I'm not sure if you mean 1/[4(x +1/2)] - 2/[3(x - 1/6)] - 1/2 , with the parentheses in the denominators, or (1/4)(x+1/2) - (2/3)(x-1/6) - 1/2, which means the fractions are simply next to the parentheses. I'm going the guess you mean the second one! Also, there's no equal sign here, so you can't solve for x, you can only simplify.
So, let's simplify:
(1/4)(x+1/2) - (2/3)(x-1/6) - 1/2
A lot of students panic in the face of fractions, but you don't need to! If this quesion looked like this:
2(x + 3) - 4(x - 5) - 6
you'd probably know what to do, right? You'd distribute the number in front of the parentheses into each term inside the parentheses. We're going to do the same thing with the fractions... don't forget the negative attached to the 2/3 !:
(1/4)x + (1/4)(1/2) + (-2/3)x - (-2/3)(1/6) - 1/2
... I know that looks messy in text. If you're having trouble, you might try following along on your own paper, just because written down fractions are easier to look at than typed up fractions.
Okay, so now you have to actually multiply those fractions! When you have a fraction times a non-fraction, like (1/4)x, you can just multiply the non-fraction by the top of the fraction, and leave the bottom part alone:
(x/4) + (1/4)(1/2) + (-2x/3) - (-2/3)(1/6) - 1/2
When you have a fraction times another fraction, like the (1/4)(1/2), you simply multiply the top times the top, and the bottom times the bottom - no common denominators needed!
(x/4) + (1/8) + (-2x/3) - (-2/18) - 1/2
At this point, we should check for any fractions we can reduce. This problem only has one: 2/18 can be 1/9:
(x/4) + (1/8) + (-2x/3) - (-1/9) - 1/2
Alright, all that's left is addition and subtraction! Again, if the fractions get too confusing and you're not sure what to do next, think of a non-fraction example:
x + 2 + (-3x) - (-4) - 5
What would you do next for that problem? The same steps apply to the problem with the fractions!
The first thing I'd do it simplify those negative a bit. Remember, plus a negative is the same as subtraction, and minus a negative is the same as addition:
(x/4) + (1/8) - (2x/3) + (1/9) - 1/2
If this were a non-fraction problem, we'd combine like terms and be done! That's exactly what we're going to do... but with fractions it does get a little more complicated to combine like terms. First let's just shuffle our fractions around a little, so the like terms are at least next to each other - don't forget that negatives are attached to the number on the right!:
(x/4) - (2x/3) + (1/8) + (1/9) - (1/2)
Alright, now the hard part.
To combine like terms means we should add all the fractions with an x together, and all the fractions without an x together. Unfortunately, adding fractions involves finding common denominators - don't try to add fractions that don't have common denominators, it doesn't work!
A common denominator of (x/4) and (2x/3) is 12 - I got that by multiplying the bottoms of the two fractions together. So, we want the denominator of each fraction to be 12. To get (x/4) to have a denominator of 12, we have to multiply the top and the bottom by 3 (12 / 4 = 3), so now we have (3x/12). To get (2x/3) to have a denominator of 12, we have to multiply by 4 - don't forget to multiply by the TOP and the bottom! We get (8x/12).
So that's our x terms, ready to add. Leave those for a minute and let's get the non-x terms ready to add. Our non-x terms are (1/8), (1/9), and (1/2). The least common denominator is 72 - I'm not going to go into too much detail on how to find LCD here, because this is getting too long as it is. Suffice to say that 8, 9, and 2 all go into 72, so we can use it. The same was as we did with the x terms, we now convert each fraction to have a denominator of 72, and we end up with (9/72), (8/72), and (36/72).
Now, let's put our converted fractions back in the problem:
(3x/12) - (8x/12) + (9/72) + (8/72) - (36/72)
NOW we can combine like terms! Remember, when you're adding fractions with common denominators, add the tops and leave the bottoms unchanged:
((3x - 8x)/12) + ((9 + 8 - 36)/72)
(-5x/12) + (-19/72)
Check if you can reduce any of your fractions -- in this case, you can't. So, you're done!
