Why is the common denominator 8x(x - 1.5) and not 8x(2x-3) ?

Question

In the equation: 4/8x -(2/2x-3) = 1/x

Why is the common denominator 8x(x - 1.5) and not 8x(2x-3) ?

If i list the factors of the denominators i get this :

8x = 2*2*2*x

x = x

(2x-3) = (2x-3)

Then the common denominator should be 8x(2x-3) since (2x-3) has no common factors. What am I missing here?

Answer 1

If you had two fractions, 1/8x(x-1.5) and 1/8x(2x-3)

and wanted to add them, combine them into one fraction, you'd multiply the both numerator and denominator by 8x(2x-3) the least common multiple to get

[8x(2x-3)+ 8x(x-1.5)]/8x(2x-3) =

(16x^2 -24x + 8x^2 + 12x)/8x(2x-3)=

(24x^2 -12x)/(16x^2-24x) = (6x-3)(4x-6) = (3/2)(2x-1)/(2x-3)

where 8x(2x-3) = 16^2x -24x is the least common multiple

But you didn't say "least common multiple", LCM, you just said common denominator, CD, not LCM

factor 8x(x-1.5) into 2^3(x)(x-1.5) and factor 8x(2x-3) into 2^4(x)(x-1.5)

the common factors between the two are 2^3 and x and (x-1.5), multiplied together = 8x(x-1.5)

maybe that explains what they're looking for

BUT with 4/8x - 2/(2x-3) = 1/x

the only common factor among the 3 denominators is x

and the LCM is 8x(2x-3)

is there more to the problem to indicate what they're getting at?

Answer 2

2x - 3 = 2(x - 1.5)

The 2 is a common factor with the 2 in the 8.