When I look at problems with variables, it is often useful to think about an easier problem without variables first. Consider the following problem: 4/5 - 1/7. We need to determine the LCD. Since both numbers are prime, we can multiply them together to get
the LCD. 5(7) =35. The next step is to rewrite the problem using equivalent fractions. We need to ask, what is the equivalent fraction of 4/5 with a denominator of 35, and what is the equivalent fraction for 1/7 with a denominator of 35?

4/5 is equivalent to 28/35, and 1/7 is equivalent to 5/35. The problem can be rewritten as 28/35 - 5/35 = (28-5)/35 = 23/35.

With this in mind, let's think about your problem. Since both denominators are prime, we can multiply them together to get the LCD, just like we did above.

Your problem was 4/x - 1/x+2 = 2/x.

The LCD is the two different denominators multiplied together x(x+2).

Recall that the next step is to rewrite the problem using equivalent fractions. We need to ask, what is the equivalent fraction for 4/x with a denominator of x(x+2), what is the equivalent fraction for 1/x+2 with a denominator of x(x+2), and what is the
equivalent fraction for 2/x with a denominator of x(x+2)?

4/x has an equivalent fraction of 4(x+2)/x(x+2)

1/x+2 has an equivalent fraction of 1x/x(x+2)

2/x has an equivalent fraction of 2(x+2)/x(x+2)

Now if we rewrite the problem we get: 4(x+2)/x(x+2) - x/x(x+2) = 2(x+2)/x(x+2)

Before we continue, simplify the numerator of the first fraction by distributing.4(x+2) = 4x + 8

We would also like to do the same for the last fraction. 2(x+2) = 2x+4

4x+8/x(x+2) -x/x(x+2) = 2x+4/x(x+2)

We can rewrite the first two fractions as one fraction, since they have the same denominator:

(4x+8 - x)/x(x+2) = 2x+4/x(x+2)

Combine like terms to get:

3x+8/x(x+2) = 2x+4/x(x+2)

Since both sides are being divided by x(x+2), we can multiply each side by x(x+2) so that we are left with just the numerators, (Note: x(x+2)/x(x+2) = 1)

3x+8 = 2x+4

To solve, we need the variables on one side and the constants on yhe other side.

Subtract 2x from both sides to get: x+8 = 4

Next subtract 8 from both sides to get: x = (-4)

With practice, you might notice a short cut way to do these types of problems. (Math is all about patterns!)

The shortcut is to multiply both sides by the LCD.

[x(x+2)] [ 4/x - 1/(x+2)] = [x(x+2)] [2/x]

We then get the following when we multiply:

(4x(x+2))/x - x(x+2)/(x+2) = (2x(x+2))/x

When we simplify we get:

4(x+2) - x = 2(x+2)

We then need to distribute to get:

4x+8-x = 2x+4

3x+8 = 2x+4

Solving for x again would give us x = (-4)

Of course, when you finish solving a problem, ask if the answer is reasonable and check your answer. In this case you can check by subsituting (-4) into the original equation for x.