How to simplify expression where x is always in denominator?

[Skippy]

Is there a way to simplify an expression like the following where x is only present in the denominators?

1/x - 2/(4x - 2)?

 

[Steven G]

It doesn't matter where x is. How do you usually add fractions? Use that method. Also I would reduce the 2nd fraction first. According to our guideline you need to show us your work so we know how to help you.

 

[Deleted member 4993]

Is there a way to simplify an expression like the following where x is only present in the denominators?

1/x - 2/(4x - 2)?

Can you simplyfy:

1/5 - 2/13

 

[JeffM]

What do you mean by "simplify"?

[MATH]\dfrac{1}{x} - \dfrac{2}{4x - 2} = \dfrac{1}{x} - \dfrac{2}{2(2x - 1)} = \dfrac{1}{x} - \dfrac{1}{2x - 1}.[/MATH]
That is the only transformation of the expression that is unambiguously simpler. We could turn the expression into a single fraction with a quadratic denominator and a linear numerator, but why is that simpler than fractions with linear denominators?

 

[Skippy]

OK, so normally to add/subtract 2 fractions I would use the LCD.

I tried that but it only made the whole expression more complex.

Like the 1/5 - 2/13 suggestion, I would simply and calculate using a common denominator of 5 x 13.

I can’t see how to do this in this example.

 

[Deleted member 4993]

OK, so normally to add/subtract 2 fractions I would use the LCD.

I tried that but it only made the whole expression more complex.

Like the 1/5 - 2/13 suggestion, I would simply and calculate using a common denominator of 5 x 13.

I can’t see how to do this in this example.

Exactly same here - the LCD is x * (4x - 2)

and continue.....

 

[JeffM]

OK, so normally to add/subtract 2 fractions I would use the LCD.

I tried that but it only made the whole expression more complex.

Like the 1/5 - 2/13 suggestion, I would simply and calculate using a common denominator of 5 x 13.

I can’t see how to do this in this example.

And that exactly is my point. What is simpler is sometimes in the eye of the beholder.

 

[HallsofIvy]

Exactly same here - the LCD is x * (4x - 2)

and continue.....

Better is that the least common divisor is x(2x- 1) since, as Jomo suggested, \(\displaystyle \frac{2}{4x-2}= \frac{1}{2x-1}\)

 

[Skippy]

Exactly same here - the LCD is x * (4x - 2)

and continue.....

Right... so if I use that as the LCD it gives me:

4x -2- 2x
____________
x(4x - 2)

or 2x - 2
_________
x(4x - 2)

which doesn’t look any simpler.

Am I missing something?

 

[Harry_the_cat]

What do you mean by "simplify"?

[MATH]\dfrac{1}{x} - \dfrac{2}{4x - 2} = \dfrac{1}{x} - \dfrac{2}{2(2x - 1)} = \dfrac{1}{x} - \dfrac{1}{2x - 1}.[/MATH]
That is the only transformation of the expression that is unambiguously simpler. We could turn the expression into a single fraction with a quadratic denominator and a linear numerator, but why is that simpler than fractions with linear denominators?

I think when rational expressions (fractions) are involved, "simplify" means to express with only one vinculum and no common factors.

 

[JeffM]

I think when rational expressions (fractions) are involved, "simplify" means to express with only one vinculum and no common factors.

Almost certainly you are correct, but it is a usage that makes no sense to a student. who likely does not know (and does not need to know) the word "vinculum." The problem would be clearer if it said "For each expression below, find a single fraction that is equivalent in value and simplify that fraction if possible.