Question

Simplify the expression. $$\frac{2 x+1}{3 x-1}-\frac{x+4}{x-2}$$

Short answer

The simplified expression is \[ \frac{-x^2+8x-6}{(3x-1)(x-2)} \].

Step-by-step solution

Identify the common denominator

In order to add or subtract fractions, they must have the same denominator. The common denominator can be found by multiplying the two denominators of the given fractions. In this case, the common denominator is \( (3x-1)(x-2) \).

Transform the fractions

Each fraction should be transformed so that it has the common denominator. For the first fraction, it should be multiplied by \(x-2\) both on the top and bottom. For the second fraction, it should be multiplied by \(3x-1\) both on top and bottom. Our expression becomes: \[\frac{(2x+1)(x-2)}{(3x-1)(x-2)} - \frac{(x+4)(3x-1)}{(3x-1)(x-2)} \].

Simplify the fractions

Now carry out the multiplication inside the new numerators and simplify: \[ \frac{2x^2-4x+1x-2}{(3x-1)(x-2)} - \frac{3x^2-x+12x-4}{(3x-1)(x-2)} = \frac{2x^2-3x-2-3x^2+11x-4}{(3x-1)(x-2)} \]. Now, combine like terms inside the numerator: \[ \frac{-x^2+8x-6}{(3x-1)(x-2)} \].

Check if the fraction can be further simplified

In order to see if the fraction can be further simplified, we can check if the numerator and the denominator have any common factors. In this case, they do not, so we have arrived at our simplified fraction: \[ \frac{-x^2+8x-6}{(3x-1)(x-2)} \].