Simplifying an expression written as the sum of three fractions

Question

Specifically, I don't know what to do first given the following expression:

$$ \frac{4x - 2}{6} - \frac{2 - x}{4} + \frac{x + 3}{3} $$

So I think of it as $\frac 16(4x-2) - \frac 14(2-x) + \frac 13(x+3)$?

That just gives me more fractions.

Answer 1

Find the common denominator first, noting that the least common multiple of $6, 4, 3$ is $12: $12 = 2\times 6 = 3\times 4 = 4\times 3$. This way, you can write the entire expression as one fraction:

$$\frac{4x - 2}{6} - \frac{2 - x}{4} + \frac{x + 3}{3} = \frac{2(4x-2) - 3(2-x) + 4(x+3)}{12}$$

Now expand the factors in the numerator on the right-hand side, and then simplify (add [or subtract] the factors of $x$, and do the same for the constants. Doing this will give you a final result of the form $\dfrac{ax + b}{12}$, where $a, b$ are nonzero constants.