in the form $\,kx\,$: