(X-2)/3+(X-2)/4=7/24

Question

PLEASE SHOW ME THE STEPS TO SOLVE THIS EQUATION. I KNOW THE ANSWER IS X=5/2

Answer 1

The first thing is to do is find the Least Common Denominator (LCD) of the three fractions.

We find the LCD by finding the Least Common Multiple(LCM) of the Denominators, 3, 4, and 24. The LCM is the smallest multiple other than 0 that a set of numbers share in common.

To find the LCM just do a prime factorization of the numbers, and see what multiples you can add to make them all the same number. This works because every counting number is either prime number or product of primes numbers.

3 is already prime so it cannot be factored anymore.
4 factos into 2x2.
24 factors into 2x12 which factors into 2x2x6 which factors into 2x2x2x3.
This shows that 24 is a multiple of 3, since 3 is in its prime factorization, and that 24 is also a multiple of 4 since the prime factorization of 4 (2x2) is also in the prime factorization of 24. This means 24 is the LCM of all three numbers, and hence it is the LCD.

Now that we have the LCD we multiply all fractions accross the equation by 24,

24* ((x-2)/3) + 24*((x-2)/4) = 24*(7/24)
3 divides 24 eight times, 4 divides 24 six times, and 24 divides 24 once.
In other words the multiplying by 24 we are multiplying to each fraction can be divided by their respective denominators first.

We are left with:

8(x-2) + 6(x-2) = 7

Next we distribute,

8x - 16 + 6x - 12 = 7

add like terms,

14x - 28 = 7

add 28 on both sides

14x = 35

x = 35/14 = 5/2

and that is how you solve rational equations.... Remember, always, always look for the LCD, some denominators will come as expressions, but you may still factor the expressions to creat an LCM.

Answer 2

If someone tells you that the answer is x=5/2 then they should be able to show you how to get that answer.  My answer is different.

(x-2)/3 + (x-2)/4 = 7/24  I approach this by expressing everything as 24ths.  Thus I must multiply the first term by 8/8 and the second term by 6/6.  Note that 8/8 = 1  and 6/6 = 1

{(x-2)/3} {8/8} + {(x-2)/4} {6/6} = 7/24

{8(x-2)}/(3)(8) + {6(x-2)}/(4)(6) = 7/24

{8x-16}/24 + {6x-12}/24 = 7/24  now we are adding 24ths and the numerators must add

8x-16+6x-12 = 7

14x -28 = 7  add 28 to both sides

14x = 35  divide both sides by 14

x = 35/14  divide top and bottom by 7

x = 5/2