7

Mathematics
Quarter 2 – Module 1:
Measurements
Mathematics – Grade 7
Quarter 2 – Module 1: Measurements
First Edition, 2020

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 7

Mathematics
Quarter 2 – Module 2:
Measurements
Introductory Message

For the facilitator:

As a facilitator, you are expected to orient the learners on how to use
this module. You also need to keep track of the learners' progress while
allowing them to manage their own learning at home. Furthermore, you are
expected to encourage and assist the learners as they do the tasks included
in the module.

For the learner:

As a learner, you must learn to become responsible of your own
learning. Take time to read, understand, and perform the different activities
in the module.
As you go through the different activities of this module be reminded of
the following:
1. Use the module with care. Do not put unnecessary mark/s on any part
of the module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer Let Us Try before moving on to the other
activities.
3. Read the instructions carefully before doing each task.

4. Observe honesty and integrity in doing the tasks and checking your
answers.
5. Finish the task at hand before proceeding to the next.

6. Return this module to your teacher/facilitator once you are done.

If you encounter any difficulty in answering the tasks in this module,

do not hesitate to consult your teacher or facilitator. Always bear in mind that
you are not alone. We hope that through this material, you will experience
meaningful learning and gain deep understanding of the relevant
competencies. You can do it!
 Let Us Learn
Have you ever wondered how primitive societies measure things like
their houses, clothing and different raw materials? Or have you been in a
grocery market buying items that needed to be measured? In this module,
you will learn how to approximate the measurements of things without using
measuring devices, the Two Standard Systems of Measurement and on how to
convert one unit to another in both Metric and English Systems.

At the end of the lesson, you are expected to

1. approximate the measures of quantities particularly length,
weight/mass, volume, time, angle and temperature and rate (M7ME-
IIa-3),
2. converts measurements from one unit to another in both Metric and
English systems (M7ME-IIb-1) and
3. solves problems involving conversion of units of measurement (M7ME-
IIb-2).

Let Us Try

Direction: Read the following questions carefully and write the letter of the
correct answer on a separate sheet of paper.

1. Which of the following is NOT true about measurement?

a. Measurement is a process of comparing an unknown quantity to a
standard known quantity.
b. Body parts were used as bases of measurements during the ancient
periods.
c. Handspans and palms are called non-standard units of
measurement.
d. Three standard systems of measurement were developed.

2. What is the widely used system of measurement?

a. ISO b. Metric c. English d. U.S Customary

3. The average newborn weighs about 7 lbs. About how many kg is that?
a. 2.2 b. 3.2 c. 4.2 d. 5.2

4. Which of the following is equivalent to 1 ½ liters?

a. 1 500 mL c. 15 000 mL
b. 1 500 cL d. 15 000 cL

5. Which of the following is true?

a. 100 mg = 1 g c. ¼ kg = 2 500 g
b. 1 000 mg = 1 g d. 1 kg = 500 g

6. Which of the following is the approximate value for 1 pound?

a. 5 g b. 50 g c. 500 g d. 5000 g

2
 7. What is the most appropriate unit of length for the width of a street?
a. Millimeter b. centimeter c. meter d. kilometer

1
8. Which Latin prefix means “ ”?
1000
a. kilo b. milli c. centi d. deci

9. Which of the following units is the lightest ?

a. pound b. kilogram c. ton d. milligram

10. A rope is 5,500 millimeters long. How long is it in meters?

a. 0.55 m b. 5.5 m c. 55 m d. 550 m

11. What is the most appropriate unit of measure for the distance between
Davao City and Tagum City ?
a. millimeter b. centimeter c. meter d. kilometer

12. Jessica is measuring two line segments. The first line segment is 30cm long.
The second line segment is 500mm long. How long are the two line segments
together, in centimeters?
a. 80 cm b. 90 cm c. 50 cm d. 60 cm

13. James has 14,500 g of sand in his sandbox. He brings home another 7,400
g of sand from the beach to add to his sandbox. How many kilograms of sand
does James have in his sandbox now?
a. 20.0 kg b. 21.9 kg c. 31.9 kg d. 29.1 kg

14. A woman’s high jump is 2.1 m, while a man’s high jump is 2.35 m. By how
many centimeters is the man’s jump higher than that of the woman?
a. 0.25 cm b. 2.5 cm c. 25 cm d. 250 cm

15. The smallest mammal, the bumblebee bat, is 1.1 inches long. How long is it
in centimeters?
a. 1.794 cm b. 2.879 cm c. 2.794 cm d. 2.974 cm

Let Us Study
To understand the lesson let’s have hands-on activities.

Activity 1: Measure Me!

Direction: Determine the dimension of the following using only parts of your
arms. Records your results in the table below.

Door Dining Table Window

Length Width Length Width Length Width

Arm part used*

Measurement

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 ➢ For the arm part, please use any of the following only: the palm, the
hand span and the forearm length.
Important terms to Remember:

➢ Palm - the width of one’s hand excluding the thumb

➢ Handspan - distance from the tip of the thumb to the tip of the little
finger of one’s hand with fingers spread apart.

➢ forearm length - the length of one’s forearm: the distance from the elbow
to the tip of the middle finger.
Questions to Ponder:

1. What was your reason for choosing which arm part to use? Why?

2. Did you experience any difficulty when you were doing the actual
measuring?

3. How can you apply your learnings in approximating length in real life
scenario?
4. Do you think learning how to approximate length is important? Why
or why not?

Activity 2:

Read and Understand Me!

Direction: Read and understand the concepts to help you to accomplish the
next activities.

Approximating Length

History of Measurement

Measurement is a process of comparing an unknown quantity to a standard

known quantity and the unit of measurement was one of the earliest
instruments that human beings invented. People used to need measurement
to decide how long or large things are; things they needed to build their homes
or make clothes. Later, measurement units were used in commerce and trade.
People used their body parts in 3rd century BC Egypt to determine
measurements of things; the same body parts you used to measure the
assigned things to you.
The forearm length was called a cubit. The handspan was considered
a half cubit while the palm was considered 1/6 of a cubit. Go ahead, check
out how many handspans your forearm length is. The Egyptians came up
with these units to be more accurate in measuring different lengths.
However, using these units of measurement had a disadvantage. Not
everyone had the same forearm length. Discrepancies arose when the people
started comparing their measurements to one another because
measurements of the same thing differed, depending on who was measuring

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it. Because of this, these units of measurement are called non-standard units
of measurement which later on evolved into what is now the inch, foot, and
yard, basic units of length in the English system of measurement.

Approximating Mass/ Weight

In common language, mass and weight are used interchangeably
although weight is the more popular term. Oftentimes in daily life, it is the
mass of the given object which is called its weight. However, in the scientific
community, mass and weight are two different measurements. Mass refers to
the amount of matter an object has while weight is the gravitational force
acting on an object.
Weight is often used in daily life, from commerce to food production.
The base SI unit for weight is the kilogram (kg) which is almost exactly equal
to the mass of one liter of water. For the English System of Measurement, the
base unit for weight is the pound (lb.). Since both these units are used in
Philippine society, knowing how to convert from pound to kilogram or vice
versa is important. Some of the more common metric units are the gram (g)
and the milligram (mg), while another commonly used English unit for weight
is ounces (oz.). Here are some of the conversion factors for these units:

1 kg = 2.2 lb. 1 g = 1000 mg

1 kg = 1000 g 1 lb. = 16 oz.

1 metric ton = 1000 kg

Approximating Volume

Volume is the amount of space an object contains or occupies. The

volume of a container is considered to be the capacity of the container. This
is measured by the number of cubic units or the amount of fluid it can
contain and not the amount of space the container occupies. The base SI unit
for volume is the cubic meter (m 3). Aside from cubic meter, another
commonly used metric unit for volume of solids is the cubic centimeter (cm 3
or cc) while the commonly used metric units for volume of fluids are the liter
(L) and the milliliter (mL).
Here under are the volume formulas of some regularly-shaped
objects:

Cube: Volume = edge x edge x edge (V = e3)

Rectangular prism: Volume = length x width x height (V = lwh)

Triangular prism: Volume = ½ x base x height of the triangular base x

height of the prism V = (1/2 bh)(H)

Cylinder: Volume = 𝜋 x (radius) 2 x height of the cylinder

(V = 𝜋r2h)

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Other common regularly-shaped objects are the different pyramids, the cone,
and the sphere. The volumes of different pyramids depend on the shape of
their base.

Here are their formulas:

Square-based pyramids:
Volume = 1/3 x (side of base) 2 x height of pyramid (V = 1/3 s 2h)

Rectangle-based pyramid:
Volume =1/3 x length of the base x width of the base x height of pyramid
(V=1/3 lwh)

Cone: Volume = 1/3 x (radius)2 x height

Approximating Time and Rate

The concept of time is very basic and is integral in the discussion of
other concepts such as speed. Currently, there are two types of notation in
stating time, the 12-hr notation (standard time) or the 24-hr notation
(military or astronomical time). Standard time makes use of a.m. and p.m. to
distinguish between the time from 12 midnight to 12 noon (a.m. or ante
meridiem) and from 12 noon to 12 midnight (p.m. or post meridiem). This
sometimes leads to ambiguity when the suffix of a.m. and p.m. are left out.
Military time prevents this ambiguity by using the 24-hour notation where
the counting of the time continues all the way to 24. In this notation, 1:00
p.m. is expressed as 1300 hours or 5:30 p.m. is expressed as 1730 hours.

Speed is the rate of an object’s change in position along a line. Average

speed is determined by dividing the distance travelled by the time spent to
cover the distance (Speed = distance /time or S = d/t, read as “distance per
time”). The base SI unit for speed is meters per second ( m/s). The commonly
used unit for speed is kilometers/hour (kph or km/h) for the metric system
and miles/hour (mph or mi/hr) for the English system.

Approximating Angle

Derived from the Latin word angulus, which means corner, an angle is
defined as a figure formed when two rays share a common endpoint called
the vertex. Angles are measured either in degree or radian measures. A
protractor is used to determine the measure of an angle in degrees. In using
the protractor, make sure that the cross bar in the middle of the protractor
is aligned with the vertex and one of the legs of the angle is aligned with one
side of the line passing through the cross bar. The measurement of the angle
is determined by its other leg.

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Approximating Temperature

Temperature is the measurement of the degree of hotness or coldness of

an object or substance. While the commonly used units are Celsius (˚C) for the
metric system and Fahrenheit (˚F) for the English system, the base SI unit for
temperature is the Kelvin (K). Unlike the Celsius and Fahrenheit which are
considered degrees, the Kelvin is considered as an absolute unit of measure and
therefore can be worked on algebraically.

Hereunder are some conversion factors:

5 𝟗
℃ = ( ) (℉ − 32) ℉ = ( ) (℃) + 32 K = ℃ + 273
9 𝟓

SYSTEMS OF MEASUREMENT
Two standard systems of measurement were developed: the US customary or
the “English” system and the metric system. The United States uses the customary
system and some metric, while the rest of the world uses the metric system
exclusively.

THE ENGLISH SYSTEM

The English system of measurement is based on the British Imperial system.

Its units evolved from nonstandard units of measure in medieval times like the foot,
yard and inch. Since the English System does not use a base unit and prefixes, it
requires a lot of memorization and is not convenient to use. Each unit has a separate
name. For example, 1 foot = 12 inches and 1 yard = 3 feet. In order to convert one
unit to another, you need to know the conversion factor or the number of units that
another unit is equal to. The table shows the common units in the English System.

Quantity Unit Symbo Equivalence

l
inch in (”)
𝑓𝑡
Length foot ft (’) 12 in
yard yd 3 ft
mile mi 5 280 ft
ounce oz
Weight 𝑙𝑏
pound lb 16 oz
ton ton 2 000 lb
pint pt 2 cups
Capacity quart qt 2 pt
gallon gal 4 qt

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THE METRIC SYSTEM
The widely used standard system of measurement is the metric system. It is based
on decimals just like our numeration and monetary systems. Since its introduction
in 1970 by a Frenchman named Gabriel Mouton, the metric system has proven to be
convenient and ideal to use. The system is based on multiples of 10, counting and
performing mathematical operations are easier. It is no surprise then that the metric
system is used in most countries all over the world, including the Philippines.
The metric system uses Greek and Latin prefixes to make conversion from one
unit to another easier. A prefix is a power-of-10 exponent or multiplier that precedes
the unit. To illustrate what is meant by a prefix, let us take the standard unit for
mass in the metric sytem - the kilogram. In the terms kilogram and milligram, kilo
and milli are the prefixes while gram is the unit.

The table below shows the Greek and Latin prefixes that are most frequently used
in the metric system.

Origin Prefixes Symbol

tera T
giga G
Greek
mega M
kilo k
hector h
deka da
deci d
centi c
Latin
milli m
micro μ
nano n
pico p

CONVERSION IN THE METRIC SYSTEM

The metric system is a system of measuring. It is used for three basic units of
measure: meters (m), liters (L) and grams (g).

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LENGTH

Length is a physical quantity that is a measure of distance. The basic unit of

length in the SI system is meter (m). Ruler, meterstick, and tape measure are often
used to measure the length of an object.

A. Converting Metric Units by Moving the Decimal Point:

The most commonly used metric units of length are kilometer (km), meter (m),
centimeter (cm), and millimeter (mm). These units of length are related as follows:

10 mm = 1 cm
100 cm = 1 m
1000 m = 1 km
One of the advantages of the Metric System, aside from its universality, is that it
uses base 10, which makes conversion from one unit to another much easier.

To multiply or divide a number by multiples of 10, we simply move the decimal

point to the left or to the right.
Study these examples:

÷1000 ÷100 ÷10 Base x10 x100 x1000

Unit
kilo- hecto- deka- deci- centi- milli-
0.008 km 0.08 hm 0.8 dam 8m 80 dm 800 cm 8 000 mm
0.500 kg 5.00 hg 50.0 dag 500 g 5 000 dg 50 000 cg 500 000 mg

In converting metric units, we simply move the decimal point.

Example 1. Convert 8m into km.
Beginning at the base unit meter (m), we have to move the decimal point
three (3) times to the left as the unit is converted from smaller to a larger unit.

The result of moving the decimal point to the left is that 8 m = 0.008 km.

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Example 2. Convert 500 m into cm.

Beginning at the base unit meter (m), move the decimal point two (2) times
to the right as the unit is converted from larger to a smaller unit. Add two zeros after
the value of 500 to accommodate the two decimal point shifts.

The result is that 500 m = 50 000 cm.

B. Converting Metric Units using Conversion Factors

A conversion factor uses your knowledge of the relationships between units to

convert from one unit to another. It is a number used to change one set of units to
another, by multiplying or dividing. When a conversion is necessary, the appropriate
conversion factor to an equal value must be used. The number is usually given as a
numerical ratio or fraction that can be used as a multiplication factor. The ratio is
equivalent to 1 where the numerator (top) of the fraction contains the units of the
unit you want to convert while the denominator contains the old unit you want to
convert. Just remember that you’re converting from one unit to another so cancelling
same units would guide you in how to use your conversion factors.

Here are the steps:

1. Identify the units used.

2. Determine the relationship between the units.
3. Determine the conversion factor (in fraction form). The denominator
should have the same unit as the original measurement.
4. Multiply the original measurement by the conversion factor.

Example 3: Convert the following measurements to the indicated units.

a. 7 cm to mm b. 8 m to cm c. 9000 m to km d. 2.5 m to mm

Solutions:

a. b.

c. d.

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CONVERSION IN THE ENGLISH SYSTEM

To convert English units, you must use a conversion factor.

A unit conversion factor is a fraction that is equal to 1. The numerator (top) of the
fraction contains the units of the unit you want to convert to while the denominator
contains the old unit you want to convert.
Here are some conversion factors you may use as a guide:

Table of Conversion

12 inches = 1 foot
36 inches = 1 yard
5280 feet = 1 mile
1760 yards= 1 mile
1 inch = 2.54 cm
1 foot = 30.5 cm
1 foot = 0.305 m
1 yard = 3 feet
1 yard = 0.915 m
1 mile = 1.6 km

Converting from one unit to another might be tricky at first, so an organized

way of doing it would be a good starting point. As the identity property of
multiplication states, the product of any value and 1 is the value itself. Consequently,
dividing a value by the same value would be equal to one. Thus, dividing a unit by
its equivalent in another unit is equal to 1. For example: 1 foot /12 inches = 1 ; 3
feet/1 yard = 1.
These conversion factors may be used to convert from one unit to another.
Just remember that you’re converting from one unit to another so cancelling same
units would guide you in how to use your conversion factors.

Example 1. Convert 30 inches to feet.

Solution:
Step 1. Identify the units used.
The units are inches (in) and feet (ft).

Step 2. Determine the relationship between the units.

12 in = 1 ft.

Step 3. Determine the conversion factor.

1𝑓𝑡
The conversion factor is
12𝑖𝑛

Step 4. Multiply the original measurement by the conversion factor.

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Example 2.

a. Convert 116 inches to feet. b. Convert 25 miles to yards.

c.12.5 inches to centimeter. d. 76 centimeter to inches. Solutions:

Solutions:

CONVERSION OF ENGLISH-TO-METRIC AND METRIC TO

ENGLISH

Similar to English-to-English conversion, English-to-Metric and Metric-to-

English conversions require familiarization of the conversion table. The relationship
between a metric and an English unit is mostly in decimal form, thus great attention
to digits is a must in order to prevent conversion errors.
Below is a conversion table between some English and Metric Units.

Length
1 in = 2.54 cm
1 ft = 30.48 cm
1 yd = 0.9144 m
1 mi = 1.609 km
Volume
1 gal = 3.785 L
Mass
1 kg = 2.2 lbs
1 oz = 28.35 g
Ares
1 ha = 2.47 acres

The method of English-to-Metric and Metric-to-English conversion is similar to

the English to-English conversion.

Here are the steps:

1. Identify the units used.

2. Determine the relationship between the units.
3. Determine the conversion factor (in fraction form).
4. Multiply the original measurement by the conversion factor.

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Example 1: Convert 3 feet to centimeters

1. Identify the units used.

The units are feet (ft) and centimeters (cm).
2. Determine the relationship between the units.
1 ft = 30.48 cm
3. Determine the conversion factor (in fraction form).
30.48𝑐𝑚
The conversion factor is
1𝑓𝑡

4. Multiply the original measurement by the conversion factor.

Therefore, 3 ft = 91.44 cm

Example 2: Convert 45 kilograms to pounds.

1. Identify the units used.

The units are kilograms (kg) and pounds (lb).
2. Determine the relationship between the units.
1 kg = 2.2 lbs
3. Determine the conversion factor (in fraction form).
2.2𝑙𝑏𝑠
The conversion factor is
1𝑘𝑔

4. Multiply the original measurement by the conversion factor.

Therefore, 45 kg = 99lbs.

Example 3: Convert 4 gallons to liters.

1. Identify the units used.

The units are gallons (gal) and liters (L).
2. Determine the relationship between the units.
1 gal = 3.785 L
3. Determine the conversion factor (in fraction form).
3.785𝐿
The conversion factor is
1𝑔𝑎𝑙

4. Multiply the original measurement by the conversion factor.

Therefore, 4 gal = 15.14 L.

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MASS/WEIGHT

Mass and weight are often interchangeably used but their meanings are not
exactly the same. Weight relates to the gravitational pull of the earth upon a mass.
Thus a person’s weight on the moon varies with his weight on Earth. Mass refers to
the amount of matter contained in an object. It remains the same regardless of
location. The standard unit of mass is kilogram (kg), which is the weight of one liter
of water at 4 ͦ C. The gram is of a kilogram. The instruments commonly used to
measure mass are weighing scale and platform balance.

Examples: Convert the given measure to the indicated unit.

1. 18 kg = ____g 2. 500 mg = ___g

Solutions:

1. 2.

