# Adding and subtracting rational expressions

### Adding and subtracting rational expressions

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: $\frac{{{x^2} - 5x + 6}}{{{x^2} - 2x - 3}} - \frac{{{x^2} + 9x + 20}}{{{x^2} + 7x + 10}}$