x = 3/8
Step-by-step explanation:
1. There are probably different ways to solve this, but I personally start by making all the fractions have the same denominator.
First, let's take a look at the denominators:
2 - x, 4 - 2x, x - 2
If we look at the first two, we can see that if we multiply (2 - x) by 2, we get 4 - 2x.
The third one is a little trickier; we need to realise that if we multiply (x - 2) by -2 we will get -2x + 4 = 4 - 2x.
Thus, we will now try to make all the fractions have a denominator of (4 - 2x) by multiplying the first and third fractions by 2 and -2, respectively.
Note that when I say multiply the fractions, I mean multiplying both the numerator and the denominator: Eg. 3/(2 - x) * (2/2) = 6/(4 - 2x)
This is crucial to remember since we are not changing the value of the fraction (as 2/2 = 1, the same with -2/-2 or 100/100) - think of it as writing 3/5 as 6/10 or 9/15.
Ok, now that we have that covered, we can start working with the equation:
3/(2 - x) + 5/(4 - 2x) - 1/(x - 2) = 4
3/(2 - x) * (2/2) + 5/(4 - 2x) - 1/(x - 2) * (-2/-2) = 4
6/(4 - 2x) + 5/(4 - 2x) + 2/(-2x + 4) = 4
(With the third fraction, note that (-1)*(-2) = + 2)
6/(4 - 2x) + 5/(4 - 2x) + 2/(4 - 2x) = 4
2. Now, since the fractions all have the same denominator, we can add their numerators together. Thus, we get:
(6 + 5 + 2)/(4 - 2x) = 4
13/(4 - 2x) = 4
3. Now, we need to isolate x and solve for it. The first step to achieving this is to remove any fractions from the equation; to do this, we can multiply both sides by (4 - 2x):
13/(4 - 2x) * (4 - 2x)/1 = 4 * (4 - 2x)
13(4 - 2x)/(4 - 2x) = 4(4 - 2x)
(Here on the left, since we have the same value in the numerator and denominator, they are cancelled out - think of this as being like (3/2) * 2 = 3)
13 = 16 - 8x (We have expanded on the right)
13 - 16 = 16 - 8x - 16 (Subtract 16 from both sides)
-3 = -8x
-3/-8 = -8x/-8 (Divide both sides by -8)
3/8 = x (A negative value divided by a negative value is a positive one)
Thus, x = 3/8
I understand this question was asked quite a long time ago and you probably have already figured out how to solve it, but I hope someone here on might perhaps still find it useful :)
The key here is to just work through the problem step by step and not get too overwhelmed by the number of fractions - look for relationships between them instead. Also always pay special attention to positive and negative numbers when you are working with them.