TIME
The unit of time is second (s). Second is defined as the time occupied by 9
192 631 770 vibrations of the light emitted by a Cesium-133 atom. Time is
introduced with clocks and calendars.
UNITS IN TIME MEASURE

Unit Equivalence in other

Units
1 century 100 years
1 score 20 years
1 decade 10 years
1 year
12 months or 365 days
1 week 7 days
1 day 24 hours
1 hour 60 minutes
1 minute 60 seconds

Examples

1. 2 centuries = ____ years

2. 2.5 hours = _____minutes
3. 132 hours = ____days
4. 216 000 minutes = ____hr
5. 5 months = ____days

14
Solution:

Temperature

Temperature refers to the degree of hotness and coldness of a body. The metric
unit of temperature is degree Celsius ( ͦ C). In the Celsius scale, 0 ͦ C is the freezing
point of water, and 100 ͦ C is the boiling point of water. The instrument used for
measuring temperature is the thermometer. Another unit of temperature is degree
Fahrenheit ( ͦ F).

Example: Convert

a. 50 ͦ F to ͦ C b. 40 ͦ C to ͦ F

Solutions:

a. ͦ C = ( ͦF − 32) b. ºF = ͦC + 32

= ( 50 − 32) = ( ) (40) + 32

= ( 18) =(72) + 32

= 10 ͦC = 104 ͦ F

Area, Volume, Speed

Area is the surface included within a particular set of dimensions. The basic
unit is square meter (m² or sq. m). A hectare is equivalent to 10 000 m².

Volume and Capacity are used interchangeably, although strictly speaking,

they mean different things. Volume refers to how much space a region takes up. It
is measure in three dimensions: length x width x height. On the other hand, capacity
refers to how much a container will hold. The commonly used metric units for
capacity are the liter (L) and the millimeter(mL). The commonly used metric units for
volume are the cubic meter (m³ or cu. m) and the cubic centimeter (cm³ or cu. cm).
A volume of 1000 cubic centimeter is equal to one liter, which is the metric unit of
liquid volume.

15
Example:

A 1 𝑚 𝑥 𝑚 𝑥 𝑚 aquarium is to be filled with water to the brim. How many

liters of water are needed?

Solution: Convert measurements to cm

Find the volume of the aquarium.

V= 150 cm x 75 cm x 50 cm = 562 500 cm³
Using 1 000 cm³ = 1 L, convert to liter.

Speed is the rate of distance travelled per unit time.

Example: A horse runs 120 km in 3 hours.

a. Find its average speed.

b. At this rate, how far does it run in 2 hours?
c. How far can it run in 20 minutes?

16
 SOLVING PROBLEMS INVOLVING MEASUREMENT

Example 1:
John rode 2 kilometers on his bike. His sister Sally rode 3000 meters on her
bike. Who rode the farthest and how much father did they ride? (answer in
kilometer)

Given: John’s covered distance: 2km

Sally’s covered distance: 3000 m

Solution:
Convert Sally’s distance to km, 3000m to km. We use the conversion factor
1km= 1000m. We place the old unit, 1000 m , in the denominator and 1 km in the
numerator, since km will be our new unit.

Sally rode farther because she rode 3km while John only rode 2km. Subtract
their distance covered we arrived with the difference:

3𝑘𝑚 − 2𝑘𝑚 = 1𝑘𝑚

Example 2:

17
Example 3:

Example 4:

Example 5:

Let Us Practice

“WHAT AM I?”

Riddle: “I am tall when I am young, and I am short when I am old.”

Direction: Follow the steps below to answer the riddle.

18
 Step 1: Give what is being ask in each item.

Step 2: Encircle the letters that contains an answer.

Step 3: Answer the riddle below by arranging the encircled letters.

Step 4: Write your answer in the space provided below.

Questions:
1. Identify the most reasonable unit to measure the volume of a tumbler.
2. Which is the closest to the weight of a tomato?
3. Identify the most reasonable unit to measure the volume of a small
bottled water.
4. Identify the most reasonable unit to measure the time it takes to fly
from Davao to Cebu.
5. Jeffrey is driving his car. Approximate the amount of time it takes to
drive from San Pedro Cathedral to Gaisano mall.
6. Which is the closest to the height of a door?

7. Estimate the measurement of the angle: ∟

Q W L
B X A L M N P
3 1 1
millim second liter millise Kilogr Kilom 3
inch 0 0
eter s cond am eter mile es hou seco
s rs nds
L E
H C F A G N R D
100° 4 5 3 90°
1kg Milligr milliliter Hours 10 met
5 0
am ° ° min ers

Answer: _________________________________________________________

Let Us Practice More

“LET’S DO THE METRIC”

Direction: Convert the following measurements to the units indicated.
a. 25 m to cm b. 6 m to mm

c.. 4 kg to cg d. 300 mg to g

e. 62.8 L to mL

19
 “LET’S CONVERT”
Direction: Convert the following measurements to the indicated units.
a. 5 ft to inches b. 15 ft to yards

c. 2.5 lbs to ounces d. 10,000 lbs to tons

e. 5 quarts to pints

“COMPUTE ME RIGHT”
Direction: Convert the following measurements to the indicated units up to 2 decimal
places, if possible.
a. 15 lb = ____kg b. 4 gallons = ____L

c. 5 years = ____ hours d. 334 min. = ____sec

e. 77 ͦ F = ____ ͦ C

Let Us Remember

• Measurement – is a process of comparing an unknown quantity to a

standard known quantity

• Approximation- Not exact, but close enough to be used

• English System- A system of measurement that is based on the British

Imperial system. The English System does not use a base unit and prefixes, it
requires a lot of memorization and is not convenient to use.

• Metric System- A system of measurement that is based on decimals just

like our numeration and monetary systems. The metric system uses Greek and
Latin prefixes to make conversion from one unit to another easier.

• In converting units in the metric system, we have to follow the rules. First, to
convert to a smaller unit, multiply by the indicated power of 10. Second, to
convert to a larger unit, divide by the indicated power of 10.

20
 • To convert English units, you must use a conversion factor. A unit conversion
factor is a fraction that is equal to 1. The numerator (top) of the fraction
contains the units of the unit you want to convert to while the denominator
contains the old unit you want to convert.

Let Us Assess
Direction: Read the following questions carefully and write the letter of the
correct answer on a separate sheet of paper.

1. Which of the following is NOT true about measurement?

a. Measurement is a process of comparing an unknown quantity to a
standard known quantity.
b. Body parts were used as bases of measurements during the ancient
periods.
c. Handspans and palms are called standard units of
measurement.
d. Two standard systems of measurement were developed.

2. Which of the following units is the heaviest ?

a. ton b. kilogram c. pound d. milligram

3. What is the most appropriate unit of length for a bottle of softdrink?

a. Milliliter b. liter c. kiloliter d. ton

4. Which Greek prefix means “1,000” ?

a. kilo b. hecto c. deka d. milli

5. What is the system of measurement used in the Philippines today?

a. U.S Customary b. ISO c. Metric d. English

6. Which of the following is equivalent to 2.5 decades?

a. 2.5 days b. 2.5 years c. 25 centuries d. 25 years

7. A fully loaded Philippine Air Lines weighs about 320,000 kg. About how many
tons is that?
a.. 3.2 tons b. 32 tons c. 320 tons d. 3200 tons

8. What instrument is used to measure angles?

a. protractor b. ruler c. thermometer d. cylinder

9. What is the standard unit of mass?

a. ton b. gram c. kilogram d. milligram

10. How many milliliters of milk is in the container which says 3.5 L?
a.. 0.35 ml b. 3,500 ml c. 35 ml d. 350 ml

21
 11. If the length of the stick is 0.572 meter, how long is it in centimeters?
a. 5.72 cm b. 57.2 cm c. 572 cm d. 5720 cm

12. John started answering the activities in his module at 8:10 am. He finished
at 9:25. How many minutes did it take him to finish his work?
a. 75 min b. 65 mins c. 55 mins d. 1.15 mins

13. How much heavier is a diamond with a mass of 1.02 g than one that measures
984 mg ?
a. 982.98 mg b. 36 mg c. 882 mg d. 985.02 mg

14. Fourteen milliliters of alcohol is mixed with 4.2 liters of water. How many
milliliters are there in the mixture?
a. 4.34 mL b. 56 mL c. 4.214 mL d. 4,214 mL

15. Kim worked on a computer in the morning for 2 hours and 25 minutes, and
another 3 hours and 45 minutes in the afternoon. How many minutes did
she work on the computer on that day?
a. 370 mins b. 350 mins c. 330 mins d. 300 mins

Let Us Enhance

“CHALLENGE ME!”
Complete the table below:

Activity Approximate Actual Convert to

Measurement
1. Length of your
_____ cm _____ cm _____ m
bed
2. Height of your
_____ m _____ m _____ ft
door
3. Time in
brushing your _____ mins _____ mins _____ secs
teeth
4. Distance from
your house to
_____ m _____ m _____ km
the nearest
store

22
 “LET’S COMPARE AND CONTRAST”

Direction: Complete the graphic organizer “compare and contrast”. Write the
similarities and differences between English System and Metric System.

Let Us Reflect

Activity 10: 3-2-1

Direction: In your answer sheet, write the following in essay form:

a. 3 things that you have learned

b. 2 things you are confused

c. 1 reflection

23
 24
Let Us Enhance:
Challenge Me: Answers may vary
Let’s Compare and Contrast!
Let Us Reflect:
Answers may vary.
Let Us Practice More: Let Us Assess: Let Us Try:
“Let’s Do the Metric” 1. c 1. d
a. 2,500 cm 2. a 2. b
b. 6,000 mm 3. b 3. b
c. 400,000 cg 4. a 4. a
d. 0.3 g 5. c 5. b
e. 62,800 ml 6. d 6. c
7. b 7. c
“Let’s Convert” 8. a 8. b
a. 60 inches 9. c 9. d
b. 5 yards
10. b 10. b
c. 40 oz
11. b 11. d
d. 5 tons
e. 10 pts 12. a 12. a
13. b 13. b
“Compute Me Right” 14. d 14. c
a. 6.82 kg 15. a 15. c
b. 15.14 L
c. 43,200 hrs Let Us Practice:
d. 20,040 sec A candle
e. 25 °C
Answer Key
 References

Elizabeth R. Aseron, Angelo D. Armas, Allan M. Canonigo, Ms. Jasmin T.

Dullete, Flordeliza F. Francisco, PhD, Ian June L. Garces, PhD, Eugenia V.
Guerra, Phoebe V. Guerra, Almira D. Lacsina, Rhett Anthony C. Latonio,
Lambert G. Quesada, Ma. Christy R. Reyes, Rechilda P. Villame, Debbie Marie
B. Verzosa, PhD, and Catherine P. Vistro-Yu, PhD,Mathematics
Measurement Learner’s Material: The Department of Education Publications,
2014

Pierce, Rod, "About Math is Fun". Math Is Fun, accessed June 14 2020.
http://www.mathsisfun.com/aboutmathsisfun.html
Pierce, Rod. "Definition of Temperature". Math Is Fun, accessed 14 2020.
June http://www.mathsisfun.com/definitions/temperature.html

Pierce, Rod. "Definition of Measurement". Math Is Fun. accessed 14 2020,

June http://www.mathsisfun.com/definitions/measurement.html

Pierce, Rod, "Mass Definition (Illustrated Mathematics Dictionary)". Math Is

Fun.accessedJune14,2020,http://www.mathsisfun.com/definitions/mass.
html

Pierce, Rod. "Rate Definition (Illustrated Mathematics Dictionary)". Math Is

Fun.accessedJune14,2020,http://www.mathsisfun.com/definitions/rate.ht
ml

Morin, Amanda, “Kid Science: How to Make Your Own Balance Scale”.
ThoughtCo., accessed June 14 2020, https://www.thoughtco.com/kid-
science-make-a-balance- scale-2086574

Learner’s Materials
Retrieved November 2020 from www.purplemath.com
Retrieved November 2020 from https://www.ck12.org/section/convertingmetric-
units-%3A%3Aof%3A%3A-using-dcimals/

Retrieved November 2020 from www.ipracticemath.com/learn/measure/

Retrieved November 2020 from www.khanacademy.org/math/

Retrieved November 2020 from americanhistory.si.edu/collections/search/object

25
For inquiries or feedback, please write or call:

Department of Education – Region XI

F. Torres St., Davao City

Telefax:

Email Address: lrms.regionxi@deped.gov.ph

26
7
Mathematics
Quarter 2 – Module 2:
The Language of Algebra
Mathematics – Grade 7
Quarter 2 – Module 2: The Language of Algebra
First Edition, 2020

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 module 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.

Published by the Department of Education – Region XI

Regional Director: Allan G. Farnazo, CESO IV
Assistant Regional Director: Maria Ines C. Asuncion

Development Team of the Module

Writer: Collen G. Baldos
Editors: Flordelisa L. Parojinog and Jessica C. Sarmiento
Reviewer: Jessica C. Sarmiento
Illustrator:
Layout Artist:
Template Developer: Neil Edward D. Diaz
Management Team: Reynaldo M. Guillena, CESO V
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Aris B. Juanillo, PhD
May Ann M. Jumuad, PhD
Antonio A. Apat

Printed in the Philippines by _____________________________________________

Department of Education – Region XI

Office Address: DepEd Davao City Division, E. Quirino Ave.

Davao City, Davao del Sur, Philippines
Telefax: (082) 224 0100
E-mail Address: info@deped-davaocity.ph
 7

Mathematics
Quarter 2 – Module 2:
The Language of Algebra
Introductory Message

For the facilitator:

As a facilitator, you are expected to orient the learners on how to use
this module. You also need to keep track of the learners' progress while
allowing them to manage their own learning at home. Furthermore, you are
expected to encourage and assist the learners as they do the tasks included
in the module.

For the learner:

As a learner, you must learn to become responsible of your own
learning. Take time to read, understand, and perform the different activities
in the module.
As you go through the different activities of this module be reminded of
the following:
1. Use the module with care. Do not put unnecessary mark/s on any part
of the module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer Let Us Try before moving on to the other
activities.
3. Read the instructions carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your
answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are done.
If you encounter any difficulty in answering the tasks in this module,
do not hesitate to consult your teacher or facilitator. Always bear in mind that
you are not alone. We hope that through this material, you will experience
meaningful learning and gain deep understanding of the relevant
competencies. You can do it!
 Let Us Learn

At the end of this module, you are expected to :

1. illustrate and differentiate related terms in algebra:
a. an , where n is a positive integer,
b. constants and variables,
c. literal coefficients and numerical coefficients,
d. algebraic expressions, terms, and polynomials;
e. number of terms, degree of the term and degree of the polynomial;
2. identify the words/phrases that represent addition, subtraction,
multiplication and division;
3. translate English phrases to mathematical phrases and English sentences
to mathematical sentences and vice versa;

Let Us Try
Directions: Find out how much you already know about topic. Choose the letter of
the correct answer. Write your answer on a separate sheet of paper.

1. What do you call a polynomial with degree zero?

a. constant b. cubic c. linear d. quadratic

2. Which of the following is an example of a trinomial?

a. 3xy b. 7qs c. de - 3cd – 1 d. 4lm – 4

3. What is the literal coefficient of the polynomial, 8a³bc - 3 ?

a. 8a³bc -3 b. a³bc c. abc d. -3

4. What type of algebraic expression is x(6xy)?

a. monomial b. binomial c. trinomial d. polynomial

5. What is the degree of the polynomial x 3y – 2x2y2 + 5y?

a. 1 b. 2 c. 3 d. 4

6. If the polynomial 6x – 7 + 2x3 is written in standard form, which among the

following is its leading coefficient?
a. 7 b. 6 c. 3 d. 2

7. Which of the following algebraic expressions is a polynomial?

5
a. 3x-2 b. 7y c. -10m1/2 d.
𝑥

8. What is the 3rd term of the expression 2x³ -4x² + x - 6?

a. 2x³ b. -4x² c. x d. - 6

9. What do you call a symbol or a letter that represents an unknown number?

a. variable c. numerical coefficient
b. exponent d. literal coefficient

2
10. Which of the following is the expanded form of 82 ?
a. 8 + 8 c. 8 x 8
b. 2+2+2+2+2+2+2+2 d. 8 x 2

11. Which of the following is NOT a symbol for inequality ?

a. ≥ b. ≤ c. > d. =

12. Which of the following is the mathematical translation of “x less than 8”?
a. x < 8 b. x > 8 c. 8-x d. x - 8

13. Which is NOT a correct verbal translation of 4x-1?

a. The difference of four and a number decreased by one
b. Four times a number decreased by one
c. The difference of four times a number and one
d. The product of four and a number diminished by one

14. Which of the following is the mathematical translation of the phrase: “five times
a number m, increased by 2”?
a. 5 (m+2) b. 5m + 2 c. 5 + 2m d. 5m +2m

15. Which of the following pair is a correct translation of 3x ≤ 12?

I. Thrice a number is at least twelve.
II. Thrice a number x is at most twelve.
III. Thrice a number x is less than or equal to twelve.
IV. IV. Thrice a number x is greater than or equal to twelve.

a. I and II b. II and III c. III and IV d. II and IV

Let Us Study
Algebra is defined as branch of Mathematics which generalizes the facts in
arithmetic. In spoken language, letters and punctuation are used to create words
and phrases. In the language of algebra, letters along with numbers and operation
symbols are used to create expressions. The ability to translate English phrases or
sentences to Mathematical phrases or sentences and vice-versa is an initial step for
us to solve worded problems involving missing quantities.

A constant is a symbol or number that has a fixed value. It is usually referred

to as the term without a variable. Examples include 7, 4, -20 and 11.

A variable is a symbol or letter in the alphabet that may take one or more
than one value from the given replacement set. It represents an unknown value or
number. A variable is written in a small letter. The following are the common symbols
used for variables x, y, z, a, b.

A term is a constant or a variable or constants and variables multiplied

together. Terms are separated by the symbols + and -. For example: 7, y or 3xy. In
a term, the number part is called the numerical coefficient while the variable or

3
variables including the exponents, is/are called literal coefficients. In 5ab², 5 is the
numerical coefficient and ab² is the literal coefficient.

An exponent is a number or letter written above and to the right of

a mathematical expression called the base. It tells the number of times the base is
to be used as a factor. For example, x2 means x • x , where x is the base and 2 is
the exponent. In 54 , this means 5 • 5 • 5 • 5 , where 5 is the base and 4 is the
exponent.

Similar Terms (Like Terms) are terms which have the same literal
coefficients. -4y² and 7y² are similar terms because their literal coefficients, which is
y², are the same. 5y³ and 10y are not similar because their literal coefficients, which
are y³ and y, are not the same.

An algebraic expression is a constant, a variable, or a combination of

constants and variables related by atleast a fundamental operation or grouping
symbol. It is a group of terms separated by the plus or minus sign.

A polynomial is an algebraic expression where each term is a constant, a

variable, or a product of a constant and a variable. Each variable should only have
an exponent that is a non-negative integer.

An algebraic expression is NOT a polynomial if

1. the exponent of the variable is a negative integer (-1, -2, -3, …)
2. the exponent of the variable is a fraction or decimal (1/2, ¾, 0.2,
and many more)
3. the variable is inside the radical sign
4. the variable is in the denominator

The degree is the highest exponent or the highest sum of exponents of the
variables in a term.

➢ For a polynomial with one variable, the degree is the highest exponent
of that variable.

In 4x³ – 2x2 + x – 10, the degree is 3

20y + 8y2 – 3y, the degree is 2

➢ For a polynomial with more than one variable, look at each term and
add the exponents of each variable in it. The largest sum is the degree
of the polynomial.

In 5xy2 – 8x2y3 + 7x – 8y2 + 7, 5xy2 has a degree of 3 (x has an

exponent of 1, while y has 2, 1 + 2 = 3); -8x2y3 has a degree of 5 (2 + 3
= 5); 7x has a degree of 1; -8y2 has a degree of 2; and 7 has a degree
of 0 (no variable). Therefore, the degree of the given polynomial is 5.

