Answer
$-\dfrac{4(x + 6)}{3(3x + 8)}$
Restrictions: $x \ne -\frac{8}{3}, 3$
Work Step by Step
Factor all expressions in the original exercise:
$\dfrac{2(x + 6)}{3(x - 3)} \cdot \dfrac{2(3- x)}{3x + 8}$
Multiply to simplify:
$\dfrac{2(x + 6) \cdot 2(3- x)}{3(x - 3)(3x + 8)}$
Cancel common factors in the numerator and denominator:
$\dfrac{(-1)(2)(2)(x + 6)}{3(3x + 8)}$
Simplify:
$-\dfrac{4(x + 6)}{3(3x + 8)}$
Restrictions on $x$ occur when the value of $x$ makes the fraction undefined, which means that the denominator becomes $0$.
Set the factors in the denominators equal to $0$ to find restrictions:
First factor:
$x - 3 = 0$
Subtract $3$ from each side of the equation:
$x = 3$
Second factor:
$3 - x = 0$
Subtract $3$ from each side of the equation:
$-x = -3$
Divide each side of the equation by $-1$:
$x = 3$
Third factor:
$3x + 8 = 0$
Subtract $8$ from each side of the equation:
$3x = -8$
Divide both sides by $3$:
$x = -\frac{8}{3}$
Restriction: $x \ne -\frac{8}{3}, 3$
