Problem 53 - College Algebra Essentials

Question

Add or subtract as indicated. $$\frac{3 x}{x^{2}+3 x-10}-\frac{2 x}{x^{2}+x-6}$$

Step-by-Step Solution

Answer

The result is \(\frac{x(x-1)}{(x-2)(x+5)(x+3)}\)

Step 1: Factorise the denominators

The first step is to factorise the denominator. \((x^{2}+3x-10)\) factors to \((x-2)(x+5)\), and \((x^{2}+x-6)\) factors to \((x+3)(x-2)\). So, the fractions become \(\frac{3x}{(x-2)(x+5)} - \frac{2x}{(x+3)(x-2)}\).

Step 2: Find the least common denominator (LCD)

The LCD is derived as the product of all the factors of each denominator of the fractions. In this case, the different factors are \((x-2),(x+5),and (x+3)\). Therefore, the LCD is \((x-2)(x+5)(x+3)\).

Step 3: Convert each fraction to the LCD

Transform each term of the expression to have this LCD by multiplying the numerator and denominator by missing factors. We get \(\frac{3x(x+3)}{(x-2)(x+5)(x+3)} - \frac{2x(x+5)}{(x-2)(x+5)(x+3)}\).

Step 4: Perform the subtraction

Now, subtract the numerators: \(\frac{3x^2 + 9x - 2x^2 - 10x}{(x-2)(x+5)(x+3)}\). Simplify the above expression to obtain \(\frac{x^2-x}{(x-2)(x+5)(x+3)}\)..

Step 5: Simplify the result

We can further factorize the expression above by taking an x-factor common in the numerator to get \(\frac{x(x-1)}{(x-2)(x+5)(x+3)}\), which is the simplified form of the expression.