4
Type of Polynomials according to its degree

1. Constant – a polynomial of degree zero

2. Linear – a polynomial of degree one
3. Quadratic – a polynomial of degree two
4. Cubic – a polynomial of degree three
5. Quartic – a polynomial of degree four
6. Quintic – a polynomial of degree five

Type of Polynomials according to Number of Terms

Examples Type of Polynomial

A 6xy
-3ab2c3 Monomial – is a polynomial with only one term.
10lmn
B -10ab + 6a
↑ ↑
1st term 2nd term
Binomial – is a polynomial with two terms.
5mn – mn
↑ ↑
1st term 2nd term

a-7
↑ ↑
1st term 2nd term
C 4x2 - 3x + 5 Trinomial – is a polynomial with three terms.

y3 + 5y -15
A multinomial is being used to refer polynomials with two or more terms.

A polynomial is in its standard form if its terms are arranged in descending

order, from the term with the highest degree up to the term with the lowest degree.
The first term is called the leading term and the numerical coefficient of the leading
term is called the leading coefficient. For example, 5x + 3x3 – 6 + 2x2 can be written
in standard form as 3x3 + 2x2 + 5x – 6. Its leading term is 3x3 with its leading
coefficient as 3. This polynomial is cubic since its degree is 3.

There are different ways in expressing Addition, Subtraction, Multiplication,

and Division of Algebraic Expressions. The table below shows words/phrases, which
may be used in translating English phrases/sentences to mathematical
phrases/sentences, and vice versa.

Addition Subtraction Multiplication Division

＋ － ×, ( ) , · /,÷
addition, plus, subtraction, multiplied, division, the
the sum of, more difference of, less, multiplied by, quotient of ,
than, increased decreased by, product of, divided by,
by, subtracted from, twice/thrice a ratio of
total, added to diminished by, number
minus, reduced,
less than

5
Some verbal expressions leading to algebraic expressions are shown below.

Examples Phrases
the sum of two and a number x
or
Addition 2+x the total of two and a number x
or
two increased by a number x
the difference of twelve and five
or
Subtraction 12 - 5 twelve decreased by five
or
twelve less five
thrice a number x
or
Multiplication 3x three multiplied to a number x
or
the product of 3 and a number x
the quotient of a number x and 15
or
Division x ÷ 15 a number x divided by 15
or
the ratio of a number x and 15

Translation of English sentences into Mathematical sentences. This time, we

will use the following symbols:
Symbols Meaning

= equals, is equal to, is

< is less than

> is greater than

≤ is less than or equal to, is at most

≥ is greater than or equal to, is at least

Below are are examples of translation of English sentences into Mathematical

sentences.

Mathematical
English Sentences
Sentences

The difference between five and two is three. 5-2=3

The sum of five and twice y is less than eighteen. 5 + 2y < 18

6
 Ten more than a number d is greater than twenty-one. d + 10 > 21

25
The quotient of twenty-five and a number z is greater ≥ 5
𝑧
than or equal to five.
The product of nine and a number y is less than or
equal to thirty-six 9y ≤ 36

Let Us Practice

Activity 1: “Identify Me!”

Directions: Identify the constants, variables, numerical and literal coefficients, and
type of polynomials according to the number of terms.

Type of
Coefficients
Polynomial
Monomial
Algebraic Numerical Literal
Variable Constant Binomial
Expressions Coefficient Coefficient
Trinomial

3xyz2 + 8

25m² +10n – 6

-5ab²c -8ab – 2

7
 Activity 2 : “Match Me Up”

Directions: Match each verbal phrase in Column A with its corresponding

mathematical phrase in Column B. Write the letter of your answer on the space
provided before each number.

Write the Column A Column B

letter of
the
correct
answer
1. the difference of 8 and m A. 1000 - f
2. exceeds r by 20 B. 18 - n
3. 5 more than t C. c - 10
4. 9 take away d D. 9 - d
5. 1000 less f E. 8 - m
6. 18 reduced by n F. n + 11
7. 10 subtracted from c G. q + p
8. 5 less than t H. t + 5
9. q increased by p I. t - 5
10. 11 added to n J. r + 20

Let Us Practice More

Complete the table.

8
 Let Us Remember

Important Terms to Remember:

• Constants are numbers that have fixed values. It is usually referred to as the
term without a variable.
• Variable is a symbol or letter in the alphabet that may take one or more than
one value from the given replacement set. It represents an unknown value or
number.
• Term is a constant/variable or constants/variables multiplied together. The
number part is called the numerical coefficient while the variable/s
including the exponents, is/are called literal coefficients.
• Exponent is a number or letter written above and to the right of
a mathematical expression called the base. It tells the number of times the
base is to be used as a factor.
• Algebraic Expression is a group of terms separated by the plus or minus sign.
• Similar Terms (Like Terms) are terms having the same literal coefficients.
• A polynomial is an algebraic expression where each term is a constant, and
a variable. Each variable should only have an exponent that is a non-negative
integer.
• Degree is the highest exponent or the highest sum of exponents of the
variables in a term.

• Type of Polynomials according to the number of terms:

1. Monomial – a polynomial with only one term
2. Binomial – a polynomial with two terms
3. Trinomial – a polynomial with three terms
4. Multinomial – a polynomial with two or more terms

• Type of Polynomials according to its degree:

1. Constant – a polynomial of degree zero
2. Linear – a polynomial of degree one
3. Quadratic – a polynomial of degree two
4. Cubic – a polynomial of degree three
5. Quartic – a polynomial of degree four
6. Quintic – a polynomial of degree five

Words/Phrases used for Addition, Subtraction, Multiplication, and Division

Addition Subtraction Multiplication Division

＋ － ×, ( ) , · /,÷
addition, plus, subtraction, difference multiplied, division, the
the sum of, more of, less, decreased by, multiplied by, product quotient of ,
than, increased subtracted from, of, twice/thrice a divided by,
by, diminished by, minus, number ratio of
total, added to reduced, less than

9
Symbols Used for Mathematical Sentences

Symbols Meaning

= equals, is equal to, is

< is less than

> is greater than

≤ is less than or equal to, is at most

≥ is greater than or equal to, is at least

Let Us Assess

Directions: Choose the letter of the correct answer. Write your answer on the
separate sheet of paper.

1. Which of the following represents a constant?

a. number of brothers and sisters c. number of hours in a day
b. number of students in each class d. number of friends in school

2. In 5πr , what is the variable ?

a. 5π b. πr c. 5 d. r

3. Refer to no. 2. Which one is the numerical coefficient?

a. 5π b. πr c. r d. 5

4. What do you call a polynomial with degree three?

a. constant b. cubic c. linear d. quadratic

5. What is the degree of 3x2y2 - x²y + xy - 65?

a. 3 b. 4 c. 5 d. 6

6. Which of the following polynomials is a binomial?

𝑥+ 4
a. x(y + 3) b. c. ab d. m - 3
5

7. Which of the following algebraic expressions is NOT a polynomial?

𝑦−4
a. 2x0.5 b. -10b c. 8𝑐√3 d.
5

8. Your classmate evaluated 55 ● 52 as 257. Is this correct?

a. No, because the exponents should also be multiplied.
b. No, because the base of the answer should be 5.
c. Yes, because the exponents are added.
d. Yes, because the bases are multiplied.

10
9. What kind of polynomial is 3(x + 4) ?
a. monomial b. binomial c. trinomial d. constant

10. What do you call a number or letter that tells the number of times the base is to
be used as a factor?
a. constant b. variable c. exponent d. coefficient

11. Multiplication indicates a multiplying action. Which of the following express

multiplication?

I. Multiplied by , of III. The product of, thrice a number

II. Times, the ratio of IV. The difference of, the product of

a. 1 and 2 b. 1 and 3 c. 1 and 4 d. 1, 2, 3, 4

12. Which of the following is the mathematical translation of “x less 10”?

a. x – 10 b. 10 – x c. x > 10 d. x < 10

13. What is the correct translation of “four subtracted from three times a number
b”?
a. 4 - 3b b. 4b - 4 c. 3b - 4 d. 3 - 4b

14. Which of the following phrases is the correct translation of x² y?

a. the square of x and y
b. the product of a square of x and y
c. the square of x added to a number y
d. twice a number x multiplied to y

15. Which of the statement below is the correct translation of 2x + 3 ≥ 12 ?

a.Twice a number x added to three is greater than or equal to twelve.
b.Twice a number x added to three is less than or equal to twelve.
c.The product of two and a number x is greater than or equal to twelve
d.The sum of twice a number x and three is greater than or equal to twelve

Let Us Enhance

Translate the mathematical sentence 2(x-3) = 8 in at least two ways. Use the
different words/phrases that refers to the given sentence.

1. _____________________________________________________
2. _____________________________________________________

11
Let Us Reflect

SKILLS ACQUIRED

Write the skills you have acquired in this competency

Example:
I can translate verbal phrases to mathematical phrases
and vice versa.

One example above is done for you, Now it’s your turn;
1. _______________________________________________________.

2. _______________________________________________________.

3. _______________________________________________________.

4. _______________________________________________________.

5. _______________________________________________________.

12
 13
Let Us Practice
Activity 2:
“Match Me Up!”
1. E
2. J
3. H
4. D
5. A
6. B
7. C
8. I
9. G
10. F
Let Us Practice Let Us Try
1. a 11. d
Activity 1: “Identify Me!” 2. c 12. c
3. b 13. a
4. a 14. b
5. d 15. b
6. d
7. b
8. c
9. a
10. c
Answer Key
 14
Let Us Enhance: (Possible
Answers)
2(x-3) = 8
• Twice the difference of x
and three is equal to eight.
• The product of two and the
difference of x and three is
eight.
• Two times the difference of
x and three is equal to
eight.
Let Us Assess:
1. c 11. b
2. d 12. a
3. a 13. c
4. b 14. b
5. b 15. d
6. d
7. a
8. b
9. a
10. c
 References

Jisela N. Ulpina and Edna D. Licardo, Math 7 Builders, JO-ES Publishing House,
Inc., 2014, 406 – 415.

“MySecretMathTutor”, youtube.com, accessed February 24, 2021,

https://www.youtube.com/watch?v=amLYLq73RvE

“Welcome to Statistics How To!” From StatisticsHowTo.com: Elementary Statistics

for the rest of us, accessed February 17, 2021, https://www.statisticshowto.com/

15
For inquiries or feedback, please write or call:

Department of Education – Region XI

F. Torres St., Davao City

Telefax:

Email Address: lrms.regionxi@deped.gov.ph

16
7
Mathematics
Quarter 2 – Module 3:
Evaluation of Algebraic
Expressions, Addition and
Subtraction of Polynomials
Mathematics – Grade 7
Quarter 2 – Module 3: Evaluation of Algebraic Expressions, Addition and Subtraction
of Polynomials
First Edition, 2020

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 module 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.

Published by the Department of Education – Region XI

Regional Director: Allan G. Farnazo, CESO IV
Assistant Regional Director: Maria Ines C. Asuncion

Development Team of the Module

Writer: Anthony C. Amorio
Editors: Mirasol O. Fabuna and Jessica C. Sarmiento
Reviewer: Jessica C. Sarmiento
Illustrator:
Layout Artist:
Template Developer: Neil Edward D. Diaz
Management Team: Reynaldo M. Guillena, CESO V
Jinky B. Firman, PhD, CESE
Marilyn V. Deduyo
Alma C. Cifra, EdD
Aris B. Juanillo, PhD
May Ann M. Jumuad, PhD
Antonio A. Apat

Printed in the Philippines by Davao City Division Learning Resources Management

Development System (LRMDS)

Department of Education – Region XI

Office Address: DepEd Davao City Division, E. Quirino Ave.

Davao City, Davao del Sur, Philippines
Telefax: (082) 224 0100
E-mail Address: info@deped-davaocity.ph
 7

Mathematics
Quarter 2 – Module 3:
Evaluation of Algebraic
Expressions, Addition and
Subtraction of Polynomials
Introductory Message

For the facilitator:

As a facilitator, you are expected to orient the learners on how to use
this module. You also need to keep track of the learners' progress while
allowing them to manage their own learning at home. Furthermore, you are
expected to encourage and assist the learners as they do the tasks included
in the module.

For the learner:

As a learner, you must learn to become responsible of your own
learning. Take time to read, understand, and perform the different activities
in the module.
As you go through the different activities of this module be reminded of
the following:
1) Use the module with care. Do not put unnecessary mark/s on any part
of the module. Use a separate sheet of paper in answering the exercises.
2) Don’t forget to answer Let Us Try before moving on to the other
activities.
3) Read the instructions carefully before doing each task.
4) Observe honesty and integrity in doing the tasks and checking your
answers.
5) Finish the task at hand before proceeding to the next.
6) Return this module to your teacher/facilitator once you are done.
If you encounter any difficulty in answering the tasks in this module,
do not hesitate to consult your teacher or facilitator. Always bear in mind that
you are not alone. We hope that through this material, you will experience
meaningful learning and gain deep understanding of the relevant
competencies. You can do it!

ii
 Let Us Learn

Hello learners! Welcome to our lesson for this week which is all about
Evaluation of Algebraic Expressions for given values of the variables (M7AL-IIc-4),
Addition and Subtraction of Polynomials (M7AL-IId-2).
Specifically, at the end of the lesson, you are expected to:

• identify the steps in evaluating algebraic expressions

• evaluate algebraic expressions
• add and subtract polynomials
• solve problems involving adding and subtracting polynomials.

Let Us Try

Multiple Choice: Choose the letter of the correct answer and write your answer on a
separate sheet of paper.

I. Evaluate each algebraic expression using the given values.

1) 3d + 5, where d = 5
a. -20 b. 20 c. 15 d. -15

2) y3 – 8, where y = 4
a. 56 b. -56 c. 50 d. -50

3) 2xy + y2, where x = 12 and y = 8

a. 250 b. 252 c. 256 d. 258

3−𝑥
4) , where x = 10 and y = 2.
4𝑦
a. -7/8 b. 7/8 c. -8/7 d. 8/7

5) y2 + x3 + 7, where y = 9 and x = 5
a. 210 b. 211 c. 212 d. 213

II. Simplify each polynomial expression.

1) (5p2 – 3) + (2p2 – 3p3)
a. -3p3 + 7p2 + 3 c. 3p3 + 7p2 – 3
b. 3p3 + 7p2 + 3 d. -3p3 + 7p2 – 3

2) (4 + 2n3) + (5n3 + 2)
a. 7n3 + 6 c. -7n3 + 6
b. 7n3 – 6 d. -7n3 – 6

1
 3) (a3 – 2a2) – (3a2 – 4a3)
a. 5a3 + 5a2 c. 5a3 – 5a2
b. -5a3 + 5a2 d. -5a3 – 5a2

4) (13n2 + 11n – 2n4) + (-13n2 – 3n – 6n4)

a. -8n4 + 8n c. -8n4 – 8n
b. 8n4 + 8n d. 8n4 – 8n

5) (7 – 11x3 – 11x) – (4 x3 + 8 – 4x5)

a. -4x5 – 15x3 + 11x – 1 c. 4x5 + 15x3 + 11x + 1
b. 4x5 + 15x3 – 11x + 1 d. 4x5 – 15x3 – 11x – 1

Let Us Study

Evaluating Algebraic Expressions

Is 5y +4 equal to 5(y + 4)? Let us find out if we assign 3 as a value of y. In

evaluating algebraic expressions, you simply substitute or replace variables by
numbers and carry out the operations following the order of operations:

Order of Operations

1. Simplify the expressions inside the grouping symbols, such as braces, parenthesis
or brackets and as indicated by fraction bars.
2. Evaluate all powers and extract roots.
3. Simplify products and quotients in order from left to right.
4. Simplify sums and differences in order from left to right.

Let us now evaluate the above expressions. We just need to substitute 3 as a

value of y, we now have
5y + 4 = 5 (3) + 4 and 5(y + 4) = 5(3 + 4)
= 15 + 4 = 5(7)
= 19 = 35

Therefore, 5y + 4 ≠ 5(y +4).

Let us try.

Evaluate 8x2 + 3 when x = 3.

Solution:
8x2 + 3 = 8(3)2 + 3
= 8(9) + 3
= 72 + 3
= 75

2
Evaluate 2x + y if x = -4 and y = 3.
Solution:
2x + y = 2(-4) + 3
= -8 + 3
= -5

Evaluate c3 – 2b + a when c = 3, b -5, and a = -12.

Solution:
c3 – 2b + a = 33 – 2(-5) + (-12)
= 27 +10 – 12
= 25

Evaluate (x + 5)(y – 4) when x = -2 and y = 8

Solution:
(x + 5)(y – 4) = (-2 + 5)(8 – 4)
= (3)(4)
= 12

Adding and Subtracting Polynomials

Algebra tiles can be used to have an easy understanding in adding and
subtracting polynomials. Let us be familiarized with the algebra tiles below.

(+1) (- 1) (+x) (-x) (+x2) (-x2)

We can represent polynomials using these tiles. Let’s say, 4x2 + x2 can be represented
as,

and 3x - 4x can be represented as,

From the above representation, we know that 4x2 + x2 can be represented by

getting 4 (+x2 tiles) and 1 more (+x2 tiles). So how many (+x2 tiles) do we have in all?
There are 5 (+x2 tiles) altogether. Therefore, 4x2 + x2 = 5x2.

3
 We also know that, 3x – 4x is represented by getting 3 (+x tiles) and 4 (-x tiles).
Now, let us recall that subtraction also means adding the negative quantity. A pair
of (+x tiles) and (-x tiles) is zero. So, what tiles do we have left? There is 1 (-x tiles)
left. This means that, 3x – 4x = -x.

Let’s consider (2x2 – 5x + 2) + (3x2 + 2x). What tiles would you put together?

You should have two (+x 2), five (-x) and two (+1) tiles then add three (+x2) and
two (+x) tiles. Matching the pairs that make zero, you have in the end five (+x 2), three
(-x), and two (+1) tiles. The sum is 5x 2 – 3x + 2.

DO YOU THINK YOU CAN ADD/SUBTRACT POLYNOMIALS WITHOUT USING THE

ALGEBRA TILES?

Rules for Adding Polynomials. To add polynomials, simply combine similar terms.
To combine similar terms, get the sum of the numerical coefficients and annex the
same literal coefficients. If there is more than one term, for convenience, write similar
terms in the same column.

Example 1: Simplify (4x2 – 5x + 2) + (-2x2 + 7).

Solution:

(4x2 – 5x + 2) + (-2x2 + 7) = 4x2 - 2x2 – 5x + 2 + 7

= 2x2 – 5x + 9

Or 4x2 – 5x + 2
+ -2x2 +7
2x – 5x + 9
2

Example 2: The lengths of two ropes are (8x2 + 5x – 2) meters and (2x2 – 9x – 5)
meters. Find the sum of their lengths.

Solution: 8x2 + 5x – 2
2x2 – 9x – 5
10x2 – 4x – 7 meters

4
Rules for Subtracting Polynomials. To subtract polynomials, change the sign of the
subtrahend then proceed to the addition rule. Also, remember what subtraction
means. It is adding the negative of the quantity.

Example 1: Subtract 8x2 + 4x – 3 from 6x2 – 7x.

Solution:
“Subtract b from a” means a – b. We have,
(6x2 – 7x) – (8x2 + 4x – 3) = (6x2 – 7x) + (-8x2 - 4x + 3) definition of subtraction
= 6x2 - 8x2 - 7x – 4x + 3
= -2x2 – 11x + 3

Or 6x2 – 7x 6x2 – 7x
-(8x2 + 4x – 3) + -8x2 – 4x + 3
-2x2 – 11x + 3

Example 2: A boy has a stick that is (8x – 3) feet long. Represent the length of the
stick after (2x – 5) feet have been cut off.

Solution: 8x – 3 8x – 3
-(2x – 5) + -2x + 5
6x + 2 feet

Let Us Practice

“The EVALUATION”

1. Evaluate the polynomial 3𝑥 2 + 2x when:

a. x = 0 b. x = 3 c. x = -4

2. Find the value of each polynomial when x=8.

a. 12x b. 5 - 4x + 𝑥 2 c. x(x – 2)

4𝑥−3
3. Evaluate when:
3𝑦−4

a. x = 1 and y = -1 b. x = 2 and y = 3 c. x = -3 and y = 6

4. Complete the table given that y = 12 – 3x.

x -3 -2 -1 0 1
y

5
 “Let’s ADD and SUBTRACT Buddy!”

