5.2 Adding and subtracting rational expressions
In previous chapter on Number system and radicals, we learned about the different systems of numbers, one of the numerous number systems is called Rational numbers and these are defined as any number that can be expressed as a fraction p/q where q will never be equal to zero. In this chapter, instead of discussing rational numbers, we will discuss Rational Expression. From the definition we have for rational numbers, we would deduce that a rational expression is simply a ratio of polynomial expressions.
Like any other expressions, Rational Expressions can also be simplified and be combined through the four operations. We all know that in solving mathematical problems and equations, the more simplified the expressions are, the easier for us to get the answers correctly, so it is crucial for us to understand how to simplify them. We will learn how to simplify rational expressions by applying our knowledge of prime factorization and factoring polynomial expressions. The aim in simplifying rational expressions is simply to reduce both the numerator and denominator as much as possible.
This chapter has two parts. In the first part of this chapter we will look at how to simplify rational expressions and also be able to look into the restriction in simplifying them. If you would review the definition of rational numbers, one of the criteria that needs to be satisfied is that the denominator should not be equal to zero. This is also true for Rational expressions, which is why there are nonpermissible values. Nonpermissible values are numbers that would make the denominator be equal to zero. For the last part of the chapter, we will learn how to add and subtract rational expressions.
Adding and subtracting rational expressions
Lessons

2.
Simplify:

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

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


3.
Simplify:

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


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


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

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


6.
State any restrictions on the variables, then simplify:

a)
$\frac{4}{{x  3}}  \frac{{5  x}}{{{x^2}  2x  3}}$

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