Wize High School Grade 11 Math Textbook > Rational Functions

Adding and Subtracting Rational Expressions
Rational expressions are quotients of polynomials, and they look like fractions. So, we can add and subtract rational expressions the same way we add and subtract fractions! 
First, we find a common denominator
Then we add or subtract the numerators only, keeping the denominator as the common denominator

Example
Simplify the expression 2x+3x2−1+x+3x+1\dfrac{2x+3}{x^2-1}+\dfrac{x+3}{x+1}x2−12x+3​+x+1x+3​

First, we factor fully:

=2x+3(x+1)(x−1)+x+3x+1=\dfrac{2x+3}{(x+1)(x-1)}+\dfrac{x+3}{x+1}=(x+1)(x−1)2x+3​+x+1x+3​

Then, we find a common denominator: (x+1)(x−1)(x+1)(x-1)(x+1)(x−1)
=2x+3(x+1)(x−1)+x+3x+1×x−1x−1=\dfrac{2x+3}{(x+1)(x-1)}+\dfrac{x+3}{x+1}\colorTwo{\times\dfrac{x-1}{x-1}}=(x+1)(x−1)2x+3​+x+1x+3​×x−1x−1​

=2x+3(x+1)(x−1)+(x+3)(x−1)(x+1)(x−1)=\dfrac{2x+3}{(x+1)(x-1)}+\dfrac{(x+3)(x-1)}{(x+1)(x-1)}=(x+1)(x−1)2x+3​+(x+1)(x−1)(x+3)(x−1)​

=2x+3(x+1)(x−1)+x2+2x−3(x+1)(x−1)=\dfrac{2x+3}{(x+1)(x-1)}+\dfrac{x^2+2x-3}{(x+1)(x-1)}=(x+1)(x−1)2x+3​+(x+1)(x−1)x2+2x−3​

=(2x+3)+(x2+2x−3)(x+1)(x−1)=\dfrac{(2x+3)+(x^2+2x-3)}{(x+1)(x-1)}=(x+1)(x−1)(2x+3)+(x2+2x−3)​

=x2+4x(x+1)(x−1)=\dfrac{x^2+4x}{(x+1)(x-1)}=(x+1)(x−1)x2+4x​

=x(x+4)(x+1)(x−1)=\dfrac{x(x+4)}{(x+1)(x-1)}=(x+1)(x−1)x(x+4)​

Restrictions: x≠1,−1x\neq1, -1x=1,−1

 Practice: Adding & Subtracting Rational Expressions
Simplify the expression 3x2x−5−2x−3\dfrac{3x}{2x-5}-\dfrac{2}{x-3}2x−53x​−x−32​ and state all restrictions.

Enter the simplified expression (to enter

\dfrac{a}{b}

, type a/ b)

Check the solutions to see if you got the restrictions right.

I don't know

Practice: Sum & Difference of Functions
If m(x)=13x−4m(x)=\dfrac{1}{3x-4}m(x)=3x−41​ and n(x)=1x+1n(x)=\dfrac{1}{x+1}n(x)=x+11​, determine:

m(x)+n(x)

B. The domain of

m(x)+n(x): x\neq{}

(smaller answer)

C. The domain of

m(x)+n(x): x\neq{}

(larger answer)

m(0)+n(-2)

m(1)+n(1)

I don't know

Practice: Adding & Subtracting Rational Expressions
Simplify 3x2−9+4x−5x2−2x−3\dfrac{3}{x^2-9}+\dfrac{4x-5}{x^2-2x-3}x2−93​+x2−2x−34x−5​ and state all of its restrictions.

Enter the simplified expression (to enter

\dfrac{a}{b}

, type a/ b)

Check the solutions to see if you got the answer right.

I don't know

Practice: Adding & Subtracting Rational Expressions
A train can travel 130x130x130x km in x+1x+1x+1 hours. A car can travel 50+20x50+20x50+20x km in x−2x-2x−2 hours.

a) Write an expression to represent the speed of the train.

b) Write an expression to represent the speed of the car.

c) Write an expression to represent the average speed of the train and the car.

a) Enter the speed of the train as a rational expression (to enter

\frac{a}{b}

, type a / b)

b) Enter the speed of the car as a rational expression (to enter

\frac{a}{b}

, type a / b)

c) Enter the average speed as a rational expression (to enter

\frac{a}{b}

, type a / b)

I don't know