I. Add the following polynomials.

1. x + 5y
2. 4xy + 10xy
3. (-6x3) + (-2x3)
4. (-6x – 3) + (3x –9)
5. (5x2 +10xy + 4y2) + (7x2 +8xy - 6y2)

II. Subtract the following polynomials.

1. ab – 8ab
2. 23x2 y2 - 35 x2 y2
3. -8c2d - 4 c2d
4. (5x + 4) – (7x + 2)
5. (a2 – 6a + 7) – (5a2 - 5a – 2)

III. Solve the problem.

If you have (100x3 – 5x2 + 3) pesos in your wallet and you spent (80x3 –
2x2 + 9) pesos in your school canteen, how much money is left in your pocket?

Let Us Practice More

“WHAT’S THE SECRET?”

Directions: Simplify the following polynomials. Match that answer to the correct
letter of the alphabet. Enter that letter of the alphabet on the blank corresponding to
the problem number to find out the secret message.

1. 2x + 3y

2. 10xy + 8xy

3. 20x2 y2 + 30 x2 y2

4. 5x + 3x + 8x + 6x

5. (-5x3) + (-4x3)

6. 10xy - 8xy

7. 20x2 y2 - 30 x2 y2

8. 5x - 3x - 8x - 6x

9. (3x + 6y) – (8y – 2x)

6
 ___ ___ ___ ___ ___ ___ ___ ___ ___
5 9 6 2 7 3 4 8 1

-12x 5x2 + 10x +

6

Let Us Remember

In evaluating algebraic expressions, you have to remember that after

substituting the given values, you must follow the PEMDAS (Parenthesis, Exponent,
Multiplication, Division, Addition, Subtraction) rule in order for you to arrive with a
correct answer.

In adding and subtracting polynomials, you always have to remember that

you can only add or subtract like terms. If there are many terms, you can write
similar terms in column for your convenience.

Let Us Assess

Multiple Choice: Choose the letter of the correct answer and write your answer on a
separate sheet of paper.

I. Evaluate each algebraic expression using the given values.

1. 6x + 7, where x = 6
a. 43 b. 44 c. 45 d. 46

2. b2 + 2b – 3, where b = 11
a. -140 b. -160 c. 140 d. 160

7
 3. -2y2 + x3 + 7, where y = 4 and x = 5
a. 99 b. -99 c. 100 d. -100

12−𝑥
4. , when x = 20 and y = 4
5𝑦
a. -2/5 b. -5/2 c. 2/5 d. 5/2

5. (x2 + 5) – (y2 – 5), where x = 6 and y = 4

a. -29 b. -30 c. 29 d. 30

II. Direction: Read and understand the problem carefully. Complete the table by
choosing the corresponding letter of your answer from the choices.

You are walking with your friends at Davao Bolton Bridge. While talking and
chatting with each other, one of your friends accidentally knocked your phone into
the river. Your friend Jaz, a math enthusiast, wants to know the height (in feet) of
your phone above the river at t seconds. So, he calculated it and he found out that
it can be represented by the equation h = -14t2 +1,230. What is the height of your
phone above the river at t = 0, 1, 2, 3, 4? Record your data on the table below.

t h = - 14t2 + 1, 230
1. 0 a. 1230 b. 1320 c. 1023 d. 1203
2. 1 a. 1621 b. 1612 c. 1216 d. 1261
3. 2 a. 1147 b. 1174 c. 1417 d. 1714
4. 3 a. 1140 b. 1410 c. 1014 d. 1104
5. 4 a. 1006 b. 1060 c. 1600 d. 6100

III. Simplify each polynomial.

1. (-6x – 3) + (3x –9)
a. 3x + 12 c. 12 – 3x
b. -3x – 12 d. 12 + 3x

2. (5x + 4) – (7x + 2)
a. -2x + 2 c. 2x + 2
b. 2x – 2 d. -2x – 2

3. (5x2 +10xy + 4y2) + (7x2 +8xy - 6y2)

a. 12x2 + 18xy – 2y2 c. -12x2 + 18xy + 2y2
b. -12x2 + 18xy – 2y2 d. 12x2 + 18xy + 2y2

4. Solve: What must be added to 4x + 8 to get a result of 7x – 2?

a. 3x + 10 c. 3x – 10
b. -3x + 10 d. -3x – 10

8
 5. Solve: Mark saved 3x2 + 17x – 10 from his allowance. How much did he have
after buying a gift worth of 12x + 6 for his sister’s birthday?
a. 3x2 + 5x – 16 c. 3x2 + 5x + 16
b. -3x2 + 5x – 16 d. -3x2 + 5x + 16

Let Us Enhance

Mang Juan’s Polynomial Vegetable Garden

Mang Juan is planning to plant vegetables in his garden. He wants to plant
Tomatoes, Pechay, Okra, Eggplant, and Garlic. His plan for the vegetable garden
layout in meters is shown in figure below.

x 7x

N
Garlic xy + 1
5x Tomatoes Okra

W E

4x – 3 Eggplant Pechay
x+5
S

5x + 4 x2 – 10 x2 – 6x + 8

Do the following and write your answer on a separate sheet of paper.

1. Write an expression that represents the length of the south side of the field.
Simplify your answer.
2. Simplify the polynomial expression that represents the north side of the field.
3. Write the simplest form of the polynomial expression that represents the
perimeter of the Pechay field.
4. Write and simplify the polynomial expression of the perimeter of the garlic field
if x = 3 and y = 7. Show your solution.

9
 Let Us Reflect

Heart and Share

On the Heart React, write three things that you have learned about the lesson.
On the Share React, write two real-life situations where you can apply the concept
of algebraic expressions

______________________________ ______________________________
______________________________ ______________________________
______________________________ ______________________________
______________________________ ______________________________
______________________________ ______________________________
______________________________ ______________________________
______________________________ ______________________________

10
 11
Let Us Enhance Let us Assess
I.
1. (5x +4) + (x2-10) + (x2-6x+8) or 2x2 - x + 2
2. x2 + 2x + 8
1. a
3. (x2-6x+8) + (x2-6x+8) + (x+5) + (x+5) or 2. c
2x2 - 10x + 26 3. c
4. (x2-6x+8) + (x2-6x+8) + (xy+1) + (xy+1) or
2x2 – 12x +2xy + 18
4. a
5. d
Substituting x=3 and y=7. We have
Perimeter: 2(3)2 – 12(3) +2(3)(7) + 18
2(9) - 12(3) + 2(21) + 18 II.
18 - 36 + 42 + 18 1. a
42 meters 2. c
3. b
4. d
5. a
III.
Let Us Practice 1. b
2. a
The Evaluation
3. a
4. c
5. a
c. 48
1
A.-
7
Let Us Practice More
Let’s Add and Subtract Buddy! Let Us Try
I. I.
1. x + 5y 1. b
2. 14xy 2. a
3. -8x3 3. c
4. – 3x – 12 4. a
5. 12x2 + 18xy – 2y2 5. d
II.
1. -7ab II.
2. -12 x2y2 1. d
3. -12c2d 2. a
4. - 2x + 2 3. c
5. - 4a2 – a + 9 4. a
5. d
III. 20x3 – 3x2 – 6
Answer Key
 References

Elizabeth R. Aseron et al., Mathematics – Grade 7 Learner’s Material First Edition

Pasig: DepEd-IMCS, 2013, 113-117; 131-134.

12
For inquiries or feedback, please write or call:

Department of Education – Region XI

F. Torres St., Davao City

Telefax:

Email Address: lrms.regionxi@deped.gov.ph

13
7
Mathematics
Quarter 2 – Module 4:
Laws of Exponents,
Multiplication and Division of
Polynomials, Special Products
Mathematics – Grade 7
Quarter 2 – Module 4: Laws of Exponents, Multiplication and Division of Polynomials
and Special Products
First Edition, 2020

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 module 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.

Published by the Department of Education – Region XI

Regional Director: Allan G. Farnazo, CESO IV
Assistant Regional Director: Maria Ines C. Asuncion

Development Team of the Module

Writer: Vivian A. Selim and Alex M. Luscares
Editors: Flordelisa L. Parojinog and Jessica C. Sarmiento
Reviewer: Jessica C. Sarmiento
Illustrator:
Layout Artist:
Template Developer: Neil Edward D. Diaz
Management Team: Reynaldo M. Guillena, CESO V
Jinky B. Firman, PhD, CESE
Marilyn V. Deduyo
Alma C. Cifra, EdD
Aris B. Juanillo, PhD
May Ann M. Jumuad, PhD
Antonio A. Apat

Printed in the Philippines by _______________________________________________

Department of Education – Region XI
Office Address: DepEd Davao City Division, E. Quirino Ave.
Davao City, Davao del Sur, Philippines
Telefax: (082) 224 0100
E-mail Address: info@deped-davaocity.ph
 7

Mathematics
Quarter 2 – Module 4:
Laws of Exponents,
Multiplication and Division
Of Polynomials, Special Products
Introductory Message

For the facilitator:

As a facilitator, you are expected to orient the learners on how to use
this module. You also need to keep track of the learners' progress while
allowing them to manage their own learning at home. Furthermore, you are
expected to encourage and assist the learners as they do the tasks included
in the module.

For the learner:

As a learner, you must learn to become responsible of your own
learning. Take time to read, understand, and perform the different activities
in the module.
As you go through the different activities of this module be reminded of
the following:
1. Use the module with care. Do not put unnecessary mark/s on any part
of the module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer Let Us Try before moving on to the other
activities.
3. Read the instructions carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your
answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are done.
If you encounter any difficulty in answering the tasks in this module,
do not hesitate to consult your teacher or facilitator. Always bear in mind that
you are not alone. We hope that through this material, you will experience
meaningful learning and gain deep understanding of the relevant
competencies. You can do it!
 Let Us Learn
At the end of this module, you are expected to:

1. Derive the laws of exponents (M7AL-IId-e-1),

2. Multiply and divide polynomials (M7AL-IIe-2), and
3. Use models and algebraic methods to find the: (a) product of two
binomials; (b) product of the sum and difference of two terms; (c)
square of a binomial; (d) cube of a binomial; and (e) product of a
binomial and a trinomial (M7AL-IIe-g-1).

Let Us Try!

Directions: Find out how much you already know about our lesson. Choose the letter
of the correct answer and write it on a separate sheet of paper.

1. Which of the following is a shortcut method of multiplying two binomials?

a. Vertical Method c. Horizontal Method
b. FOIL Method d. DPMA
2. Which of the following laws of exponent states that “to multiply two exponents
with the same base, you keep the base and add the exponents”.
a. Product of a Power c. Quotient of Powers
b. Power of a Product d. Power of a Quotient
3. Which of the following laws of exponent states that “when dividing powers with
the base, exponents are subtracted”.
a. Product of a Power c. Quotient of Powers
b. Power of a Product d. Power of a Quotient

4. Which of the following shows how to simplify “power of a product”?

a. 10a ∙ 10b = 10a+b c. (x4)2 = x (4)(2) = x8
b. (m ) = m
2 5 10 d. (x2y3)5 = x(2)(5) y(3)(5) = x10 y15

5. What will be the quotient if 106 is divided by 104?

a. 10,000,000,000 c. 100,000
b. 100 d. 1,000,000,000

6. What is the product if 3x + 4 and x – 2 are multiplied?

a. 3x2 + 2x - 8 c. 3x2 + 6x - 8
b. 3x – 2x - 8
2 d. 3x2 – 6x - 8

7. When 4m3 – 6m2 + 2m is divided by 2m, what is the quotient?

a. 4m2 + 3m - 1 c. 2m2 – 3m + 1
b. 2m3 – 3m2 + m d. 2m2 + 3m + 1

8. What will be the result if (5x3)2 is simplified?

a. 10x5 b. 10x6 c. 5x5 d. 25x6

1
 9. Which of the following states that “in raising a product to a certain power, each
factor is raised to the indicated power”.
a. Product of a Power c. Quotient of Powers
b. Power of a Product d. Power of a Quotient

10. When 8x2 + 22x – 21 is divided by 2x + 7, what is the quotient?

a. 4x – 3 b. 4x + 3 c. 8x + 3 d. 8x – 3

11. What are the Outer Terms if we are to multiply 2x + 3 and x – 4 using the
FOIL method?
a. 2x and – 4 b. 3 and x c. 2x and x d. 3 and – 4

12. What is the product of 3m + 2 and 3m – 2?

a. 6m2 – 4 b. 6m2 + 4 c. 9m2 – 4 d. 9m2 + 4
13. What is the first step when squaring a binomial?
a. Combine like terms. c. Cube the first term.
b. Square the first term. d. Square the last term.
14. Find (2s + 3)3.
a. 2s3 + 9 c. 8s3 - 36s2 + 54s – 27
b. 2s3 + 27 d. 8s3 + 36s2 + 54s + 27
15. Which of the following special products formula is applicable to find the
product of (5m + 2)2?
a. Cube of a Binomial c. Product of Two Binomials
b. Square of a Binomial d.Product a Trinomial and a Binomial

Let Us Study

LAWS OF EXPONENTS
Exponents follow certain rules that help in simplifying exponential
expressions which are also called its laws.

A. Product of a Power
Consider the following examples:
1. (x5)(x4) = (x)(x)(x)(x)(x)(x)(x)(x)(x) = x5+4 = x9
2. (am)(an) = am+n
3. 23 22 = (2)(2)(2 )(2)(2) = 23+2 = 25= 32
4. 10a 10b = 10a+b
5. (2x3) (3x2) = (2)(3) (x)(x)(x) (x)(x) = 6x3+2 = 6x5

To multiply two exponents with the same base, you keep the
base and add the exponents. In symbols, (x a)(xb) = xa+b

2
B. Power of a Power
Consider the following:
1. (ax)y = axy
2. (x4)2 = x (4)(2) = x8
3. (33)3 = 3(3)(3) = 39 = (3)(3)(3)(3)(3)(3)(3)(3)(3) = 19, 683
4. (m2)5 = m10
5. (a2x)y = 22xy

To raise a power to another power, multiply the inner and

outer exponents. In symbols, (xa)b = xab

C. Power of a Product

The following examples show how to simplify power of a product.

1. (3•5)2 = (3)(5)(3)(5) = (3)(3)(5)(5) = (32)(52) = 9 (25) = 225
2. (x2y3)5 = x(2)(5) y(3)(5) = x10 y15
3. (ab2)3 = a3 . b (2)(3) = a3 b6
4. (3p2)3 = (33)(p2)3 = 27p6
5. (a2b3)(x + 3) = a2(x+3) b3(x+3) = a2x+6 b3x+9

In raising a product to a certain power, each factor is raised to the indicated

power. For any number x and y and any positive integer a, (xy)a = xaya

D. Quotient of Power
Examples.

1. 26 = 26-4 Copy the same base then subtract the exponents. The
24 exponent of the numerator is greater than the exponent
= 22 of the denominator.
=4 Simplify.

2. 18 x4 y5 z3 = (6•3) x4-2 y 5-2 z3-1 Get the factors. Cancel common factors
24 x2 y2 z (6•4)

= 3x2 y3 z2 Copy the remaining factors. Subtract the exponents

4 of terms with the same base.

3
3. b5 = b5-3 = b2 Copy the same base then subtract the exponents.
b3

4. 24 = 24-6 = 2-2 = 1 = 1 The exponent of the numerator is less than

26 22 4 the exponent of the denominator.

5. m6 = m6-6 = m0 = 1 Any number raised to zero is always equal to one.

m6

When dividing powers with the base, exponents are subtracted.

For integers a and b:
xa = xa-b If a > b and x ≠ 0
xb

xa = 1 If a < b and x ≠ 0
xb xa-b

xa = xa-b = x0 = 1 If a = b, and x ≠ 0
xb

D. Power of a Quotient

Power of a quotient is similar to a power of a product. Consider the following

examples.

1. x 4 = x4
y y4

2. 3x 2 = (32)(x2) = 9x2
2y (22)(y2) 4y2

3. m5n2 4 = m(5)(4) n(2)(4) = m20n8

p2q4 p(2)(4) q(4)(4) p8 q16

4. 2 2 = 22 = 4
3 32 9

5. x2 2 = x2(2) = x4
5 52 25

For power of a quotient, both numerator and denominator are raised to

the indicated power such that
𝑥 a = xa
𝑦 𝑦a , y ≠ 0.

4
 MULTIPLICATION AND DIVISION OF POLYNOMIALS

Multiplication Of A Polynomial By A Monomial

You can get the product of a polynomial and a monomial by multiplying each
term of the polynomial by the monomial. The laws of exponent for multiplication and
the Distributive Property of Multiplication over Addition or Subtraction (DPMA) can
be used.
Examples:
Multiply:
1.) 4a2 (3a + 5)
Solution:
4a2 (3a + 5) = 4a2(3a) + 4a2 (5) Multiply 4a2 to each term of the binomial
= 12a3 + 20a2 inside the parenthesis

2.) 9y (5y3 – 3y + 5)
Solution:
9y (5y3 – 3y + 5) = 9y (5y3) + 9y (– 3y) + 9y (5)
= 45y4 – 27y2 + 45y

3.) -m6 (70 – 30m – 20m4)

Solution:
-m6 (70 – 30m – 20m4) = -m6 (70) +(-m6) (-30m) +(-m6) (-20m4)
= -70m6 + 30m7 + 20m10

Multiplication Of A Polynomial By Another Polynomial

The product of two polynomials is obtained by taking one term of the

multiplier and multiplying the multiplicand at a time. In writing down the partial
products, see to it that similar terms must fall on the same column, then combine
like terms to express the product in the simplest form. Multiplication can be done
vertically or horizontally.
Examples:
Vertical Form
1. (4x – 4)(2x + 5)
Solution:
4x – 4
2x + 5
20x – 20 Multiply 5 by -4, then 5 by 4x
8x2 – 8x Multiply 2x by -4, then 2x by 4x
8x2 + 12x – 20 Align like terms, then add.

5
 2. (8x2 + 7x – 11) (2x – 3)
8x2 + 7x – 11
2x – 3
-24x2 – 21x + 33 Multiply 8x2 + 7x – 11 by -3
16x3 +14x2 – 22x Multiply 8x2 + 7x – 11 by 2x
16x3 – 10x2 – 43x + 33 Align like terms, then add.

Horizontal Form

1. (4x – 4) (2x + 5)
Multiply each term of the first polynomial to each term of the second
polynomial. Combine like terms
(4x – 4) (2x + 5) = (4x) (2x + 5) + (-4) (2x + 5)
= (4x) (2x) + (4x) (5) + (-4) (2x) + (-4) (5)
= 8x2 + 20x – 8x – 20
= 8x2 + 12x – 20 (final product)

2. (8x2 + 7x – 11) (2x – 3)

= 2x (8x2 + 7x – 11) + (-3) (8x2 + 7x – 11)
= 2x (8x2) + 2x (7x) + 2x (-11) + (-3) (8x2) + (-3) (7x) + (-3) ( -11)
= 16x3 + 14x2 – 22x – 24x2 – 21x + 33
= 16x3 + 14x2 – 24x2 – 22x – 21x + 33
= 16x3 – 10x2 – 43x + 33 (final product)

Division Of A Polynomial By A Monomial

To divide a polynomial by a monomial, divide each term of the polynomial by

the monomial. The law of exponents for division is applied in simplifying the powers
appearing in each term.

Examples:
1. Divide (4x8 – 4x4 + 8x3) by (2x2)
Solution:
Step 1: Rewrite the problem. Write each term in the polynomial as dividend
of the monomial.

