SHS

General
Mathematics
Quarter 1 – Module 11:
Solving Problems Involving Rational
Function, Rational Equation and
Rational Inequality
General Mathematics – SHS
Quarter 1 - Week 3, Module 11: Solving Problems Involving Rational Function,
Rational Equation and Rational Inequality

Republic Act 8293, section 176 states that: No copyright shall subsist in any work
of the Government of the Philippines. However, prior approval of the government agency or
office wherein the work is created shall be necessary for exploitation of such work for profit.
Such agency or office may, among other things, impose as a condition the payment of
royalties.

Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names,
trademarks, etc.) included in this book are owned by their respective copyright holders.
Every effort has been exerted to locate and seek permission to use these materials from
their respective copyright owners. The publisher and authors do not represent nor claim
ownership over them.

Regional Director: Gilbert T. Sadsad

Assistant Regional Director: Jessie L. Amin

Development Team of the Module

Writer: Mary Doll P. Villanueva
Editor: Desiree R. Euste
Reviewers: Sarah Christine P. Godoy
Simeon D. Brillantes
Illustrator: Kenji Ryan F. Amaro
Layout Artist: Jhomar B. Jaravata
Cover Art: Jed Adra

ii
SOLVING PROBLEMS INVOLVING RATIONAL FUNCTION,
RATIONAL EQUATION AND RATIONAL INEQUALITY

Introduction
This module aims to help you solve word problems involving rational function,
rational equation, and rational inequality. This module also provides tasks and
exercises that you will perform for you to be equipped with the necessary
competency.

Objectives

At the end of this module, you will be able to solve problems involving rational
functions, rational equations, and rational inequalities. (M11GM-Ic-3)

Vocabulary List

The following important terms will be used in this module. They are
defined as follows:
𝑝(𝑥)
• Rational Function- is a function of the form 𝑓(𝑥) = 𝑞(𝑥) where
𝑝(𝑥) and 𝑞(𝑥) are polynomial functions and 𝑞(𝑥) is not the zero
polynomial (i.e., 𝑞(𝑥) ≠ 0).

• Vertical asymptote- a vertical line x = a where the graph tends

toward positive or negative infinity as the inputs approach a.

• Horizontal asymptote- a horizontal line y= b where the graph

approaches the line as the inputs increase or decrease without
bound.

• Y-intercept - the function value when x = 0.

• Rational equation- an equation involving rational expressions.

• Rational inequality- an inequality involving rational

expressions.

Reference: General Mathematics Learner’s Material and Teacher’s Material, 1 st Edition 2016

1
 Pre-Test

Direction: Answer the following questions. Choose the letter of the best answer.

400
1. If the function 𝑠(𝑡) = 𝑡
represents the speed of a runner, how long will it take by
a runner to travel the distance if his speed is 25 meters per second?

1. 15 3. 18
2. 16 4. 20

2. Three saltwater crocodiles were discovered in Palawan island group that includes
Balabac as the reptiles’ shrinking habitat leads to repeated attacks and their
50(𝑡+1)
population growth can be approximated by the function 𝑃(𝑡) = , where P
𝑡+4
represents the crocodile population in t year since they were discovered. How many
crocodiles will there be after 4 years?
a. 13 c. 20
b. 17 d. 31

3. There are two numbers. The second number is 1 larger than the first number. The
sum of the first number and the reciprocal of the second number is greater than 1.
What is the smallest positive integral value of the second number?

a. -1 b. 0 c. 1 d. 2

For item number 4-5,

Seve went to a 25 mile trip to the city. On his way back, due to road construction he
had to drive 13 mile per hour slower and made the trip 1 hour longer.

4. If r be Seve’s rate, which of the following may represent the equation?

𝑟 𝑟−13 25 25
a. 25
= 25
+ 1 c. = +1
𝑟 𝑟−13
25 25 25 25
b. = − 1 d. 𝑟
+ 1 = 𝑟−13
𝑟 𝑟−13

5. How fast did Seve drive? (use scientific calculator and quadratic formula for
finding r)
a. 12.66 mph c. 15.34 mph
b. 25.34 mph d. 25.66 mph

Your Pre-Test ends here. -

2
 Learning Activities

Learn from the succeeding example on how to solve problems involving rational
functions.

