Combining Rational Fractions

Question

Jim ran a total of x/x^2 +2x+1 miles in the gym and x+2/x+1 miles outside. How many more miles did he run inside?

Answer 1

The key words "how many more" tells us that we have to subtract the rational fractions.

 

He ran a certain amount in the gym, and we need to find how many more miles he ran inside.  Since the gym is inside, we can write  word expression that represents this. This is because he ran more miles inside.

 

miles he ran inside = miles he ran in the gym - miles he ran outside

 

Now, we just plug in the appropriate rational expressions into this logic.

 

(x / (x + 1)(x + 1)) - ((x + 2) / (x + 1))

 

 

The fractions have difference denominators, so we must find the LCD.  The LCD is (x + 1)(x + 1).

 

[x / (x + 1)(x + 1)] - [(x + 2)(x + 1) / (x + 1)(x + 1)] =

 

(x - (x + 2)(x + 1)) / ((x + 1)(x + 1)) =

 

(x - (x² + 3x + 2)) / ((x + 1)(x + 1)) =

 

(x - x² - 3x - 2) / (x + 1)(x + 1) =

 

(-x² - 2x - 2) / (x + 1)(x + 1) =

 

(-x² - 2x - 2) / (x² + 2x + 1)

 

 

He ran  (-x² - 2x - 2) / (x² + 2x + 1) more miles inside.