4x8 – 4x4 + 8x3 = 4x8 - 4x4 + 8x3

2x2 2x2 2x2 2x2

6
 Step 2: Follow the laws of exponents and operations on integers and divide
each term by the monomial. Therefore,

4x8 – 4x4 + 8x3 = 2x6 – 2x2 + 4x

2x2

2. Find the quotient : 16a3b2 c5 – 24a5bc4 + 44a7 b6c6

4a2bc3

= 16a3b2c5 – 24a5bc4 + 44a7 b6c6

4a2bc3 4a2bc3 4a2bc3

= 4abc2 – 6a3c + 11a5b5c3

Division Of A Polynomial By Another Polynomial

In finding the quotient of two polynomials, use the same procedure in doing
ordinary division with whole numbers. Make sure that both the dividend and the
divisor are in the standard form. It guarantees that their exponents are in decreasing
order from left to right. Also, insert zero(s) for missing terms.
Examples:
1. (6x2 -2x – 28) ÷ (2x + 4)
Solution:
Step 1: Divide the leading term of the dividend by the leading term of the
divisor.
Step 2: Place the partial quotient on top.

Step 3: Now take the partial quotient you placed on top, 3x, and distribute
into the divisor (2x + 4).

Step 4: Position the product of (3x) and (2x + 4) under the dividend. Make
sure to align them by similar terms.

7
Step 5: Perform subtraction by switching the signs of the bottom polynomial.

Step 6: Proceed with regular addition vertically. Notice that the first column
from the left cancels each other out.

Step 7: Carry down the next adjacent “unused” term of the dividend.

Step 8: Next, look at the bottom polynomial, −14x − 28, take its leading term
which is −14x and divide it by the leading term of the divisor, 2x.

Step 9: Again, place the partial quotient on top.

Step 10: Use the partial quotient that you put up, −7, and distribute into the
divisor.

8
 Step 11: Place the product of −7 and the divisor below as the last line of
polynomial entry.

Step 12: Subtraction means you will switch the signs (in red).

Step 13: Perform regular addition along the columns of similar terms

Step 14: This is great because the remainder is zero. It means the divisor is a
factor of the dividend. The final answer is just the expression on
top of the division symbol.
Therefore, the quotient is 3x – 7.

2. Divide:

Solution:
Step 1: Divide the leading term of the dividend by the leading term of the
divisor.
Step 2: Place the partial quotient on top.

9
Step 3: Now take the partial quotient you placed on top, x 2, and distribute
into the divisor (x + 2).
Step 4: Position the product of (x2) and (x + 2) under the dividend.

Step 5: Perform subtraction by switching the signs of the bottom polynomial.

Step 6: Proceed with regular addition vertically. Notice that the first column
from the left cancels each other out.

Step 7: Carry down the next adjacent “unused” term of the dividend.

Step 8: Next, look at the bottom polynomial, −6x2 + 2x, take its leading term
which is −6x2 and divide it by the leading term of the divisor, x.
Step 9: Again, place the partial quotient on top.

Step 10: Use the partial quotient that you put up, −6x, and distribute into the
divisor.
Step 11: Place the product of −6x and the divisor below as the last line of
polynomial entry.

Step 12: Subtraction means you will switch the signs.

Step 13: Perform regular addition along the columns of similar terms. Bring
down the next term.

10
Step 14: Repeat the same process. Divide the leading term of the bottom
polynomial by the leading term of the divisor. In this case, we have
14x divided by x which is +14.

Step 15: Multiply (or distribute) the answer obtained in the previous step by
the polynomial in front of the division symbol. In this case, we need
to multiply 14 and (x + 2).

Step 16: Subtract and notice there are no more terms to bring down.

Step 17: Write the final answer. The term remaining after the last subtraction
step is the remainder and must be written as a fraction in the final
answer.

Therefore, the quotient is

11
 SPECIAL PRODUCTS
Multiplying two polynomials together, if they have more than three terms and
in their terms have several variables, can be very tedious and you will take a lot of
time to finish.
Would you like to know how to identify when you have a special product and
how to apply its formula?
In this module it will be explained to you, step-by-step, the different special
product formulas. Examples were answered so that you will learn how to apply them
in your exercises.

I. Product Of Two Binomials

Study the following algebra tiles:

Example 1: Use these tiles to find the product of (2x)(x).

12
Example 2: Illustrate the product of (x + 1)(x - 2) using algebra tiles.

Using the concept of Algebra tiles, observe the figure below. Given (a + b) (c + d),

To get the total area of the rectangle, we simply add ac + bc + ad + bd. This
is the general form of multiplying two binomials: (a + b)(c + d) = ac + bc + ad + bd.
This form is also known as the FOIL Method.

FOIL Method - This is a shortcut method in multiplying two binomials.

It stands for "First, Outer, Inner, Last"

It is the sum of:

· multiplying the First terms,
· multiplying the Outer terms,
· multiplying the Inner terms, and
· multiplying the Last terms

13
Example 1: Multiply: (5x – 3) (2x + 7)

Step 1: Multiply each term in the first

binomial with each term in the second
binomial using the FOIL method as shown.

Step 2: Combine like terms.

Example 2: Multiply: (4x – 5) (3x – 8)

Step 1: Multiply each term in the first

binomial with each term in the second
binomial using the FOIL method as shown.

Step 2: Combine like terms.

II. PRODUCT OF THE SUM AND DIFFERENCE OF TWO TERMS

The product of the sum of two terms (a + b) and the difference of the same
terms (a – b) is the difference of the squares of the two terms.

Example
Problem Multiply the binomials. (2n – 5)(2n + 5)

(2n)2 = 4n2 Square the first term, including the coefficient.

(5)2 = 25 Square the last term.
4n2 - 25 Take the difference.
Answer (2n – 5)(2n + 5) = 4n2 – 25

14
 More Examples:

1. (x + y )(x – y) = x2 – y2
2. (2c + 3)(2c – 3) = 4c2 – 9
3. (m + n)(m – n) = m2 – n2
4. (3x – y)(3x + y) = 9x2 – y2
5. (6p – 2r)(6p + 2r) = 36p2 – 4r2

III. SQUARE OF A BINOMIAL

The general form for the square of a binomial are:

Case 1. Squaring a Binomial Sum

To square a binomial, do the following:
➢ Square the first term.
➢ Add the product of the two terms, times two.
➢ Add the square of the last term.
This process is illustrated by: (a + b)2 = a2 + 2ab + b2

Example
Problem Square the binomial. (2x +6)2

(2x)2 = 4x2 Square the first term.

(2x) (6)(2) = Multiply the two terms together and double the
24x product.
62 = 36 Square the last term.
4x2 + 24x +36 Combine the terms.
Answer (2x +6)2 = 4x2 + 24x +36

More Examples:

1. (a + 1)2 = a2 + 2a + 1
2. (y + 3)2 = y2 + 6y + 9
3. (2a + 4)2 = 4a2 + 16a + 16
4. (3x + 5)2 = 9x2 + 30x + 25
5. (7m + 6n)2 = 49m2 + 84mn + 36n2

15
 Case 2. Squaring a Binomial Difference
To square a binomial difference, do the following:
➢ Square the first term
➢ Subtract the product of the two terms, times two
➢ Add the square of the last term.

This process is illustrated by: (a – b)2 = a2 – 2ab + b2

Example
Problem Square the binomial. (x – 7)2
(x)2 Square the first term.
(x)(-7)(2) = -14x Multiply the two terms together and double the
product.
(-7)2 = 49 Square the last term.
x2 – 14x + 49 Combine the like terms.
Answer (x – 7)2 = x2 – 14x + 49

More Examples:

1. (k - 1)2 = k2 – 2k + 1
2. (t - 8)2 = t2 – 16t + 64
3. (6h - 3)2 = 36h2 – 36h + 9
4. (2x - 10)2 = 4x2 – 40x + 100
5. (2m - 7n)2 = 4m2 – 28mn + 49n2

IV. CUBE OF A BINOMIAL

For cubing a binomial, we need to know the formulas for the sum of cubes
and the difference of cubes.
The general form for the cube of a binomial are:

Case 1. Sum of Cubes

The sum of a cube of two binomials is equal to the cube of the first term, plus
three times the square of the first term times the second term, plus three times the
first term times the square of the second term, plus the cube of the second term.

(a + b)3 = a3 + 3a2b + 3ab2 + b3

16
 Examples:
1. (x + y)3 = (x)3 + 3(x)2(y) + 3(x)(y)2 + (y)3 = x3 + 3x2y + 3xy2 + y3
2. (x + 4)3 = (x)3 + 3(x)2(4) + 3(x)(4)2 + (4)3 = x3 + 12x2 + 48x + 64
3. (2b + 3c)3 = (2b)3 + 3(2b)2(3c) + 3(2b)(3c)2 + (3c)3 = 8b3 + 36b2c + 54bc2 + 27c3

Case 2: Difference of Cubes

The difference of a cube of two binomials is equal to the cube of the first term,
minus three times the square of the first term times the second term, plus three
times the first term times the square of the second term, minus the cube of the
second term.
(a – b)3 = a3 – 3a2b + 3ab2 – b3
Examples:
1. (p - r)3 = (p)3 - 3(p)2(r) + 3(p)(r)2 - (r)3 = p3 – 3p2r + 3pr2 – r3
2. (3n - 4s)3 = (3n)3 -3(3n)2(4s) + 3(3n)(4s)2 - (4s)3 = 27n3 – 108n2s + 144ns2 – 64s3
3. (t - 5)3 = (t)3 - 3(t)2(5) + 3(t)(5)2 - (5)3 = t3 – 15t2 + 75t – 125

V. PRODUCT OF A BINOMIAL AND A TRINOMIAL

We have multiplied monomials by monomials, monomials by polynomials, and
binomials by binomials. Now we’re ready to multiply a trinomial by a binomial.
Remember, the FOIL method will not work in this case, but we can use either the
Distributive Property or the Vertical Method. In special products, we will focus only
on one general form of trinomial multiplied to a binomial, and that is:

Examples:
1. (x2 – xy + y2)(x + y) = x3 + y3
2. (m2 + mn + n2)(m - n) = m3 – n3
3. (f2 – fg + g2)(f + g) = f3 + g3
4. (x2 - 4x + 16)(x + 4) = x3 + 64
5. (y2 + 5y + 25)(y – 5) = y3 – 125
6. (4x2 – 6x + 9)(2x + 3) = 8x3 + 27

Notice that the product is always the sum or difference of two cubes. This
method is only applicable if the given binomial and trinomial follows the general form,
that is, if you square the first term of the binomial you will get the first term of the
trinomial; if you multiply the first and second term of the binomial you will get the
second term of the trinomial; and if you square the second term of the binomial you
will get the last term of the trinomial. Take note also of their signs.

17
 Let Us Practice

I. Identify the law of exponent and simplify the following expressions. Write your
answers on the space provided before the number.

__________________________________ 1) (a 2bc) 3

__________________________________ 2) 4xy2 3

3wz

__________________________________ 3) m 5 ∙ m3

__________________________________ 4) (32)4

__________________________________ 5) x -9 y3
X -7 y8

II. Match given pairs of polynomial in column A with the correct product or
quotient in column B. Write the letter of your answer on the space provided
before the number.

A B

_____ 1) 2 (x + y) a. y2 – 2y + 3

_____ 2) (4x – 3) (2x + 7) b. 8x2 + 22x – 21

_____ 3) (a2 + 2a – 4) (a – 3) c. 2x + 2y
_____ 4) 10y3 – 20y2 + 30y d. x - 5
10y

_____ 5) x2 ─ 3x ─ 10 e. a3 – a2 – 10a + 12
x+2

Let Us Practice More

I. Find the product using the FOIL method. Write your answers on the spaces
provided:

1. (x + 2) (x + 7) _______________________________________
2. (x + 4) (x – 8) _______________________________________
3. (x – 2) (x – 4) _______________________________________
4. (x – 5) (x + 1) _______________________________________
5. (2x + 3) (x + 5) ____________________________________

18
 II. State the type of special product to be used in solving the following pair of
polynomials the easiest way. Write only the corresponding letter on the
space before each item.

A - Product of Two Binomials;

B – Product of the Sum and Difference of Two Terms;
C - Square of a Binomial;
D - Cube of a Binomial;
E - Product of a Binomial and a Trinomial.

_______ 1. (h2 – hk + k2) (h + k)

_______ 2. (x - 12) (x + 8)
_______ 3. (d + p)3
_______ 4. (2m + 3n)2
_______ 5. (s - 9)3
_______ 6. (8x – y) (8x + y)
_______ 7. (4a2 + 6ab + 9b2) (2a – 3b)
_______ 8. (c + 11) (c - 2)
_______ 9. (4p + r) (4p – r)
_______ 10. (7b + c)3

Let Us Remember

Let us summarize important concepts discussed in the lesson.

Laws of Exponents

1. Product of a Power (xa)(xb) = xa+b

2. Power of a Power (xa)b = xab

3. Power of a Product (xy)a = xaya

4. Quotient of a Power xa = xa-b If a > b and x ≠ 0
xb

xa = 1 If a < b and x ≠ 0
xb xa-b

xa = xa-b = x0 = 1 If a = b, and x ≠ 0
xb

5. Power of a Quotient 𝑥 a = xa
𝑦 𝑦a where y ≠ 0

19
Multiplication and Division of Polynomials
In multiplying a polynomial by a monomial, multiply each term. The Laws of
Exponents and the Distributive Property of Multiplication over Addition or
Subtraction can be used.
In multiplying a polynomial by a polynomial, take one term of the multiplier
and multiply the multiplicand at a time. See to it that the similar terms must fall on
the same column in writing partial product, then combine similar terms to express
the product in simplest form.
To divide a polynomial by a monomial, divide each term of the polynomial by
the monomial. The law of exponents for division is applied in simplifying the powers
appearing in each term.
To divide a polynomial by a polynomial, use the same procedure in doing
ordinary division with whole numbers.

Special Products

1. Product of Two Binomials

(a + b)(c + d) = ac + bc + ad + bd
2. Product Of The Sum And Difference Of Two Terms
(a + b) (a – b) = a2 – b2

3. Square of a Binomial
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2

4. Cube of a Binomial
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a – b)3 = a3 – 3a2b + 3ab2 – b3

5. Product Of A Binomial And A Trinomial

(a2 – ab + b2) (a + b) = a3 + b3
(a2 + ab + b2) (a – b) = a3 – b3

20
 Let Us Assess
MULTIPLE CHOICE: Find out how much you already know about our lesson. Choose
the letter of the correct answer and write it on a separate sheet of paper.

1. Ana evaluated 55 ● 52 as 257. Is this correct?

a. No, because the exponents should also be multiplied.
b. No, because the base of the answer should be 5.
c. Yes, because the exponents are added.
d. Yes, because the bases are multiplied.

2. What law of exponent will satisfy the given equation: (m2)(m3) = m5?
a. Power of powers c. Quotient of a Power
b. Product of a Power d. Law for Zero Exponent

3. Which of the following is a shortcut method of multiplying two binomials?

a. Vertical Method c. FOIL Method
b. Horizontal Method d. DPMA
4. What is the value for □
which will make (3x2y3)4 = □x8y12 true?
a. 3 b. 7 c. 12 d. 81

5. What is 6(-3x4)3 in simplest form?

a. -162x12 b. 162x12 c. -21x7 d. -21x12

6. Find the product of 4a5b3 (a3 + 2ab – b2).

a. 4a8b6 + 8a5b3 – 4a5b6 c. 4a8b3 + 8a6b4 – 4a5b5
b. 4a6b6 + 8a2b3 – 4a2b5 d. 4a5b3 + 8a3b4 – 4a2b5

18𝑥 3 +6𝑥 2 +24𝑥

7. Simplifying the expression, .will give what result?
6𝑥
a. 3𝑥 3 + 𝑥2 + 4𝑥 c. 3𝑥 3 + 𝑥 2 + 4
b. 3𝑥 2 + 𝑥 + 4 d. 3𝑥 2 + 𝑥 2 + 4

8. What is the quotient if (6x2 + x – 1) is divided by (2x + 1) ?

a. 3x – 1 b. 3x + 1 c. 3x – 2 d. 3x + 2

9. What should be multiplied to (x + 2) to get a product of (x2 + 10x + 16) ?

a. x + 8 b. x – 8 c. x + 18 d. x – 18

10. What is the product of squaring a binomial?

a. monomial b. binomial c. trinomial d. multinomial

11. In multiplying binomials (9x – 2)(x + 5) using FOIL method. Which terms
should be multiplied first?
a. (9x)(x) b. (9x)(5) c. (x)(-2) d. (-2)(5)

12. What is the product of (7x – 3y)2 ?

a. 14x2 + 6y2 c. 49x2 – 42x + 9y2

21
 b. 49x2 + 9y2 d. 49x2 + 42x + 9y2

13. What is the last term of the product of (7x + 4) and (7x – 4) ?
a. -8 b. -16 c. -28 d. 0

14. What is the middle term of the product of (5x – 8y)2 ?

a. –13xy b. –26xy c. –40xy d. –80xy

15. Multiply (3a – 4b)3.

a. 27a3 – 108a2b + 144ab2 – 64 b3
b. 27a3 + 108a2b – 144ab2 + 64 b3
c. 27a3 – 108a2b – 144ab2 + 64 b3
d. 27a3 + 108a2b – 144ab2 – 64 b3

Let Us Enhance

“Who Is He?”
The letters will form a name of a great mathematician. He was the first Greek
mathematician who recognized fractions as numbers; thus, he allowed positive
rational numbers for the coefficients and solutions. He was also the author of a series
of books called Arithmetica.

To find out the answer, simplify and write the letter corresponding to the answer in
the boxes below.

I ( 3a2 b)2 T ( 4x2 y2)3

P ( 2a2) (2a2 b3)5 H ( 6ab2)2

D ( 5xy3)2 A ( 3a2 b)2 (2ab2)2

S ( -2xy2)2 (2x2 y)2 U ( 2xy)2 (2x2 y)3 (2xy)

O (-3x2 y2)3 N (-3a3 b2 ) 2

25x2y 9a4b - 64a12b1 36a2b 36a6b 9a6b 64x6y 64x9y 16x6y

6 2 27x6y 5 4 6 4 6 6 6
6

22
 Let Us Reflect

Complete the reaction boxes by stating the concepts you have understood in this
module under ACCOMPLISHMENTS and writing the things that still confuse you
about the lesson under CHALLENGES.

23
 24
Let Us Enhance
Let Us Assess Let Us Practice More
1. b 11. a I. 1. x2 + 9x + 14
2. b 12. c 2. x2 – 4x – 32
3. c 13. b 3. x2 – 6x + 8
4. d 14. d 4. x2 – 4x – 5
5. a 15. A 5. 2x2 + 13x + 15
6. c
7. b II. 1. E 6. B
8. a 2. A 7. E
9. a 3. D 8. A
10. c 4. C 9. B
5. D 10. D
Let Us Practice Let Us Try
I. 1. Power of a Product a6b3c3 1. b
2. a
64𝑥 3 𝑦 6 3. c
2. Power of a Quotient
27𝑤 3 𝑧 3
4. d
3. Product of a Power m8 5. b
6. b
4. Power of a Power 6,561
7. c
1
5. Quotient of a Power 8. d
𝑥2𝑦5
9. b
II. 1. c 10. a
2. b 11. a
3. e 12. c
4. a 13. b
5. d 14. d
15. b
Answer Key
 References

https://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/f
oil_method.html

http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_
RESOURCE/U11_L2_T4_text_final.html

https://www.math-only-math.com/cube-of-a-binomial.html

https://brainly.ph/question/683476
https://courses.lumenlearning.com/prealgebra/chapter/multiplying-a-trinomial-
by-a-binomial/

https://mathbitsnotebook.com/Algebra1/Polynomials/POpolynomial.html
Elizabeth R. Aseron et. Al, Mathematics 7 Learner’s Material: Pasig City: DepEd
IMCS, 2013, 153-161

25
For inquiries or feedback, please write or call:

Department of Education – Region XI

F. Torres St., Davao City

Telefax:

Email Address: lrms.regionxi@deped.gov.ph

26
 7
Mathematics
Quarter 2 – Module 5:
Algebraic Expressions, Equations
and Inequalities
Mathematics – Grade 7
Quarter 2 – Module 5: Algebraic Expressions, Equations and Inequalities
First Edition, 2020

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 module 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.