Solving Problems involving Rational Functions

Sample Problem #1: The number of students, N, at five (5) secondary schools in Partido
Area who have had flu t months after the school year started can be modelled by the formula
550
𝑁(𝑡) = 600 − .
2 + 4𝑡
1. What is the initial number of students who already have had the flu?
2. How many students have had the flu 2 months after the school year begins?
3. How long (in month/s) will it take until 450 students contract the flu?
4. What is the horizontal asymptote of the function (𝑡) ? What does it mean within the
context of the problem?

Solution:

1. To find the number of students who already have had the flu during the beginning of
550
the school year, evaluate the function 𝑁(𝑡) = 600 − at t = 0.
2+4𝑡

550
𝑁(0) = 600 −
2 + 4(0)
550
𝑁(0) = 600 −
2+0
𝑁(0) = 600 − 275
𝑵(𝟎) = 𝟑𝟐𝟓

It means that at the beginning of the school year or 0 month, the initial number of
the students who already have had the flu is 325.
2. To find the number of students who have had the flu 2 months after the school year
550
begins, evaluate the function𝑁(𝑡) = 600 − at t = 2.
2+4𝑡
550
𝑁(2) = 600 −
2 + 4(2)
550
𝑁(2) = 600 −
2+8
550
𝑁(2) = 600 −
10
𝑁(2) = 600 − 55
𝑵(𝟐) = 𝟓𝟒𝟓

It means that after 2 months, the number of the students who had the flu rose to
545.

3
 3. To find the time (in month/s) it takes 450 students to contract the flu, let 𝑁(𝑡) be
equal to 450. Substitute it to the function
550
𝑁(𝑡) = 600 −
2 + 4𝑡

550
450 = 600 −
2 + 4𝑡
Algebraically solve for t,
600(2 + 4𝑡) − 550 (1)
450 =
2 + 4𝑡
1200 + 2400𝑡 − 550
450 =
2 + 4𝑡
650 + 2400𝑡
450 =
2 + 4𝑡
650 + 2400𝑡
(2 + 4𝑡)450 = (2 + 4𝑡)
2 + 4𝑡
900 + 1800𝑡 = 650 + 2400𝑡

900 − 650 = 2400𝑡 − 1800𝑡

250 = 600𝑡

𝒕 = 𝟎. 𝟒𝟐

Thus, it takes 0.42 month or almost half a month for the 450 students to contract the
flu.

4. To find the horizontal asymptote, rewrite the rational function into a single rational
function if it is not written as a single fraction.
550
𝑁(𝑡) = 600 −
2 + 4𝑡

600(2 + 4𝑡) − 550 (1)

𝑁(𝑡) =
2 + 4𝑡

1200 + 2400𝑡 − 550

𝑁(𝑡) =
2 + 4𝑡

650 + 2400𝑡
𝑁(𝑡) =
2 + 4𝑡

4
 Identify the degree of the numerator and the degree of the denominator. Since the
degree of the numerator is equal to the degree of the denominator which is 1, identify
the leading coefficient of the numerator (a) and the denominator (b). Thus,

𝒂 𝟐𝟒𝟎𝟎
𝒚=𝒃= = 𝟔𝟎𝟎.
𝟒

It means that, as month goes by, the number of students who might contract flu will not
exceed 600.

Sample Problem #2: A rare species of insect was discovered in Mt. Isarog. To protect the
species, environmentalists declared the insect endangered and transplanted the insect into a
protected area. The population P of the insect t months after being transplanted is

76(2 + 0.1𝑡)
𝑝(𝑡) =
2 + 0.02𝑡

1. How many insects were discovered? (In other words, the number of population
when t = 0.)
2. What will the population of the insects be after 5 months of being transplanted into
a protected area?
3. What does the horizontal asymptote mean in the context of the problem?

