Manipulating Rational Expressions

We'll multiply the rational expression by

x

.

\frac{x + 8}{x}

\frac{a}{b} = \frac{a \cdot x}{b \cdot x}

\frac{( x + 8 ) \cdot x}{x \cdot x}

DistrDistribute

x

\frac{x \cdot x + 8 \cdot x}{x \cdot x}

\frac{x ^{2} + 8 x}{x ^{2}}

We have squared

x

in both the numerator and denominator. These terms can never be negative but how about the term

8 x ?

Yes, if

x

is negative. Thus, if

x^{2} - 8 x < 0

the whole expression will be negative. Lets' try with the smallest negative integer,

x = - 1 .

\frac{x ^{2} + 8 x}{x ^{2}}

x = - 1

\frac{( - 1 ) ^{2} + 8 ( - 1 )}{( - 1 ) ^{2}}

Simplify power and product

\frac{1 - 8}{1}

SubTermSubtract term

\frac{- 7}{1}

\frac{a}{1} = a

- 7

Therefore, Ron-Jon is incorrect as we found a negative value.

Manipulating Rational Expressions 1.21 - Solution