Published by the Department of Education – Region XI

Regional Director: Allan G. Farnazo, CESO IV
Assistant Regional Director: Maria Ines C. Asuncion

Development Team of the Module

Writer: Teresita T. Manambay and Ruthsie M. Ponte
Editors: Flordelisa L. Parojinog and Jessica C. Sarmiento
Reviewer: Jessica C. Sarmiento
Illustrator:
Layout Artist:
Template Developer: Neil Edward D. Diaz
Management Team: Reynaldo M. Guillena, CESO V
Jinky B. Firman, PhD, CESE
Marilyn V. Deduyo
Alma C. Cifra, EdD
Aris B. Juanillo, PhD
May Ann M. Jumuad, PhD
Antonio A. Apat

Printed in the Philippines by ________________________________________

Department of Education – Region XI
Office Address: DepEd Davao City Division, E. Quirino Ave.
Davao City, Davao del Sur, Philippines
Telefax: (082) 224 0100
E-mail Address: info@deped-davaocity.ph
 7

Mathematics
Quarter 2 – Module 5:
Algebraic Expressions, Equations
and Inequalities
Introductory Message
For the facilitator:
As a facilitator, you are expected to orient the learners on how to use
this module. You also need to keep track of the learners' progress while
allowing them to manage their own learning at home. Furthermore, you are
expected to encourage and assist the learners as they do the tasks included
in the module.

For the learner:

As a learner, you must learn to become responsible of your own
learning. Take time to read, understand, and perform the different activities
in the module.
As you go through the different activities of this module be reminded of
the following:
1. Use the module with care. Do not put unnecessary mark/s on any part
of the module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer Let Us Try before moving on to the other
activities.
3. Read the instructions carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your
answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are done.
If you encounter any difficulty in answering the tasks in this module,
do not hesitate to consult your teacher or facilitator. Always bear in mind that
you are not alone. We hope that through this material, you will experience
meaningful learning and gain deep understanding of the relevant
competencies. You can do it!

ii
 Let Us Learn

At the end of this module, you are expected to:

1. Solve problems involving algebraic expressions (M7AL-IIg-2);

2. Differentiate algebraic expressions, equations and inequalities;
3. Illustrate linear equation and inequality in one variable (M7AL-IIh-4).

Let Us Try
Directions: Find out how much you already know about this module. Choose the
letter of the correct answer. Write your answer on a separate sheet of paper.

1. In a bowl of fruits, there are a mangoes and b papayas. In a paper bag, there
are 5 mangoes and 9 papayas. Write the total number of fruits as an
expression.
A. a + b + 5 B. a + b + 14 C. a + b + 9 D. a + b – 14

2. Mary has 50 mangoes. If she sells x mangoes per day for 7 days, how many
mangoes will he have left? Write an expression.
A. 7x - 50 B. 7x + 50 C. 50 - 7x D. 50 + 7x

3. Find the perimeter of an isosceles triangle whose two legs has the measurement
of x + 3 and the base is x.
A. 3x + 6 B. x + 6 C. 6 +3x D. 6 + x

4. Ann is m years old. John is five years younger than Ann. How old will John
be in 10 years?
A. m + 5 B. m – 5 C. m + 10 D. m - 10

5. Johnny takes h hours and m minutes to complete the typing job. His friend
Andrew takes as twice as long to finish typing. Write an algebraic expression
for Andrew’s typing speed.
A. 2 hm B. h + 2m C. 2h + m D. 2h + 2m

6. Angeline has 80 more pesos than Lily has. If c equals the amount of money
Lily has, which of the following expressions represents the amount Angeline
has?
A. 80c B. 80 - c C. c +80 D. c- 80

7. It is a statement in which two expressions, at least one containing the variable

are equal.
A. Equation B. Expression C. Inequality D. Equivalent

8. It is a mathematical sentence which states that the two expressions are

unequal.
A. Equation B. Expression C. Inequality D. Equivalent

3
 9. Which of the following is not an example of linear equations?
A. 2x + 10 = 30 B. 5(x- 8) = 60 C. y + 35 ≠ 10 D. 4x - 5 = 11

10. These symbols (>, <, ≤, ≥, ≠) indicates_______________.

A. Equation B. Expression C. Inequality D. Equivalent

11. Which of the following is not an example of linear inequality?

A. 2x + 10 = 30 C. y + 35 ≠ 10
B. x + 8 < 12 D. x ≤ 20

12. A mathematical sentence is to an equation, as a mathematical phrase is to

a/an __________________.
A. Inequality C. Variable
B. Polynomial D. Algebraic Expression

13. Which of the following is a linear equation in one variable?

A. x2 + x – f = 3 C. 8x = 40
B. 4 – x = y D. x3 = 8

14. It is a mathematical phrase which can be a variable, a constant or a

combination of both.
A. Equation C. Variable
B. Inequality D. Algebraic Expression

15. Which of the following statements describes a linear equation in one variable?
A. It has one or more algebraic terms in a phrase.
B. It has a definite solution.
C. It has more than one solution.
D. There is “>” sign.

Let Us Study

Solving Problems involving Algebraic Expressions

An Algebraic Expression is a mathematical phrase which combines numbers,

variables and operators to show the value of something. Learning how to translate
verbal phrases into mathematical phrases is very useful in solving worded problems
involving algebraic expressions. We must also remember to follow the steps in solving
word problems in order for us to solve it easily.

These are the steps in solving problems involving algebraic expressions:

1. Read the problem carefully. You may not be able to visualize all the details,
but you should gain a mental picture of what is generally being discussed.
2. Determine what the problem is asking for. A problem may provide you with
enough details to calculate all sorts of parameters, but the problem probably
will only be asking for one or two. Thus, figure out what you are trying to find
and write it down.

4
3. Identify the variables. As you have learned from the previous lesson, the
variables are symbols or letters in the alphabet. Use this to represent the
missing quantity.
4. Translate the problem into a mathematical expression. In order to solve the
expression, you have to translate from verbal phrases/sentence into a
mathematical expression.
5. Solve the problem and check your results. You can now solve the given
expression and make sure to check thoroughly the result or answer.

EXAMPLES:

1. If the length of a rectangular flower garden is thrice a number increased by 6

and its width is a number decreased by 8, find the perimeter of the rectangular
garden.

STEP 1 Read the problem carefully.

STEP 2 Determine what the problem is asking for.

Find the perimeter of the rectangular garden.

STEP 3 Identify the variables

Let x be the unknown variable.

STEP 4 Translate the problem into a mathematical expression.

3x + 6 thrice the number increased by 6 (for the length l )
x–8 the number decreased by 8 (for the width w )

STEP 5 Solve the problem and check your results.

To get the perimeter of a rectangular garden, we will simply add
twice the length and twice the width, since Perimeter = 2l + 2w.

2(3x + 6) Multiply by 2 for the length

6x + 12 Answer

2(x – 8) Multiply by 2 for the length

2x – 16 Answer

Then, combine similar terms:

P = 6x + 12 + 2x – 16
P = 8x -14 Final Answer.

2. Beverly has c chocolates and d drinks. Ethan has five fewer

chocolates than Beverly, but twice as many drinks. Write an
expression for Ethan snacks.

STEP 1 Read the problem carefully.

STEP 2 Determine what the problem is asking for.
Write an expression for Ethan snacks.
STEP 3 Identify the variables

5
 Let c – the number of chocolates and d – number of drinks

STEP 4 Translate the problem into a mathematical expression.

Replace the words with numbers and symbols.

Five fewer chocolates c-5
Twice as many drinks 2d

STEP 5 Solve the problem and check your results.

Ethan snacks
c-5 Ethan has five fewer chocolates than Beverly
2d and twice as many drinks

and combine, we have the answer c – 5 + 2d

EQUATION

An equation is a mathematical sentence indicating that two expressions are

equal. The symbol “equal sign” ( = ) is used to denote equality. The equation which
contains only one variable x, is called a linear equation in one variable. Examples are
4a = 20, b – 8 = 10, and 15 – 3c = 18. A linear equation in one variable is one which
can be written in the form ax + b = 0, where a and b are real numbers and a ≠ 0.
Also, linear equations are first degree polynomials. It means that the highest
exponent of its variable is 1.

A mathematical statement containing one variable can be classified according

to the nature of its solution.

An equation may either be true of false.

8 + 10 = 18 This is true equation, since both sides have the same value.
x + 8 = 15 This equation can either be true or false, depending upon the value
of x. This is Linear Equation.

Three types of linear equations

Linear equations in one variable can be classified into three types: identity,
conditional, or inconsistent.

An identity equation is true for all values of the variable. Here is an example
of an identity equation.
5x = 4x + x

The solution set consists of all values that make the equation true. For this equation,
the solution set is all real numbers because any real number substituted for x will
make the equation true.

6
 A conditional equation is true for only some values of the variable. For
example, if we are to solve the equation 4x + 2 = 3x - 1, we will arrive with:

4x + 2 = 3x – 1

4x - 3x = -1 - 2

x=-3

The solution set consists of one number : {-3}. It is the only solution and, therefore,
we have solved a conditional equation.

An inconsistent equation results in a false statement. For example, if we are

to solve 3x – 15 = 3 (x-4), we have the following:

3x - 15 = 3x - 4

3x – 15 – 3x = 3x – 12 – 3x subtract 3x from both sides.

-15 ≠ - 12 false statement

Indeed, -15 ≠ -12. There is no solution because this is an inconsistent equation

INEQUALITY

An inequality is a mathematical sentence indicating that two expressions are

not equal. The unequal sign ( ≠ ) is a symbol for inequality. If two expressions are
unequal, symbols are used to denote that one quantity is larger or smaller in value
than another. Symbols can be any of the following:

> greater than

≥ greater than or equal to
< less than
≤ less than or equal to

Examples of Linear Inequality in One Variable:

x ≥ 20 2n + 5 > 33 x - 7 < 24 y + 3 ≤ 25

Differences Between Algebraic Expressions, Equations and

Inequalities
Algebraic Expression Equation Inequality
• An equation is a
• An algebraic • An inequality is a
mathematical sentence
expression is a mathematical
indicating that two
mathematical phrase sentence indicating
expressions are equal.
which combines, that two
numbers, variables expressions are not
and operators to show equal.
the value of something.

7
 • It uses an equal sign (=) as • It uses an
• It only has operating
relation symbol. inequality sign (<,
symbols and no
relation symbols. ≤, >, ≥) as
relation symbol.

• It has one definite • It may have

• There is no solution. It
solution. several solutions.
can only be simplified
in the form of an
expression or a
numerical value.
• Example: • Example:
• Example:
6x - 2(3x + 7) 9x - 5 = 31 y<6

Let Us Practice

Activity 1: “I belong to you!”

Direction: Classify the following as algebraic expressions, equations and inequalities.

a+b+c=0 54x -x + 2 = 18
8x +5 c – 1 > -4 y > 15
-y+ 7 x≤5 x = -5
3+m<2 2x + 6 = 24 6(n + 8 )
x+y+z 5 – b ≤ -4 6(3) + 1 = x

Algebraic Expressions Equations Inequality

8
 Activity 2: “Identity, Conditional or Inconsistent Equation?”
A. Direction: Determine whether the given equation is identity, conditional or
inconsistent linear equation. Write your answer on the space provided before
the number.

_________1. 4x - x = 3x _______ 6. 2(x – 3) = 2x - 6

_________2. 2x + 1 = 2x - 8 ________7. x + 1 = -3

_________3. 4(x-2) = 4x-8 ________8. 6x – 20 = 6x + 3

_________4. 5x – 2 = 3x + 2x - 2 _______ 9. 3a – 5 + a = 6a + 5 - 2a

_________5. 2x-1=8 ________10. x+10=2x-3

Let Us Practice More

Direction: Solve the following problems. Write your answer as an expression.

1. Find the perimeter of a square if the side is x.
2. Find the perimeter of a rectangle if the longest side is 3x and the shorter side
is x.
3. The price of a repair and maintenance of the laptop is P1000.00. If the
technician finds more damages, an additional charge of P 500.00 per damage
to the bill. If the technician finds n damages, how much is the bill?
4. There are b players on the XYZ basketball team. The team scored a total of 70
points on the game. One of the players, Alex, scored 5 more points than the
average per player. How many points did Alex score?
5. Johnny earns P 35.00 per hours in working as a part-timer in a fast-food. He
worked for x hours this week. However, he was deducted P 40.00 for he was
late on Wednesday. How much money did he earn this week? Write your
answer as an expression.

Let Us Remember

1. Solving word problems involving algebraic expressions is easy when you

always remember to follow steps.

2. An algebraic expression is a mathematical phrase which combines

numbers, variables and operators to show the value of something. An
equation is a mathematical sentence indicating that two expressions are equal
while an inequality is a mathematical sentence that shows two expressions
are not equal.

9
 3. An equation uses an equal sign “=” while an inequality uses symbols such as
<, ≤,>, ≥ and ≠.
4. A linear equation in one variable is one which can be written in the form ax +
b = 0, where a and b are real numbers and a ≠ 0. It has three types: identity,
conditional and inconsistent.

Let Us Assess

Directions: Choose the letter of the correct answer. Write your answer on the separate
sheet of paper.

1. The following are examples of an equation except:

A. 4a + 7 = 24 B. x = 80 C. 6b – 10 D. 4x + 9 = 20

2. Which of the following is NOT an example of inequality?

A. 2x + 5 < 42 C. y + 35 ≠ 10
B. -2x + 30 > 5 D. 2a = 4

3. In a box, there are x white chocolates and y dark chocolates. In another box,
there are 5 white chocolates and 7 dark chocolates. What is the total number
pieces of chocolates combined?
A. x + 12 B. y + 12 C. 5x +0 D. x + y + 12

4. Which of the following does not belong to the group?

A. c + 5d B. a2x3+z C. 7x – 2y D. 2a = 4

5. It uses an equal sign (=) as relation symbol that shows equality between two
expressions.
A. Equation B. Expression C. Inequality D. Equivalent

6. Emelio takes h hours and m minutes to complete a mini- triathlon. His friend
Aguinaldo takes thrice as long to finish the race. Write and algebraic
expression for Aguinaldo’s race time.
A.3 hm B. 3h + 3m C. 3h + m D. h + 3m

7. What type of equation has no solution?

A. identity C. inconsistent
B. consistent D. none of the above

8. Adrian has r red marbles and b blue marbles. Mark keeps losing his marble,
and has half as many red marbles, and 5 less blue marbles than Adrian. Write
an expression for Mark’s total marbles.
2 𝑟 𝑟 𝑏
A. + b -5 B. + b +5 C. + b -5 D . + r -5
𝑟 2 2 2

9. Which is a mathematical sentence stating that the two expressions are

unequal?
A. Equation B. Inequality C. Expression D. Equivalent

10
 10. Which inequality symbol means “less than or equal to”?
A. < B. ≤ C. > D. ≥

11. Which of the following statements describes an inequality in one variable?

A. It has one or more algebraic terms in a phrase.
B. It has a definite solution.
C. It has more than one solution.
D. There is “ = ” sign.

12. Which of the following is a statement in which two expressions, at least one
containing the variable, are equal.
A. Equation B. Inequality C. Expression D. Equivalent

13. Sheila is eight years older than her brother. If her brother is m years old,
how old is Sheila five years ago?
A. m + 3 B. m – 3 C. m + 13 D. m – 13

14. Find the perimeter of a square whose side measures (x + 2) units.

A. 2(x + 2) B. 4(x + 2) C. x + 4 D. 2(x + 4)

15. Which of the following is an example of a linear inequality in one variable?

A. y > 2x -4 B. 3x2 + 5 ≤ 8 C. x + y < z D. 5x + 1 ≥ 16

Let Us Enhance

Write at least five verbal statements about your relationships with your classmates,
friends or parents. Translate them into mathematical statements.

1. _________________________________________________________.
2. _________________________________________________________.

3. _________________________________________________________.

4. _________________________________________________________.
5. _________________________________________________________.

Let Us Reflect
Comment on this statement:

“Equality is a boundary between less than and greater than”

11
 Answer Key

C 10.
C 9.
C 8.
A 7.
C 6.
B 15. D 5.
D 14. A 4.
C 13. A 3.
D 12. C 2.
A 11. B 1.
Try Let Us Let Us Practice: Activity 1

Conditional 10.
Inconsistent 9.
Inconsistent 8. B 10.
Conditional 7. 35x - 40 5. B 9.
Identity 6. C 8.
𝑏
Conditional 5. +5 4. C 7.
70
Identity 4. B 6.
Identity 3. 1000 + 500n 3. D 15. A 5.
Inconsistent 2. 6x + 2x = 8x 2. B 14. D 4.
Identity 1. A 13. D 3.
4x 1. A 12. D 2.
Activity 2 C 11. C 1.
Let Us Practice: Let Us Practice More Us Assess Let

References

Accessed July10,2020, https://www.mathnasium.com/lakeboone-news-problem-

solving-with-algebra
Accessed June 21 ,2021, http://www.differencebetween.net/language/difference-
between-inequalities-and-equations/

Accessed June 23 ,2021, https://keydifferences.com/difference-between-

expression-and-equation.html

Julieta G. Bernabe, Elementary Algebra, SD Publications, Inc., Quezon City, 2009.

12
For inquiries or feedback, please write or call:

Department of Education – Region XI

F. Torres St., Davao City

Telefax:

Email Address: lrms.regionxi@deped.gov.ph

7
Mathematics
Quarter 2 – Module 6:
Solving Linear Equations and
Inequalities in One Variable
Mathematics – Grade 7
Quarter 2 – Module 6: Solving Linear Equations and Inequalities in One Variable
First Edition, 2020

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 module 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.

Published by the Department of Education – Region XI

Regional Director: Allan G. Farnazo, CESO IV
Assistant Regional Director: Maria Ines C. Asuncion

Development Team of the Module

Writer: John Paul N. Paculdo and Jessica C. Sarmiento
Editors: Mirasol O. Fabuna and Niño Lito R. Salvan
Reviewer: Niño Lito R. Salvan
Illustrator:
Layout Artist:
Template Developer: Neil Edward D. Diaz
Management Team: Reynaldo M. Guillena, CESO V
Jinky B. Firman, PhD, CESE
Marilyn V. Deduyo
Alma C. Cifra, EdD
Aris B. Juanillo, PhD
May Ann M. Jumuad, PhD
Antonio A. Apat

Printed in the Philippines by Davao City Division Learning Resources Management

Development System (LRMDS)

Department of Education – Region XI

Office Address: DepEd Davao City Division, E. Quirino Ave.

Davao City, Davao del Sur, Philippines
Telefax: (082) 224 0100
E-mail Address: info@deped-davaocity.ph
 7

Mathematics
Quarter 2 – Module 6:
Solving Linear Equations and
Inequalities in One Variable
Introductory Message

For the facilitator:

As a facilitator, you are expected to orient the learners on how to use
this module. You also need to keep track of the learners' progress while
allowing them to manage their own learning at home. Furthermore, you are
expected to encourage and assist the learners as they do the tasks included
in the module.

For the learner:

As a learner, you must learn to become responsible of your own
learning. Take time to read, understand, and perform the different activities
in the module.
As you go through the different activities of this module be reminded of
the following:
1. Use the module with care. Do not put unnecessary mark/s on any part
of the module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer Let Us Try before moving on to the other
activities.
3. Read the instructions carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your
answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are done.
If you encounter any difficulty in answering the tasks in this module,
do not hesitate to consult your teacher or facilitator. Always bear in mind that
you are not alone. We hope that through this material, you will experience
meaningful learning and gain deep understanding of the relevant
competencies. You can do it!

ii
 Let Us Learn
After going through this module, you are expected to:

1. Find the solution of linear equation or inequality in one variable (M7AL-IIi-1).

2. Solve linear equation or inequality in one variable involving absolute value
by: (a) graphing; and (b) algebraic methods (M7AL-IIi-j-1).

Specifically, you will;

1. find the solution of linear equations or inequalities in one variable by using

the following:
a. guess and check
b. guess and check from a given replacement set
c. algebraic method
2. show solution set of linear equations or inequalities in one variable by
graphing.