Solution:

1. The number of insects discovered is the number of the population when t = 0. Thus,
76(2+0.1𝑡)
evaluate the function 𝑝(𝑡) = 2+0.02𝑡
at t = 0.

76(2 + 0.1𝑡)
𝑝(𝑡) =
2 + 0.02𝑡
76[2 + 0.1(0)]
𝑝(0) =
2 + 0.02(0)
76(2 + 0)
𝑝(0) =
2+0
𝒑(𝟎) = 𝟕𝟔
There were 76 discovered insects.

5
2. To find the number of the population of insects 5 months after being transplanted into
76(2+0.1𝑡)
a protected area, evaluate the function𝑝(𝑡) = 2+0.02𝑡
at t = 5.

76(2 + 0.1𝑡)
𝑝(𝑡) =
2 + 0.02𝑡

76[2 + 0.1(5)]
𝑝(5) =
2 + 0.02(5)

76(2 + 0.5)
𝑝(5) =
2 + 0.1

76(2.5)
𝑝(5) =
2.01
190
𝑝(5) =
2.01
𝑝(5) = 94.53
𝒑(𝟓) = 𝟗𝟓
The population of insects increases to 95 after five months into a protected area.

76(2+0.1𝑡)
3. To find the horizontal asymptote, simplify the function 𝑝(𝑡) = 2+0.02𝑡
and identify
the degree of the numerator and the degree of the denominator.

76(2 + 0.1𝑡)
𝑝(𝑡) =
2 + 0.02𝑡

152 + 7.6𝑡
𝑝(𝑡) =
2 + 0.02𝑡

The degree of the numerator is equal to the degree of the denominator which
𝒂
is 1. Thus, the equation of the horizontal asymptote is 𝒚 = where a is the leading
𝒃
coefficient of the numerator and b is the leading coefficient of the denominator.
𝑎
𝑦=
𝑏
7.6
𝑦=
0.02
𝒚 = 𝟑𝟖𝟎
It means that the number of insects that a protected area can sustain is 380.

6
Sample Problem # 3: The cost C in pesos to sell p% of dressed native chicken in a public
610𝑝2
market is given by 𝐶(𝑝) = , 0 ≤ 𝑝 < 100 . What is the horizontal asymptote?
50− 2𝑝

Solution: To find the horizontal asymptote, identify the degree of the numerator and the
610𝑝2
degree of the denominator of the function 𝐶(𝑝) = .
50− 2𝑝

The degree of the numerator is 2 greater than the degree of the denominator which is 1.
Thus, there is no horizontal asymptote.

Sample Problem # 4: The concentration C of an antihistamine drug in patient’s bloodstream

2𝑡−1
t minutes after injection is given by 𝐶(𝑡) = 2 . Find the horizontal asymptote. What does
𝑡 +5
the horizontal mean in the context of the problem?

Solution: To find the horizontal asymptote, identify the degree of the numerator and the
2𝑡−1
degree of the denominator of the function 𝐶(𝑡) = 𝑡 2 +5 .

The degree of the numerator is 1 less than the degree of the denominator which is 2. Thus,
the horizontal asymptote is 𝒚 = 𝟎. It means that, as time goes by, the concentration C of an
antihistamine drug in patient’s bloodstream after injection approaches to 0.

Solving Problems involving Rational Equations and Rational Inequalities

In solving problems involving rational equation and rational inequality, we must follow these
steps:

Steps on Solving Problems involving Rational Equation and Rational Inequality

I. Read through the entire problem.

II. Determine what the problem requires you to do. Represent it through a variable.
III. Construct the equation or inequality.
IV. Solve the equation or inequality.
V. Interpret the result.

Learn from the succeeding examples on how to solve problems involving

rational equations and rational inequalities.

Examples:
A. Motion Problem
Sean walks 4 miles to the market to buy his new bike and returns home on a bike. He
averages 3 miles per hour faster when cycling than when walking, and the total time for
both trips is two hours. Find Sean’s walking speed.

