# How do you simplify and restricted value of ((a - 4) (a + 3)) / (a^2 - a) div( 2 (a - 4)) /( (a + 3) (a - 1))?

Jul 3, 2017

See a solution process below:

#### Explanation:

First rewrite the denominator of the fraction on the left as:

$\frac{\left(a - 4\right) \left(a + 3\right)}{a \left(a - 1\right)} \div \frac{2 \left(a - 4\right)}{\left(a + 3\right) \left(a - 1\right)}$

Next, rewrite the expression as:

$\frac{\frac{\left(a - 4\right) \left(a + 3\right)}{a \left(a - 1\right)}}{\frac{2 \left(a - 4\right)}{\left(a + 3\right) \left(a - 1\right)}}$

Now, use this rule for dividing fractions:

$\frac{\frac{\textcolor{red}{a}}{\textcolor{b l u e}{b}}}{\frac{\textcolor{g r e e n}{c}}{\textcolor{p u r p \le}{d}}} = \frac{\textcolor{red}{a} \times \textcolor{p u r p \le}{d}}{\textcolor{b l u e}{b} \times \textcolor{g r e e n}{c}}$

(color(red)((a - 4)(a + 3))/color(blue)(a(a - 1)))/(color(green)(2(a - 4))/color(purple)((a + 3)(a - 1))) = (color(red)((a - 4)(a + 3)) xx color(purple)((a + 3)(a - 1)))/(color(blue)(a(a - 1)) xx color(green)(2(a - 4)))

Next, cancel common terms in the numerator and denominator:

$\frac{\textcolor{red}{\cancel{\left(a - 4\right)} \left(a + 3\right)} \times \textcolor{p u r p \le}{\left(a + 3\right) \cancel{\left(a - 1\right)}}}{\textcolor{b l u e}{a \cancel{\left(a - 1\right)}} \times \textcolor{g r e e n}{2 \cancel{\left(a - 4\right)}}} =$

${\left(a + 3\right)}^{2} / \left(2 a\right) =$

$\frac{{a}^{2} + 6 a + 9}{2 a}$

Excluded values for:

$a \left(a - 1\right)$ are: $a \ne 0$ and $a \ne 1$

$\left(a + 3\right) \left(a - 1\right)$ are: $a \ne - 3$ and $a \ne 1$

$2 a$ are $a \ne 0$

Therefore the excluded values are: $0 , 1 , - 3$