Let Us Try!

MULTIPLE CHOICE: Read and understand each question carefully. Then, write the
letter of your answer on a separate sheet of paper.

1. What value of x satisfies the equation 3x - 7 = 8?

a. 1/3 b. 3 c. – 5 d. 5

2. Which equation has x =15 as a solution?

a. 3x = 12 c. 12x = 3
b. x - 3 = 12 d. 3x = 12 + 5

3. Given the replacement (- 20, - 8 , 0, 8, 20), which of the following is/are the
solution/s of the equation – ½ x + 7 = 3?
a. 8 b. - 8 c. 20 d. - 20

4. Given the replacement set ( - 2, -1, 0, 1, 2, 3), which of the following is/are
the solution/s of the inequality 3x - 1 > 2?
a. x = (-3, -1, 0, 2) c. x = (2, 3)
b. b. x = (0, 1, 2, 3) d. x = ( -2,-1, 0)

5. Consider this statement: “If milk + tea = milk tea, then milk tea = milk +
tea.” What property of equality does it show?
a. Reflexive Property c. Transitive Property
b. Symmetric Property d. Addition Property

1
6. What is the first step in solving absolute value equations in one variable?
a. Solve both equations.
b. Let the expression on one side of the equation consist only of a single
absolute value expression.
c. If the absolute value of an expression is equal to a positive number, say
a, then the expression inside the absolute value can either be a or –a.
Equate the expression inside the absolute value sign to a and to –a.
d. Determine if the number on the right side of the equation is negative or
not. If its negative then there is no solution for the given, otherwise,
proceed to step 3.

7. What is the solution of a linear equation whose absolute value is equal to a

negative number?
a. There is no solution.
b. There is only one solution.
c. There is a positive and a negative solution.
d. The solution is the set of all negative numbers.

8. By guess-and-check, what value would make the equation |x| + 7 = 10 , true?

a. -17 b. -3 c. 0 d. 17

9. In the equation |a + 10| = 4 , what is a + 10 equal to?

a. -4 and 4 b. -10 and 10 c. 4 and 10 d. 6 and 14

10. Which of the following is NOT a solution of the inequality |3b + 9| ≤ 12

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

11. Which of the following is the solution set of |y - 2| > 5 ?

a. { y|y < -3 or y > 7 } c. { y|y < -7 or y > 3 }
b. { y|y < 3 or y > -7 } d. { y|y < 7 or y > -3 }

12. Which of the following graphs shows the solution of the equation 2x + 9 = 16?

a.

b.

c.

d.

2
 13. Which of the following graph shows the solutions of the inequality 2x - 5 >
6?
a.

b.

c.
d.

14. Which of the following graphs represents the solution set of |x + 2| = 7 ?

a.

b.

c.

d. *

15. Which of the following graphs shows the solution set of |x + 3| ≤ 4 ?

a. *

b.

c.

d.

Let Us Study
Finding the Solution of Linear Equation in One Variable

Method 1: Guess and Check

We can find the solution of a linear equation through Guess and Check.

3
Solution of a linear equation is a value, such that, when you replace the variable with
it, it makes the equation true. Linear equation in one variable can be written in the
form of ax + b = 0 where a and b are real numbers and a ≠ 0.

Example: Find the solution of the linear equation x + 20 = 50.

x + 20 = 50 Guess a number that when

you add to 20 the sum is 50.

30 + 20 = ? We guess it is 30.

50 = 50 Therefore 30 is a solution.

Method 2: Guess and Check from a given Replacement Set.

Example: Given, 2x – 3 = 17, find the solution from the given replacement set
{– 9, – 6, 0, 4, 10}.

Solution:
2x – 3 = 17
For x = – 9: For x = – 6: For x = 0: For x = 4: For x = 10:
2(– 9) - 3 = ? 2(– 6) – 3 = ? 2(0) – 3 = ? 2(4) – 3 = ? 2(10) – 3 = ?
– 18 – 3 = 17 – 12 – 3 = 17 0 – 3 = 17 8 – 3 = 17 20 – 3 = 17
– 21 ≠ 17 – 15 ≠ 17 – 3 ≠ 17 5 ≠ 17 17 = 17

Thus, – 9 is Thus, – 6 is Thus, 0 is not Thus, 4 is a Thus, 10 is a

not a not a solution. a solution. not solution. solution.
solution.

Method 3: Algebraic Method

Before we use algebraic method, let us discuss first the different properties of
equality. These properties are important in order for us to arrive with solutions for
linear equation.

Properties of Equality
1. Reflexive Property of Equality
For each real number a, a = a.
Examples: a) 19 = 19 b) – d = – d c) x + 5 = x + 5
2. Symmetric Property of Equality
For any real numbers a and b, if a = b then b = a.
Examples: a) If 10 + 9 = 19, then 19 = 10 + 9. b) If x – 3 = 5, then 5 = x – 3.
3. Transitive Property of Equality
For any real numbers a, b, and c, If a = b and b = c, then a = c.
Examples: a) If 7 + 3 = 10 and 10 = 6 + 4, then 7 + 3 = 6 + 4.
b) If x – 25 = y and y = 19, then x – 25 = 19.

4
4. Substitution Property of Equality
For any real numbers a and b: If a = b, then a may be replaced by b, or b may be
replaced by a, in any mathematical sentence without changing its meaning.
Examples: a) If p + d = 19 and p = 10, then 10 + d = 19.
b) If 16 – x = 10 and x = 6, then 16 – 6 = 10.

5. Addition Property of Equality (APE)

For all real numbers a, b, and c, a = b if and only if a + c = b + c.
If we add the same number to both sides of the equal sign, then the two sides
remain equal.
Examples:
a) 10 + 19 = 29 is true if and only if 10 + 9 + 100 = 29 + 100. It is true because the
same number, 100, was added to both sides of the equation.
b) x + 4 = 5 is true if and only if x + 4 + 10 = 5 + 10. It is true because the same
number, 10, was added to both sides of the equation.

6. Multiplication Property of Equality (MPE)

For all real numbers a, b, and c, where c ≠ 0, a = b if and only if ac = bc. If we
multiply the same number to both sides of the equal sign, then the two sides
remain equal.
Example:
a) 7 · 9 = 63 is true if and only if (7 · 9) · 3 = 63 · 3. It is true because the same
number, 3, was multiplied to both sides of the equation.
b) 5x = 19 is true if and only if 5x · 1/5 = 19 · 1/5. It is true because the same
number, 1/5, was multiplied to both sides of the equation.

Is there Subtraction or Division Property of Equality?

None, because subtracting or dividing the same number from both sides of an
equation are already covered by APE and MPE. Subtracting the same number from
both sides of an equality is the same as adding a negative number to both sides of
an equation. Dividing the same number from both sides of an equality is the same
as multiplying the reciprocal of the number to both sides of an equation.

Tips in solving Linear equation in one variable.

1. We may add, subtract, multiply, or divide an equation by a number or an
expression as long as we do the same thing to both sides of the equal sign (refer to
the Properties of Equality). Note that we cannot divide by zero.
2. Isolate the variable on one side of the equation.
3. When the variable is multiplied by a coefficient in the final stage, multiply both
sides of the equation by the reciprocal of the coefficient (MPE).

Illustrative Examples
Example 1: Solve x + 21 = 3.
Solution: x + 21 = 3 Given
x + 21 + (-21) = 3 + (-21) Add -21 to both sides of the equation (APE)
x = -18 final answer
Example 2: Solve x - 3 = 10.
Solution: x - 3 = 10 Given
x - 3 + 3= 10 + 3 Add 3 to both sides of the equation (APE)
x = 13 final answer

5
Example 3: Solve 4x = 32.
Solution: 4x = 32 Given
1 1 1
4x ∙ = 32 ∙ Multiply to both sides of the equation (MPE)
4 4 4
x=8 final answer
1
Multiplying both sides of the equation with is the same with dividing both sides
4
of the equation by 4. So, we can have an alternate solution of:

Solution: 4x = 32 Given
4x = 32 Divide both sides of the equation by 3(MPE)
4 4
x=8 final answer

More Examples
Example 1: Solve and graph the solution of the linear equation 3x + 8 = 23.

3x + 8 = 23 Given
3x + 8 + (– 8) = 23 + (– 8) Addition Property of Equality
Add both sides by negative 8
3x = 15 Get the reciprocal of the coefficient
1/3 is the reciprocal of 3.
1
∙ 3x = 15 ∙
1 Multiplication Property
3 3
Multiply both sides by 1/3
x=5 The solution is 5.

The graph shows the solution of the equation 3x + 8 = 23.

Example 2: Solve and graph the solution of the linear equation 5x + 3 = x – 13.

5x + 3 = x – 13 Given
5x + 3 + (–3) = x – 13 + (–3) Addition Property of Equality
Add both sides by negative 3 then
5x = x - 16 simplify.
Addition Property of Equality
5x + (-x) = x + (-x) - 16 Add both sides by negative x then
simplify.
4x = - 16 The reciprocal of 4 is .
1
4
1
∙ 4x = -16 ∙
1 Multiplication Property of Equality
4 4 1
Multiply both sides by
4
x=-4 The solution is - 4.
.
The graph shows the solution of the equation 5x + 3 = x – 13.

6
 Finding the Solution of Linear Inequality in One Variable

Solution of an inequality is a value, such that, when you replace the variable with
it, it makes the inequality true. Linear inequality in one variable can be written in
the form of ax + b ≥, >, ≤, < 0 where a and b are real numbers and a ≠ 0.

Method 1: Guess and Check from the given Replacement Set

Example 1. Given, x – 7 ≤ 15, find the solution/s from the given replacement set
{– 9, 0, 22, 23, 30}.

x – 7 ≤ 15
For x = – 9: For x = 0: For x = 22: For x = 23: For x = 30:
– 9 – 7 ≤ 15 0 – 7 ≤ 15 22 – 7 ≤ 15 23 – 7 ≤ 15 30 – 7 ≤ 15
–16 ≤ 15, true – 7 ≤ 15, true 15 ≤ 15, true 16 ≤ 15, false 23 ≤ 13, false

Thus, – 9 is a Thus, 0 is a Thus, 22 is a Thus, 23 is Thus, 30 is

solution. solution. solution. not a solution. not a solution.

Based on the answers, part of the solution set are -9, 0, and 22.

Example 2. Given, 2x + 1 > 11, find the solution/s from the given replacement set
{– 6, 5, 6, 12, 15}.

2x + 1 > 11
For x = – 6: For x = 5: For x = 6: For x = 12: For x = 15:
2(– 6) + 1 > 11 2(5) + 1 > 11 2(6) + 1 > 11 2(12) + 1 > 11 2(15) +1 > 11
– 12 + 1 > 11 10 + 1 >11 12 + 1 > 11 24 + 1 > 11 30 + 1 > 11
–11 > 11, false 11 > 11, false 13 > 11, true 25 > 11, true 31 > 11, true

Thus, – 6 is not Thus, 5 is not a Thus, 6 is a Thus, 12 is a Thus, 15 is a

a solution. solution. solution. solution. solution.

Based on the answers, part of the solution set are 6, 12, and 15.

Method 2: Algebraic Method

Properties of Inequalities
1. Trichotomy Property
For any number a and b, one and only one of the following is true: a < b, a =
b, or a > b.
Example: One and only one of the following is true: 3 < x, 3 = x, or 3 > x.

2. Transitive Property of Inequality

For any numbers a, b and c, (a) if a < b and b < c, then a < c, and (b) if a > b
and b > c, then a > c.
Examples:
a) If 7 < x and x < 10, then 7 < 10.
b) If 10 > x and x > 5, then 10 > 5.

7
3. Addition Property of Inequality (API)
For all real numbers a, b and c: (a) if a < b, then a + c < b + c, and
(b) if a > b, then a + c > b + c.
Adding the same number to both a and b will not change the inequality.
Examples:
a) If x + 3 < 5, then x + 3 + 10 < 5 + 10.
b) If x +15 > 5, then x + 15 + (-15) > 5 + (-15).

4. Multiplication Property of Inequality (MPI)

For all real numbers a, b and c, then all the following are true:
(a) if c > 0 and a < b, then ac < bc;
(b) if c > 0 and a > b, then ac > bc.
(c) if c < 0 and a < b, then ac > bc;
(d) if c < 0 and a > b, then ac < bc.
Examples:
1 1
a) If 2x < 5, then ∙ 2x < ∙ 5
2 2
1 1
b) If -3x < 10, then − ∙ −3x > − ∙ 10
3 3

This simply means that if we multiply a positive number to both sides of an

inequality, the inequality symbol will not change. However, if we multiply a negative
number to both sides of the inequality, the inequality symbol will change.

Is there Subtraction or Division Property of Inequality?

None, same with the equality, there is no Subtraction or Division Property of
Inequality because subtracting or dividing the same number from both sides of an
inequality are already covered by API and MPI. Subtracting the same number from
both sides of an inequality is the same as adding a negative number to both sides of
an inequality. Dividing the same number from both sides of an inequality is the same
as multiplying the reciprocal of the number to both sides of an inequality.

Tips on Solving Linear Inequalities in One Variable

Solving linear inequalities in one variable have the same procedure
with solving linear equations. We will simplify both sides, get all the terms with the
variable on one side and the numbers on the other side, and then multiply/divide
both sides by the coefficient of the variable to get the solution (Properties of
Inequality).

Example 1. Solve and graph the solutions of the linear inequality 7x – 4 ≥ 10.
7x – 4 ≥ 10 Given
7x – 4 + 4 ≥ 10 + 4 Addition Property of Inequality
Add 4 both sides then simplify.
7x ≥ 14 1
is the reciprocal of 7.
7
1 1 Multiplication Property of Inequality
∙ 7x ≥ 14∙ 1
7 7
Multiply both sides by
7

x ≥2 The solutions are all real numbers

greater than or equal to 2.

8
In symbols the solution set is {x|x ≥ 2}. The graph shows the solutions of
7x – 4 ≥ 10 which is x ≥ 2 .

Example 2. Solve and graph the solutions of the linear inequality – 3x + 5 ≤ – 16.

– 3x + 5 ≤ – 16 Given
– 3x + 5 + (–5) ≤ – 16 + (–5) Add negative 5 both sides
– 3x ≤ - 21 (- 1/3) is the reciprocal of -3.
1 1
− 3 ∙ −3x ≤ -21∙ − 13 Multiply both sides by −
3

Note: Reverse the inequality symbol

because we multiplied a negative number
x ≥7 to both sides of the equation.
The solutions are real numbers greater
than or equal to 7.

In symbols the solution set is {x|x ≥ 7}. The graph shows the solutions of
– 3x + 5 ≤ – 16 which is x ≥ 7.

As we look at the solutions of linear equations and inequalities in one variable,

we observe that linear equation in one variable may have a unique solution, but
linear inequalities in one variable may have many solutions.

Solving Linear Equation in One Variable Involving Absolute Value

by Algebraic Method and Graphing
Many absolute value equations are not easy to solve by the guess-and-check method.
An easier way may be to use the following procedure.

To solve an absolute value equation algebraically, we simply follow the steps:

Step 1: Let the expression on one side of the equation consist only of a
single absolute value expression.
Step 2: Determine if the number on the right side of the equation is negative or not.
If its negative then there is no solution for the given, otherwise, proceed to
step 3.
Step 3: If the absolute value of an expression is equal to a positive number, say a,
then the expression inside the absolute value can either be a or –a. Equate
the expression inside the absolute value sign to a and to –a.
Step 4: Solve both equations.

9
Example 1: Solve and graph |2a + 4| – 6 = 20.

Step 1: Let the

expression on one side
of the equation consist |2a + 4| – 6 = 20
only of a |2a + 4| = 20 + 6 APE
single absolute value
expression. |2a + 4| = 26

Step 2: Determine if
the number on the
Since the number on the right side, which is 26, is
right side of the
positive, we proceed to step 3.
equation is negative or
not.

Step 3: Equate the

expression inside the
2a + 4 = 26 2a + 4 = –26
absolute value sign to
26 and to –26.

Step 4: Solve both 2a + 4 = 26 2a + 4 = -26

equations. 2a = 26 - 4 2a = -26 - 4
2a = 22 2a = -30
2a = 22 2a = -30
2 2 2 2
a = 11 a = -15

We can check that these two solutions make the original equation true.
If a = 11, then |2a + 4| – 6 = |2(11) + 4| – 6 = 26 – 6 = 20.
Also, if a = -15, then |2a + 4| – 6 = |2(-15) + 4| – 6 = 26 – 6 = 20.

Graphing the solution set {-15, 11}, we have,

Example 2: Solve and graph |8x + 1| + 10 = 7.

Step 1: Let the expression on one side |8x + 1| + 10 = 7

of the equation consist only of a
single absolute value expression. |8x + 1| = 7 – 10

|8x + 1| = -3

Step 2: Determine if the number on the Since the number on the right side of
right side of the equation is negative or the equation, which is -3, is negative,
not. then the given absolute value equation
has no solution.

There is no graph since there is no solution.

10
Example 3: Solve and graph |x – 5| = |3x + 5|.

Step 1: Let the

expression on one
side of the equation The expression on the left side of the equation has
consist only of a already have a single absolute value expression.
single absolute value
expression.

Step 2: Determine if
the number on the
|3x + 5| is positive, so we proceed to step 3.
right side of the
equation is negative
or not.

Step 3: Equate the

first expression inside
x – 5 = +(3x + 5) x – 5 = -(3x + 5)
the absolute value
sign to +(3x + 5) and
to –(3x + 5).

Step 4: Solve both x – 5 = +(3x + 5) x – 5 = -(3x + 5)

equations. x – 5 = 3x + 5 x – 5 = -3x - 5
x - 3x = 5 + 5 x + 3x = -5 + 5
-2x = 10 4x = 0
-2 -2 4 4
x = -5 x=0

We can check now the two values of x if it satisfy the original equation. If x = -5 and
x = 0.
|x – 5| = |3x + 5| |x – 5| = |3x + 5|
|-5 – 5| = |3(-5) + 5| |0 – 5| = |3(0) + 5|
|-10| = |-15 + 5| |-5| = |5|
10 = 10, True 5 = 5, True

Graphing the solution set {-5, 0}, we have,

Example 4: Solve and graph|x + 5| = |x – 4|

Step 1: Let the

expression on one side of
the equation consist only The expression on the left side of the equation
of a already have a single absolute value expression.
single absolute value
expression.

11
 Step 2: Determine if the
number on the right side
|x - 4| is positive, so we proceed to step 3.
of the equation is
negative or not.

Step 3: Equate the first

expression inside the
x + 5 = +(x - 4) x + 5 = -(x - 4)
absolute value sign to
+(x - 4) and to –(x - 4).

Step 4: Solve both x + 5 = +(x - 4) x + 5 = -(x - 4)

equations. x+5=x-4 x + 5 = -x + 4
x - x = -4 - 5 x+x=4-5
0 = -9 2x = -1
2 2
This is a false statement. x = -½
There is no solution in
this equation. This is an
example of an
inconsistent equation you
have learned in the
previous topics.

The original equation is only satisfied by only one value: {-½}

Graphing the solution set {-½}, we have,

Example 5: Solve and graph |x - 7| = |7 – x|.

Step 1: Let the

expression on one side of
the equation consist only The expression on the left side of the equation
of a already have a single absolute value expression.
single absolute value
expression.

Step 2: Determine if the

number on the right side
|7 - x| is positive, so we proceed to step 3.
of the equation is
negative or not.

Step 3: Equate the first

expression inside the
x - 7 = +(7 - x) x - 7 = -(7 - x)
absolute value sign to
+(7 - x) and to –(7 - x).

12
 Step 4: Solve both x - 7 = +(7 - x) x - 7 = -(7 - x)
equations. x-7=7-x x - 7 = -7 + x
x+x=7+7 x - x = -7 + 7
2x = 14 0=0
2 2
x=7 This is an identity equation.
This will always be true no
matter what the of x is. Thus,
all real numbers are
solutions to this equation.

The solution set in the original equation is the set of all real numbers, so its graph
shows infinity.