7
 Tasks Solutions
Represent Sean’s walking speed. Let r (or you may opt to use any variable) be
Sean’s walking speed.

Represent Sean’s cycling speed. Let r + 3 be Sean’s cycling speed.

Using the relationship distance(d) = rate(r) x

𝑑
time (t), we have 𝑡 = 𝑟
distance rate time
Walking 4 r 4
a. The time it takes for Sean to walk to the 𝑟
market. Cycling 4 r+3 4
b. The time to cycle back. 𝑟+3

Construct an equation that describes Sean’s 4 4

combined time for both trips which is 2 + =2
hours. 𝑟 𝑟+3

Solve for r. 4 4
+ =2
𝑟 𝑟+3
4(𝑟+3)+ 4𝑟
=2
𝑟(𝑟+3)
4𝑟+12+ 4𝑟
=2
𝑟2 +3𝑟
8𝑟+12
=2
𝑟2 +3𝑟
(8𝑟 + 12) = 2(𝑟2 + 3𝑟)
8𝑟 + 12 = 2𝑟 2 + 6𝑟
2𝑟 2 − 2𝑟 − 12 = 0
𝑟2 − 𝑟 − 6 = 0
You may use factoring method (if factorable)
and/or other algebraic methods to solve for x.
Thus,

(𝑟 − 3)(𝑟 = 2) = 0
r = 3 or r = - 2
Interpret the result Since r represents Sean’s walking speed, it
does not make sense for r to be negative.
Thus, Sean’s walking speed is 3 miles per
hour.

8
B. Work Problem

A Mathematics teacher requires his students to get a pair and work together in a
drill. One pair of students can solve a complex problem in 3 minutes. If working
alone, first student can solve a problem in 1.5 minutes faster than the second
student. How would it take them to solve the same problem alone?

Task Solution
• Represent first student’s number of • Let x be the first student’s number of
minutes to solve a complex problem minutes to solve a complex problem
alone alone
Represent second student’s number • Since the first student can solve 1.5
of minutes to solve same problem minutes faster than the other working
alone alone, thus, let 𝒙 + 𝟏. 𝟓be the second
student’s number of minutes to solve
same problem alone

Use the concept, No. of Working

𝑟𝑎𝑡𝑒/𝑚𝑖𝑛𝑠. (𝑟) Student Solved time 𝑝
Rate/ 𝑟 =
number of problems solved (𝑝) Problem (min.) ℎ
=
working time in minutes (ℎ) 1st 1 x 1 (𝑝𝑟𝑜𝑏𝑙𝑒𝑚)
𝑝 𝑥 (𝑚𝑖𝑛. )
Thus, 𝑟=
ℎ
• Represent first student’s rate per 2nd 1 x+1.5 1 (𝑝𝑟𝑜𝑏𝑙𝑒𝑚)
minute 𝑥 + 1.5 (𝑚𝑖𝑛. )
• Represent second student’s rate together 1 3 1 (𝑝𝑟𝑜𝑏𝑙𝑒𝑚)
per minute 3 (𝑚𝑖𝑛. )

1 1 1
+ 3 =
Construct an equation that shows a pair of 𝑥 𝑥+ 3
2
students solved a complex problem together
in 3 minutes.
Solve the equation 1 1 1
+ 3 =
𝑥 𝑥+ 3
2

3 1 1 1 3
[𝑥 (𝑥 + )] + 3 = [𝑥 (𝑥 + )]
2 𝑥 𝑥+ 3 2
2

3 3 3
1[𝑥 (𝑥 + )] 1 [𝑥 (𝑥 + )] 1[𝑥 (𝑥 + )]
2 2 2
+ 3
=
𝑥 𝑥+ 3
2
3
3 𝑥2 + 2 𝑥
𝑥+ +𝑥 =
2 3

9
 3
3 𝑥2 + 2 𝑥
2𝑥 + =
2 3
3
3 𝑥2 + 2𝑥
[3 (2𝑥 + = )]
2 3
9 3
6𝑥 + = 𝑥2 + 𝑥
2 2
3 9
𝑥 2 + 𝑥 − 6𝑥 − = 0
2 2
2
9 9
𝑥 − 𝑥− =0
2 2

