When adding and subtracting rational expressions, the denominators of the expressions will dictate how we solve the questions. Different denominators in the expressions, for example, common denominators, different monomial/binomial denominators, and denominators with factors in common, will require different treatments. In addition, we need to keep in mind the restrictions on variables.

Lessons

• Introduction

• 1.
Simplify:
a)
$\frac{3}{{13}} + \frac{8}{{13}}$

b)
$\frac{3}{2} + \frac{4}{5}$

• 2.
Simplify:
a)
$\frac{x}{6} + \frac{{2x}}{3} - \frac{{5x}}{4}$

b)
$\frac{{y - 3}}{3} + \frac{{2y + 3}}{6}$

c)
$\frac{{3a - 5}}{3} - \frac{{2a - 1}}{2}$

• 3.
Simplify:
a)
$\frac{{5x - 3}}{9} + 6x - \frac{{3x - 2}}{3}$

b)
$3 - \frac{{y - 1}}{4} - \frac{{4 - 3y}}{6}$

• 4.
Adding and Subtracting with Common Denominators
State any restrictions on the variables, then simplify:
a)
$\frac{3}{x} + \frac{{12}}{x} - \frac{5}{x}$

b)
$\frac{{6a - 2}}{{3a}} + \frac{{ - 10a + 2}}{{3a}}$

c)
$\frac{{6m}}{{6m - 5}} - \frac{5}{{6m - 5}}$

d)
$\frac{{9x - 1}}{{2x - 3}} - \frac{{8 + 3x}}{{2x - 3}}$

• 5.
Adding and Subtracting with Different Monomial Denominators
State any restrictions on the variables, then simplify:
a)
$\frac{3}{{4m}} + \frac{2}{{5m}}$

b)
$\frac{5}{{4x}} - \frac{7}{6}$

c)
$\frac{{2x - 3}}{{10x}} - \frac{{3x - 2}}{{5x}}$

d)
$\frac{{y - 1}}{{3y}} - \frac{2}{{2{y^2}}}$

• 6.
Adding and Subtracting with Different Monomial/Binomial Denominators
State any restrictions on the variables, then simplify:
a)
$\frac{{x - 4}}{{3x}} + \frac{{5x}}{{x - 2}}$

b)
$\frac{5}{{3m + 2}} - \frac{1}{{4m - 7}}$

c)
$\frac{6x-1}{2x+3}-\frac{1-x}{4x+5}$

• 7.
State any restrictions on the variables, then simplify: $\frac{1}{{x + 2}} - \frac{5}{{x - 1}} + \frac{3}{x}$

• 8.
Denominators with Factors in Common
State any restrictions on the variables, then simplify:
a)
$\frac{5}{{4x}} - \frac{5}{{12x}}$

b)
$\frac{4}{{3x + 9}} + \frac{5}{{2x + 6}}$

c)
$\frac{3}{{{x^2} - 5x}} - \frac{8}{{{x^2}}}$

• 9.
Denominators with Factors in Common
State any restrictions on the variables, then simplify: $\frac{5}{{\left( {x - 1} \right)\left( {x + 3} \right)}} + \frac{4}{{\left( {x + 2} \right)\left( {x - 1} \right)}}$

• 10.
State any restrictions on the variables, then simplify: $\frac{x}{{{x^2} - 9}} + \frac{5}{{x - 3}}$

• 11.
State any restrictions on the variables, then simplify:
a)
$\frac{4}{{x - 3}} - \frac{{5 - x}}{{{x^2} - 2x - 3}}$

b)
$\frac{3}{{{a^2} - a - 2}} + \frac{5}{{{a^2} + 3a + 2}}$

c)
$\frac{1}{{{x^2} + 4x + 4}} - \frac{4}{{{x^2} + 5x + 6}}$

• 12.
State any restrictions on the variables, then simplify: $\frac{{{x^2} - 5x + 6}}{{{x^2} - 2x - 3}} - \frac{{{x^2} + 9x + 20}}{{{x^2} + 7x + 10}}$