Question

Simplify the expression. $$\frac{2(x+2)}{5(x-3)} \div \frac{4(x-2)}{5 x-15}$$

Short answer

The simplified expression is \(\frac{x^2 - 13x + 30}{4(x-3)(x-2)}\).

Step-by-step solution

Rewrite Division as Multiplication

When dealing with fractions, dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator. So the expression \(\frac{2(x+2)}{5(x-3)} \div \frac{4(x-2)}{5(x-15)}\) can be rewritten as \(\frac{2(x+2)}{5(x-3)} * \frac{5(x-15)}{4(x-2)}\).

Simplify the Expression

Find factors that are common to both a numerator and a denominator. They can be reduced (divided by the common factor) as they are all multiplication. In this expression, 2 and 4 have a common factor of 2, and \(5(x-15)\) and \(5(x-3)\) have a common factor of 5. Reducing these gives \(\frac{(x+2)}{2(x-3)} * \frac{(x-15)}{2(x-2)}\).

Multiply the Fractions

Multiplication of fractions is straightforward, as the numerators are multiplied together and the denominators are multiplied together. We have \(\frac{(x+2)(x-15)}{4(x-3)(x-2)}\).

Simplify the Result

Although it is not required for the problem, the expression can be simplified further for neatness. Expanding the multiplication in the numerator results in \(x^2 - 13x + 30\). The denominator remains as it is. So, the simplified form is \(\frac{x^2 - 13x + 30}{4(x-3)(x-2)}\).