This document discusses rational functions, equations, and inequalities. It defines rational expressions as ratios of polynomials and provides examples. Rational equations use equality symbols with rational expressions. Rational inequalities use inequality symbols. Rational functions are functions where both the numerator and denominator are polynomial functions, not including where the denominator is the zero function. The document provides additional examples comparing rational equations, inequalities, and functions.

1. z
Rational
Functions,
Equations, and
Inequalities

2. z
At the end of the lesson:
 The learner should be able to
distinguish among rational
functions, rational equations, and
rational inequalities.

3. z
Definition
 A rational expression is an expression that
can be written as a ratio of two polynomials.
 Some examples of rational expressions are
2
𝑥
,
𝑥2+2𝑥+3
𝑥+1
, and
5
𝑥−3
.

4. z
Consider the following algebraic expressions, determine
whether they are rational or not.
1.) 3𝑥2
− 5𝑥 + 2 ; Rational expression because it is a ratio
of two polynomials
2.)
2
𝑥−5
; Rational expression because 1 and x – 5 are
polynomials.
3.)
𝑥−4
2𝑥+1
; Not a rational expression since the numerator is
not a polynomial.

5. z
4.) x + 5 ; Rational expression because the numerator
x + 5 and denominator 1 are polynomials.

6. z
Read and Analyze
Call for Applicants for Contract of Service Workers
For the purpose of 2022 census of population and
housing, Philippine Statistics Authority is hiring
enumerators.
 Any male or female
 Age – at least 18 years old and at most 45 years old
 Educational Attainment – at least SHS graduate
 Salary – P700 per day
 Physically fit for work
 Interested applicants must submit application letter
and Personal Data Sheet (PDS) to PSA – Quezon
City.

7. z
The job advertisement shows the need to conduct a
census. Census is important because it will serve as
the basis for planning for the future such as public
safety, infrastructures like hospital and schools and
improving homes in the neighborhood.
This also serves as the basis in predicting the number
of people who need help in times of crisis.

8. z
However, in finding enumerators who will conduct the
census, there are different qualifications needed and
some of those are the age requirement and
educational attainment, to represent it mathematically
the use of inequality symbol was employed.

9. z
However, the salary was also mentioned and to be
able to represent it, equality symbol is used.
At this lesson, we will compare rational expressions
using equality and inequality symbols, thus we will
further classify the differences among rational
equation, rational inequality, and rational function.

10. z
Rational Equation
Definition An equation
involving rational
expression
Example 𝑥 + 4
𝑥 − 1
=
1
5

11. z
Rational Inequality
Definition An inequality involving
rational expressions
Example 𝑥 − 2
5
> 3

12. z
Rational Function
Definition A function of the form
𝑓 𝑥 =
𝑝 𝑥
𝑞 𝑥
where p(x)
and q(x) are polynomial
functions and q(x) is not
the zero function.
Example
𝑓 𝑥 =
𝑥2
+ 6𝑥 + 8
𝑥 + 4

13. z
Additional Examples
Determine whether the given sentence is a
rational function, a rational inequality or
none of these.
1.)
𝑥+5
𝑥−1
= 𝑦

14. z
Additional Examples
1.)
𝑥+5
𝑥−1
= 𝑦
This is an example of rational function
because the symbol y is also a
representation of function of 𝑥 or f(x).

15. z
Additional Examples
2.)
2
𝑥+1
≤ 3

16. z
Additional Examples
2.)
2
𝑥+1
≤ 3
None of these because
2
𝑥+1
is not a rational
expression