Answer
a. $-\frac{4x}{y}$
Restriction: $x \ne 0; y \ne 0$
b. $\frac{x + 4}{x - 3}$
Restriction: $x \ne 3, 2$
c. $-\frac{4}{x + 3}$
Restriction: $x \ne 3, -3$
Work Step by Step
a. Cancel out common factors in the numerator and denominator:
$\frac{24x}{-6y}$
Simplify the constants in the numerator and denominator by dividing both by their greatest common factor, which is $6$, in this case:
$-\frac{4x}{y}$
Restrictions occur where the denominator becomes undefined, which means the denominator equals $0$. Set the denominators equal to $0$ and solve:
First factor:
$x^3 = 0$
Take the cube root of $0$:
$x = 0$
Second factor:
$y^2 = 0$
Take the square root of $0$:
$y = 0$
Restriction: $x \ne 0; y \ne 0$
b. Factor the expressions in the numerator and denominator:
$\frac{(x + 4)(x - 2)}{(x - 3)(x - 2)}$
Cancel out common terms in the numerator and denominator:
$\frac{x + 4}{x - 3}$
Restrictions occur where the denominator becomes undefined, which means the denominator equals $0$. Set the denominators equal to $0$ and solve:
First factor:
$x - 3 = 0$
Add $3$ to each side of the equation:
$x = 3$
Second factor:
$x - 2 = 0$
Add $2$ to each side of the equation:
$x = 2$
Restriction: $x \ne 3, 2$
c. Factor the expressions in the numerator and denominator:
$\frac{4(3 - x)}{(x - 3)(x + 3)}$
The binomials $3 - x$ and $x - 3$ are reciprocals, so their product is $-1$:
$\frac{(-1)(4)}{x + 3}$
Simplify:
$-\frac{4}{x + 3}$
Restrictions occur where the denominator becomes undefined, which means the denominator equals $0$. Set the denominators equal to $0$ and solve:
First factor:
$x - 3 = 0$
Add $3$ to each side of the equation:
$x = 3$
Second factor:
$x + 3 = 0$
Subtract $3$ from each side of the equation:
$x = -3$
Restriction: $x \ne 3, -3$
