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Using the LCD how do you solve: 4/x-1/x-2=2/x

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3 Answers

Stephanie, in this case you want to first identify the LCD. Because we are dealing with fractions with variables and expressions, we have to multiply the denominators to create the LCD.  So the LCD would be x(x+2).  Now, focusing on one part of the equation at a time, you must multiply the numerator by the same value as the denominator. So, to show you, for the first term, the denominator was x and then we multiplied it by (x+2) to give us x(x+2).  So the numerator must be multiplied by (x+2) as well. When you have the numerators, you can then use the distributive property and order of operations to simplify.

Simply multiply each side by the LCD, thus eliminating the fractions.  The LCD is x(x+2). First, however, combine common terms.  4/x-1/(x+2)=2/x , if we add 1/(x+2) and subtract 2/x on each side, we get 

2/x=1/(x+2).  now, multiply both sides by x(x-2) and we have 2(x+2) = x,  distribute the 2 and we have 

2x +.4 = x,  subtract 4 and subtract x and we get x = - 4.  The check shows that this answer is correct.

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.

 

 

 

 

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