EDGE Math 8

Example 2

Divide and then simplify if possible.

a.

\frac{x - 2}{x + 3} \div \frac{x + 2}{2 x + 6}

b.

\frac{5 m - 1}{m + 4} \div \frac{4 m}{m^{2} + m - 12}

Solution

a.

\frac{x - 2}{x + 3} \div \frac{x + 2}{2 x + 6}

= \frac{x - 2}{x + 3} \cdot \frac{2 x + 6}{x + 2}

= \frac{(x - 2) (2 x + 6)}{(x + 3) (x + 2)}

Factor $2 x + 6$ in the numerator. Then cancel common factors.

\frac{(x - 2) (2 x + 6)}{(x + 3) (x + 2)} = \frac{(x - 2) (2) (x + 3)}{(x + 3) (x + 2)}

= \frac{2 (x - 2)}{x + 2}

Thus,

\frac{x - 2}{x + 3} \div \frac{x + 2}{2 x + 6} = \frac{2 (x - 2)}{x + 2}

or

\frac{2 x - 4}{x + 2} .

b.

\frac{5 m - 1}{m + 4} \div \frac{4 m}{m^{2} + m - 12}

= \frac{5 m - 1}{m + 4} \cdot \frac{m^{2} + m - 12}{4 m}

= \frac{(5 m - 1) (m^{2} + m - 12)}{(m + 4) (4 m)}

Factor $m^{2} + m - 12$ in the numerator. Then cancel common factors.

\frac{(5 m - 1) (m^{2} + m - 12)}{(m + 4) (4 m)}

= \frac{(5 m - 1) (m + 4) (m - 3)}{(m + 4) (4 m)}

= \frac{(5 m - 1) (m - 3)}{4 m}

Thus,

\frac{5 m - 1}{m + 4} \div \frac{4 m}{m^{2} + m - 12}

= \frac{(5 m - 1) (m - 3)}{4 m}

or

\frac{5 m^{2} - 16 m + 3}{4 m} .