Using the completing the square technique to

9 9
solve the equation 𝑥 2 − 2 𝑥 − 2 = 0, we have
9 9
𝑥2 − 𝑥 =
2 2
2
9 81 9 81
𝑥 − 𝑥+ = +
2 16 2 16
9 81 153
𝑥2 − 𝑥 + =
2 16 16

9 2 153
(𝑥 − ) =
4 16
9 √153
𝑥= ±
4 4

or,
𝑥1 = −0.84
𝑥2 = 5.34

Since there is no negative value for a time, we

prefer x = 5.34.
And,
x+ 1.5 = 5.34+1.5 = 6.84

Interpret the result The first student can solve a complex problem
in 5.34 minutes alone while the second student
can solve the same problem in 6.84 minutes
alone.

10
C. Number Problem
There are two numbers. The second number is 2 larger than twice the first number. The
sum of the first number and the reciprocal of the second number is greater than 1 . Find
the solution set of the first number.
Tasks Solutions
• Represent the first number • Let x be the first number.
• Represent second number • The second number is 2 larger than twice the first
number, thus, it can be represented by 2x+2.
Construct an equation that shows The inequality will be
the sum of the first number and the 1
𝑥+ >1
reciprocal of the second number 2𝑥 + 2
which is greater than1.
Solve the equation Rewrite the inequality to a single inequality with 0 on the
nother side of the inequality sign.
1
𝑥+ >1
2𝑥 + 2

𝑥(2𝑥 + 2) + 1
>1
2𝑥 + 2

2𝑥 2 + 2𝑥 + 1
>1
2𝑥 + 2

2𝑥 2 + 2𝑥 + 1
−1>0
2𝑥 + 2

(2𝑥 2 + 2𝑥 + 1) − (2𝑥 + 2)
>0
(2𝑥 + 2)(1)

2𝑥 2 + 2𝑥 + 1 − 2𝑥 − 2
>0
(2𝑥 + 2)

2𝑥 2 − 1
>0
(2𝑥 + 2)

√2 √2
-1 − 2 2

Construct a table for the interval, test point and signs.

Interval 𝑥 < −1 √2 √2
√2 √2 𝑥>−
−1 < 𝑥 < − − < 𝑥 < 2
2 2 2
Test-point −2 -0.9 0.5 1

2𝑥 2 − 1 + + - +
(2𝑥 + 2) - + + +
2𝑥 2 − 1 - + - +
(2𝑥 + 2)
Interpret the result Thus, the solution set is {𝑥|−1 < 𝑥 < −
√2
𝑜𝑟 𝑥 >
√2
}.
2 2

11
 Practice Task 1

Direction: Read the problems below then answer the questions.

For items 1-3,

The concentration C of a drug in a patient’s bloodstream t hours after injection is given by
3𝑡−1
𝐶 (𝑡) = .
𝑡2 +4

1. What is the concentration C of a drug 2,3,4 hours after injection?

2. What happens to the concentration of the drug 2 to 4 hours after injection?

3. What is the horizontal asymptote of the function C(t)? What does it mean within the
context of the problem?

4. To protect a rare species of tarantula that can be found in the Philippines, the
Philippine Animal Welfare Society (PAWS) transplanted the tarantulas into a
protected area. The population P of the tarantula t months after being transplanted is
145(0.3𝑡+2)
(𝑡) = . What is the largest population that the protected area can sustain?
0.04𝑡+1

5. The cost C in peso to take out p% of Tilapia fish from Buhi Lake is given by
1650 𝑝
𝐶(𝑝) = where 0 ≤ 𝑝 < 100. What percentage of Tilapia fish can you
100−𝑝
take out for Php 1,500?

12
 Practice Task 2

Direction: Answer the following problems completely.

1. Al can finish a job in 6 hours. Maria can finish the same job in 4 hours. How many
hours do they need to finish the job if they work together?

