Is the LCD of x/2x-4 and 5x/12x-24 not 24(x-2)? The original question is x/2x-4 + 5x/12x-24 and I factored the denominators first so that I could find the LCD. And I got:

Question

Is the LCD of x/2x-4 and 5x/12x-24 not 24(x-2)? The original question is x/2x-4 + 5x/12x-24 and I factored the denominators first so that I could find the LCD. And I got:

x/2(x-2) + 5x/12(x-2)

After that, I multiplied up to the LCD and got:

x/2(x-2) + 5x/12(x-2)
= x/2(x-2)*(12/12) + 5x/12(x-2)*(2/2)
= 12x/24(x-2) + 10x/24(x-2)
= 22x/24(x-2)

What did I do wrong?

Answer 1

2x-4 = 2(x-2)

12x-24 = 12(x-2)

So, the LCD is 12(x-2), not 24(x-2)

(6x+5x) / 12(x-2)
11x/12(x-2)

This is the same as your answer, if you cancel a factor of 2 which you did not need in the LCD.

Answer 2

I don't think I exactly understand how to figure out the LCD. So how do you know that the LCD is 12(x-2) not 24(x-2)?

Answer 3

To find the least common denominator (LCD) for the expression x/2x-4 + 5x/12x-24, you correctly factored the denominators as 2(x-2) and 12(x-2). However, you made a mistake when you multiplied up to the LCD.

Let's go through the steps again using the correct method:

1. Start with the original expression: x/2x-4 + 5x/12x-24

2. Factor the denominators: 2(x-2) and 12(x-2)

3. Find the LCD by taking the least common multiple (LCM) of the denominators, which is the product of the highest power of each prime factor. In this case, the highest power of 2 is 2, and the highest power of x-2 is 1. So the LCD is 2(x-2).

4. Rewrite the expression using the correct LCD: x/2(x-2) + 5x/12(x-2)

5. To add the fractions, they need to have the same denominator. Since we have the LCD, we can now proceed to multiply each fraction by appropriate terms to obtain the common denominator.

For the first fraction, multiply the numerator and denominator by 12:
x/2(x-2) * (12/12) = 12x/24(x-2)

For the second fraction, multiply the numerator and denominator by 2:
5x/12(x-2) * (2/2) = 10x/24(x-2)

Now that both fractions have a common denominator, we can simplify and add them:

12x/24(x-2) + 10x/24(x-2)

Here, we can see that both fractions already have the same denominator, so we can simply combine the numerators:

(12x + 10x) / 24(x-2) = 22x / 24(x-2)

So, the correct answer is 22x / 24(x-2).