Question

How do you simplify `\frac { \frac { x } { 10} - \frac { 5} { x } } { \frac { 1} { 5} + \frac { 1} { x } }`?

Answer 1

Make all sub-fractions have the same denominator; cancel this off. Factor and reduce any remaining terms common to both sides of the overall fraction.

Explanation:

Step 1: Give all terms a common denominator (in this case, `10x`).

`(x/10-5/x)/(1/5+1/x)=(x/10(x/x)-5/x(10/10))/(1/5((2x)/(2x))+1/x(10/10))`

`color(white)((x/10-5/x)/(1/5+1/x))=(x^2/(10x)-50/(10x))/((2x)/(10x)+10/(10x))`

Step 2: Cancel off this common denominator (with the condition that its value cannot be zero).

`color(white)((x/10-5/x)/(1/5+1/x))=(x^2-50)/(2x+10)"   "` (if `x!=0`)

Step 3: Factor if requested/possible.

`color(white)((x/10-5/x)/(1/5+1/x))=(x^2-50)/(2(x+5))`

Step 4: Cancel any common factors (in this case, there are none).

`color(white)((x/10-5/x)/(1/5+1/x))=(x^2-50)/(2(x+5))`