2. Dahlia hiked 7 miles from A to B at a rate of r mph. For the return trip her rate was 2
mph faster. It took her 6 hours for the entire round trip. Find Dahlia’s hiking rate.

3. The denominator of a fraction is 3 more than twice the numerator. If both the
numerator and the denominator of the fraction are increased by 5, the resulting
2
fraction is . Find the fraction.
7

4. Shannie swims 15 miles downriver in the same time she can swim 7 miles upriver.
The speed of the current is 5 miles per hour. Find the speed of the person in still
water.

13
 Practice Task 3

Direction: Answer the following questions.

1. A box of face mask with a square base is to have a volume of 27 cubic meters. Let x
be the length of the side of the square base and h be the height of the box. What are
the possible measurements of a side of the square base if the height is given by
27
>𝑥?
𝑥2

2. Matthew enters a race where he has to cycle and run. He cycles a distance of 25 km,
and then runs for 20 km. His average cycling speed is twice of his average running
speed. Matthew completes the race in less than 2.5 hours, what is his average
cycling speed?

3. A box of face mask with a square base is to have a volume of 64 cubic meters. Let x
be the length of the side of the square base and h be the height of the box. What are
the possible measurements of a side of the square base if the height should be
longer than a side of the square base?

You are now ready to take the next test.

Good luck!

14
 Post-Test

Direction: Answer the following questions. Choose the letter of the best answer.
.
For items 1-2,

The concentration C of an antihistamine drug in patient’s bloodstream t minutes after

50𝑡
injection is given by 𝐶(𝑡) = 𝑡 2 +25 .

1. What is the concentration C of a drug 2 hours after injection?

a. 2.09
b. 4.51
c. 3.45
d. 4.87

2. It takes Jane 3 hours to pick ten bushels of oranges. Llod can pick the same
amount in 4.5 hours. How long would it take them if they worked together?
a. 1.8 h c. 2.2 h
b. 2 h d. 2.7 h

For items 3-4

The number of enrollees (N) every week during enrolment period at a secondary school
in Tigaon District can be modelled by the formula
650
N(t) = 1500 − .
1 + 5t

3. What is the initial number of enrollees?

a. 600 c. 750
b. 700 d. 850

4. How long (in week/s) will it take until 1,300 students to be enrolled?
a. 0.30 c. 0.70
b. 0.45 d. 1.5

5. There are two numbers. The second number is 3 more than the first number.
7
The sum of their reciprocals is less than . Find the smallest integral value of the
10
first number.
a. 1 c. 3
b. 2 d. 4

15
 Your Post-Test ends here!
Assignment

Create your own real-life word problem involving rational functions, rational equation, and
rational inequality.

Reference: General Mathematics Learner’s Material and Teacher’s Material, 1 st Edition 2016

16
 ANSWER KEY
PRE-TEST
1. D
2. A
3. C
4. B
5. D
PRACTICE TASK I
1. 0.88,0.62,0.55, respectively
2. The concentration of the blood decreases from 2 to 4 hours .
3. y = 0, it means that as time increases the concentration of a drug to the patient’s
bloodstream approaches to 0.
4. 1, 087.5 or 1, 088 tarantulas
5. p = 47.62%

PRACTICE TASK II
1. 2.40 h
2. 1.70 mph
19
3.
29
4. 13.75 mph
PRACTICE TASK III
1. 𝟎 < 𝑥 < 3 or the side of square base should be less than 3 meters but greater than 0
meter.
2. {𝑥|(𝑥 > 13 𝑘𝑚/ℎ)} running speed; 𝑐𝑦𝑐𝑙𝑖𝑛𝑔 𝑠𝑝𝑒𝑒𝑑 > 26 𝑘𝑚/ℎ
3. 𝟎 < 𝑥 < 4 or the side of square base should be less than 4 meters but greater than 0
meter.

POST TEST
1. C 4. B
2. A 5. C
3. D

ASSIGNMENT

Students’ outputs may vary.

17
1