All Real Numbers

Solving Linear Inequality in One Variable Involving Absolute Value

by Algebraic Method and Graphing

Case 1: To solve an absolute value inequality < or ≤, we express the inequality as an

equivalent conjunction.

|x| ≤ a, where a ≥ 0, is equivalent to the conjunction x ≤ a and x ≥ - a.

Example: Solve the inequality: |2x - 5| < 9 and graph the resulting statement.
Solution: Since the inequality involves a “less than” sign, so we set up the
expression in the absolute value as less than 9 and greater than -9.
|2x - 5| < 9

2x - 5 < 9 and 2x - 5 > - 9

2x < 9 + 5 and 2x > -9 + 5
2x < 14 and 2x > - 4
2 2 2 2
x<7 and x>-2

Therefore, the solution of this inequality is {x|-2 < x < 7}. Graphing the result,
we shade the numbers that are less than 7 but greater than -2.

Note: We use an open circle if the final answer is not part of the solution set. If it is
part of the solution set, then we will use a closed circle.

13
Case 2: To solve an absolute value inequality > or ≥, we express the inequality as an
equivalent disjunction.

|x|≥ a, where a ≥ 0, is equivalent to the disjunction x ≤ - a and x ≥ a.

Example: Solve the inequality: |2x - 1| ≥ 7 and graph the resulting statement.

Solution: Since the inequality involves a “greater than” or equal to sign, so we set up
the expression in the absolute value as less than or equal to -7 and greater than or
equal to 7.

|2x - 1| ≥ 7

2x – 1 ≤ - 7 or 2x – 1 ≥ 7
2x ≤ -7 +1 or 2x ≥ 7 + 1
2x ≤ - 6 or 2x ≥ 8
2 2 2 2
x≤-3 or x≥4

Therefore, the solution of this inequality is {x|x ≤ - 3 or x ≥ 4}. Graphing the

result, we shade the numbers that are less than or equal to -3 and numbers greater
than or equal to 4. We used a closed circle since the answers are included in the
solution set.

Let Us Practice

I-Directions:
A. Determine which among the given replacement set {–3, –2, –1, 0, 1, 2, 3} is the
solution for each equation.
1) x+5=4
2) x–3=–2
3) 2x + 7 = 7
4) 5x – 13 = – 3
5) 3x + 2 = 2x - 1

B. Identify the property of equality shown in each sentence

1) If 5 · 4 = 20 and 20 = 2 · 10, then 5 · 4 = 2 · 10
2) 11 = 11
3) If x + 3 = 9, then x + 3 + (–3) = 9 + (–3)
4) If 1 + 5 = 6, then 6 = 1 + 5.
1 1
5) If 3x = 10, then ( ) (3x) = ( )(10)
3 3

14
C. Fill-in the blanks with correct expressions indicated by the property of equality
to be used.
1) If 2 + 8 = 10, then 10 = ____ (Symmetric Property)
2) (80 + 4) · 2 = 84 · ____ (Multiplication Property)
3) 11 + 8 = 19 and 19 = 10 + 9, then 11 + 8 = _____ (Transitive Property)
4) 5 + 10 + (–10) = 15 + ____ (Addition Property)
5) 10 = ____ (Reflexive Property)

D. Determine which among the given replacement set is part of the solution
for each inequality.
1) 2x + 5 > 7 ; {–6, –3, 4, 8, 10} 4) 2x ≤ 3x –1 ; { –5, –3, –1, 1, 3 }
2) 5x + 4 < –11 ; {–7, –5, –2, 0 } 5) 11x + 1 < 9x + 3 ; { –7, –3, 0, 3, 5}
3) 3x – 7 ≥ 2; { –2, 0, 3, 6 }

E. Identify the property of inequality shown in each sentence.

1) If m > 7 and 7 > n, then m > n.
2) If c > d and p < 0, then cp < dp.
3) if a < b, then a + 5 < b + 5.
4) If c > d and p > 0, then cp > dp
5) For any number a and b, one and only one of the following is true: a < b, a =
b, or a > b.

Let Us Practice More

Direction: Solve the following linear equations and inequalities in one variable.

A. Solve the following linear equations in one variable applying the Properties of
Equality. Show your solution.
1) x + 6 = 10 4) 2x - 3 = 13
2) 5x – 4 = 11 5) 5 – x = 3
3) 2x = 8 – 16

B. Solve the following inequality in one variable algebraically. Show your

solution.
1) x + 4 > 7 4) 3x + 4 ≥ 25
2) 5x – 8 < 22 5) -2x - 1 > 9
3) x + 7 ≤ 15

C. Solve the following absolute value equations and graph its solution set.
1) |m| – 3 = 7 4) |3a – 8| + 4 = 11
2) |2v| – 4 = 8 5) |10 – u| = |u – 10|
3) |4x + 2| – 3 = –7

D. Solve the following absolute value inequalities and graph its solution set.
1) |x – 4| < 5 4. |2x – 5| < 9
2) |2x + 3| > 13 5. |2x – 1| ≥ 7
3) |3x – 7| – 4 > 10

15
 Let Us Remember

Let us summarize important things about the lesson to guide us in finding the
solution of linear equations/inequalities in one variable.

➢ To find the solution of linear equations/inequalities in one variable, always

remember that you have to learn how to evaluate linear equations/inequalities in
one variable first. Second, you can use Guess and check so you can determine
whether the given replacement set is true or false. Third, apply the properties of
equality and inequality in finding the value of unknown x. Lastly, do not forget
that linear equations in one variable may have a unique solution, but linear
inequalities in one variable may have many solutions.

➢ To solve an absolute value equation algebraically, we simply follow the steps:

Step 1: Let the expression on one side of the equation consist only of a
single absolute value expression.
Step 2: Determine if the number on the right side of the equation is negative or
not. If its negative then there is no solution for the given, otherwise, proceed to
step 3.
Step 3: If the absolute value of an expression is equal to a positive number, say
a, then the expression inside the absolute value can either be a or –a. Equate the
expression inside the absolute value sign to a and to –a.

Step 4: Solve both equations.

➢ To solve an absolute value inequality < or ≤, we express the inequality as an

equivalent conjunction.

|x| ≤ a, where a ≥ 0, is equivalent to the conjunction x ≤ a and x ≥ - a.

➢ To solve an absolute value inequality > or ≥, we express the inequality as an

equivalent disjunction.
|x|≥ a, where a ≥ 0, is equivalent to the disjunction x ≤ - a and x ≥ a.

Let Us Assess
Multiple Choice. Let us assess what you have learned on the lesson by answering the
problems below. Write the letter of your answer on a piece of paper.

1. What value of x satisfies the equation 2x + 5 = 3?

b. 0 b. – 2 c. – 1 d. – 3

2. Which equation has x = 5 as a solution?

a. 2x = 10 b. x - 5 = 10 c. 2x = 5 d. 2x = 10 + 5

16
3. Which of the following is NOT a solution of the inequality |2a - 6| > 12
a. -12 b. 0 c. 10 d. 12

4. Given the replacement set (-3, - 2, -1, 0, 1, 2, 3), which of the following is/are the
solution/s of the inequality 3x + 4 < 6?
a. x = (-3, -1, 0, 2) c. x = (- 3, -2, -1, 0)
b. x = (0, 1, 2, 3) d. x = ( -2,-1, 0, 1)

5. What property of equality is applied to find the solution of a - 10 = 5?

a. Reflexive Property c. Transitive Property
b. Symmetry Property d. Addition Property

6. If – 5x > 20, then,

a. x > - 4 b. x > 4 c. x < - 4 d. x < 4

7. What is a - 25 equal to, in the equation |a - 25| = 10 ?

a. -25 and 25 b. -10 and 10 c. -15 and 35 d. 15 and 35

8. What is/are the solution of the equation |- 2x| + 50 = 10?

a. (-20, 20) b. (-30, 30) c. (-40, 40) d. no solution

9. What is/are the solution of the equation |2x + 1| = |3x - 2|?

a. (3, 1/5) b. (- 3, 1/5) c. (3, - 1/5) d. (- 3, -1/5)

10. What is/are the solution of the inequality |2x - 1| ≥ 7?

a. x ≥ 3 or x ≥ - 4 c. x ≤ - 3 or x ≥ 4
b. x ≥ 3 or x ≥ 4 d. no solution

11. If | x + 1 | < 3 , then,

a. – 3 < x < 3 c. – 4 < x < 2
b. – 3 > x > 3 d. – 4 > x > 2

12. Which of the following graphs shows the solution of the equation 3x - 2 = 10?

a.

b.

c.

d.

17
13. Which of the following graphs shows the solution of the inequality 2x - 5 ≤ 6?

a.

b.

c.

d.

14. Which of the following graph shows the solutions of the equation |2x - 7| = 3?

a.

b.

c.

d.

15. Which of the following graph shows the solutions of the inequality |3x - 8| ≥
7?

a.

b.

c.

d.

18
 Let Us Enhance

Let’s face the problem!

Procedure
1. Create a situation in real-life where linear equation or inequality in one
variable is applied. You may research some problems as your guide
in creating a problem and then solve it whichever way you want.
2. Prepare diagrams or pictures that will help us to visualize the
situation/problem that you have made. You may also prepare the
necessary table or graph to present the important data in your
situation/problem and the correct equation or inequality and steps
to solve the problem. Show that you know about the topic by using
concepts about the properties of real numbers as applied in linear
equations or inequalities in one variable to describe the situation.
3. Timely, relevant, and pressing issues or situations such as the
Covid19 pandemic, business, etc. that we face today are more
encouraged.

Rubric on Problems and Equations Formulated and Solved

Score Descriptions
5 Poses a more complex problem with two or more
solutions and communicates ideas unmistakably,
shows in-depth comprehension of the pertinent
concepts and/or processes and provides
explanation wherever appropriate.
Equations/inequalities are properly formulated
and solved correctly.
3 Poses a more complex problem and finishes all
significant parts of the solution and communicates
ideas unmistakably, shows in-depth
comprehension of the pertinent concepts and/or
processes. Equations/inequalities are properly
formulated but not all are solved correctly.
1 Poses a complex problem and finishes all
significant parts of the solution and communicates
ideas unmistakably, shows in-depth
comprehension of the pertinent concepts and/or
processes. Equations/inequalities are properly
formulated but are not solved correctly.

19
 Let Us Reflect
Reflect on what you have learned about this week’s lesson on Solving Linear
Equations and Inequalities in One Variable. Check the box of the emoticon to show
what you thought about the lesson and state why you felt that way.

How Do You Feel About This Week’s Lesson?

I feel ……………………

Great !!! Okay Not so good

because _____________________________________________
__________________________________________________
__________________________________________________
__________________________________________________
_________________________________________________.

20
 21
Let Us Practice Let Us Practice
A.
D. 1. x = -1
1. { 4, 8, 10} 2. x = 1
2. { - 7, - 5} 3. x = 0
3. {3, 6} 4. x = 2
4. {1, 3} 5. x = -3
5. {- 7, - 3, 0}
B.
E. 1. Transitive Property of Equality
1. Transitive Property of Inequality 2. Reflexive Property of Equality
2. Multiplication Property of Inequality 3. Addition Property of Equality
3. Addition of Property of Inequality 4. Symmetry Property of Equality
4. Multiplication Property of Inequality 5. Multiplication Property of Equality
5. Trichotomy Property of Inequality
C.
1. 2 + 8
2. 2
3. 10 + 9
4. (-10)
5. 10
Let Us Try
1. d 6. b 11. a
2. b 7. a 12. c
3. a 8. b 13. b
4. c 9. a 14. d
5. b 10. d 15. a
Answer Key
 22
Let Us Practice More Let Us Practice More
B. A.
1) x+4>7 1) x + 6 = 10
x + 4 + (-4) > 7+ (-4) API x + 6 + (-6) = 10 + (-6) APE
x>3 x=4
2) 5x – 8 < 22 2) 5x – 4 = 11
5x – 8 + 8 < 22 + 8 API 5x – 4 + 4 = 11 + 4 APE
5x < 30 5x = 15
1 1 1 1
( ) 5x < 30 ( ) MPI ( ) 5x = 15 ( ) MPE
5 5 5 5
x<6 x=3
3) x + 7 ≤ 15 3) 2x = 8 – 16
x + 7 + (-7) ≤ 15 + (-7) API 2x = - 8
x ≤8 1 1
( ) 2x = (- 8 )( ) MPE
2 2
x = -4
4) 3x + 4 ≥ 25
3x + 4 + (-4) ≥ 25 + (-4) API
3x ≥ 21 4) 2x - 3 = 13
1 1
(
) 3x ≥ 21 ( ) MPI
3 3 2x - 3 + 3 = 13 + 3 APE
x≥7 2x = 16
1 1
( ) 2x = 16 ( ) MPE
2 2
5) 2x - 1 > 9 x = 8
– 2x - 1 + 1 > 9 + 1 API
– 2x > 10
1 1 5) 5–x=3
(- )(– 2x) < (10)(- ) MPI
2 2
5 + (-5) – x = 3 + (-5) APE
x<-5
-x = - 2
( -1 )( -x ) = (- 2 )( -1 ) MPE
x=2
 23
Let Us Practice More
C.
1) m = {– 10, 10}
2) v =( – 6, 6)
3) There is no graph because there is no solution.
1
4) a = { 3, 5 }
5) All real numbers {u|u ∈ R} or R ( - ∞ , ∞ )
D.
1) Solution Set: {x|-1 < x < 9}
2) Solution Set: {x|x < -8} and {x|x > 5}
7
3) Solution Set: {x|x < - } and {x|x
3
4) Solution Set: {x|- 2 < x < 7}
5) Solution Set: {x|x ≤ - 3} and {x|x ≥ 4}
 15. a 10. c 5. d
14. d 9. a 4. c
13. c 8. d 3. b
12. b 7. b 2. a
11. c 6. c 1. c

Let Us Assess

Answers may vary Answers may vary

Let Us Enhance Let Us Reflect

References

Ricardo M. Crisostomo, et. al., Our World of Math: Quezon City: Vibal Publishing
House Inc., 2013, 193-242.

Orlando A. Oronce and Marilyn O. Mendoza, Worktext in Mathematics, E-math:

Manila: Rex Book Store, Inc., 2012, 233-336

24
For inquiries or feedback, please write or call:

Department of Education – Region XI

F. Torres St., Davao City

Telefax:

Email Address: lrms.regionxi@deped.gov.ph

25
7
Mathematics
Quarter 2 – Module 7:
Solving Problems Involving
Linear Equation and Inequality
in One Variable
Mathematics – Grade 7
Quarter 2 – Module 7: Solving Problems Involving Linear Equation and Inequality in
One Variable
First Edition, 2020

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Published by the Department of Education – Region XI

Regional Director: Allan G. Farnazo, CESO IV
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Printed in the Philippines by ____________________________________________

Department of Education – Region XI

Office Address: DepEd Davao City Division, E. Quirino Ave.

Davao City, Davao del Sur, Philippines
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E-mail Address: info@deped-davaocity.ph
 7

Mathematics
Quarter 2 – Module 7:
Solving Problems Involving
Linear Equation and Inequality
in One Variable
Introductory Message

For the facilitator:

As a facilitator, you are expected to orient the learners on how to use
this module. You also need to keep track of the learners' progress while
allowing them to manage their own learning at home. Furthermore, you are
expected to encourage and assist the learners as they do the tasks included
in the module.

For the learner:

As a learner, you must learn to become responsible of your own
learning. Take time to read, understand, and perform the different activities
in the module.
As you go through the different activities of this module be reminded of
the following:
1. Use the module with care. Do not put unnecessary mark/s on any part
of the module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer Let Us Try before moving on to the other
activities.
3. Read the instructions carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your
answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are done.
If you encounter any difficulty in answering the tasks in this module,
do not hesitate to consult your teacher or facilitator. Always bear in mind that
you are not alone. We hope that through this material, you will experience
meaningful learning and gain deep understanding of the relevant
competencies. You can do it!

ii
 Let Us Learn

Hello learners! Welcome to our lesson for this week which is to solve problems
involving linear equation and inequality in one variable (M7AL-IIj-2).

Specifically, you are expected to:

• recall and translate verbal sentences into mathematical equation

or inequality;
• apply the different properties of equality and inequality
• solve problems involving equations and inequalities in one
variable.

Let Us Try
Direction: Read the following questions carefully and write the letter of the
correct answer on a separate sheet of paper.

1. If a number is increased by 7, the result is 24. What is the number?

a. 15 b. 16 c. 17 d. 31

2. A number less than 15 equals 3. What is the number?

a. 5 b. 12 c. 18 d. 45

3. There are two numbers whose sum is 72. One number is twice the other. What
are the numbers?
a. 22 and 50 c. 25 and 50
b. 24 and 48 d. 28 and 44

4. The sum of two numbers is 52. If one number is -38, what is the other
number?
a. – 90 b. – 14 c. 14 d. 90

5. The sum of three consecutive integers is 90. What is the middle integer?
a. 28 b. 29 c. 30 d. 31

6. In 15 years, Richard will be 47. How old is he now?

a. 23 b. 32 c. 34 d. 42

7. A man’s age is 51 and that of his son is 7 years. In how many years will the
man be 5 times as old as his son?
a. 4 b. 6 c. 8 d. 10

8. If the weight of a sack of rice is multiplied by 2/3, the result is 32 kg. What
is the weight of the sack of rice?
a. 40 kg b. 42 kg c. 45 kg d. 48 kg

1
 9. If Manny can do twice as many push-ups as Joem, who does 48, how many
push-ups can he do?
a. 24 b. 50 c. 96 d. 100

10. Aurora is 2 times older than Jessica. In 10 years, the sum of their ages is 50
years. Find their present ages.
a. Aurora is 10 and Jessica is 20. c. Aurora is 30 and Jessica is 10.
b. Aurora is 20 and Jessica is 10. d. Both Aurora and Jessica are 25.

11. Gigi has score 82, 91, 85, 92 on four summative tests. What is the least
score she can get on the fifth test to have an average of more than 88?
a. 89 b. 90 c. 91 d. 92

12. The difference between two numbers is greater than 50. If the larger number
is 98, what is the possible smaller number?
a. 47 b. 48 c. 49 d. 50

13. There are seven less girls than boys in a class of less than sixty-one
students. How many boys could possibly be there in a class?
a. 33 b. 34 c. 35 d. 36

14. Twice a number added to 15 is at least 51. Which of the following best
describes the number?
a. The number is more than 18. c. The number is at most 18.
b. The number is less than 18. d. The number is at least 18.
15. The length of a rectangle is 3 cm more than its width. If the perimeter of the
rectangle is at most 138 cm, what is the maximum width of the rectangle?

a. 31 cm b. 32 cm c. 33 cm d. 34 cm

Let Us Study

In the previous modules, we have already studied how to translate phrases to

algebraic expressions, and sentences to equations/inequalities. We have also
learned steps in finding solutions in linear equations and inequalities in one variable.
This time, let us apply our skills in solving problems.

Steps in Solving Problems Involving Linear Equations and Inequalities

in One Variable

1. Read and understand the problem carefully.

2. Represent the unknowns using variables.
3. Formulate an equation or inequality.
4. Solve the formulated equation/inequality.
5. Check or justify your answer.

2
 LINEAR EQUATIONS

Number Relation Problem

The sum of two numbers is 36. If the second number is four more than thrice
the first, find the numbers.
Let x be the first number.
The problem tells us that the second number is “four more than thrice the first
number.” Therefore, we can represent the second number as:
second number = four more than thrice the first
3x + 4
Also, we know that the sum of the numbers is 36.
Therefore, our equation is
x + (3x + 4) = 36
Solving the equation, we have

x + (3x + 4) = 36
4x + 4 = 36 Combine like terms.
4x + 4 + (-4) = 36 + (-4) Addition Property of Equality
4x = 32 Divide both side by 4
4 4
x =8

The first number is 8. The second number is 3x + 4 = 3(8) + 4 = 24 + 4 = 28.

The two numbers are 8 and 28. Check the answer against the original words of the
problem.
Checking:
Is the sum of 8 and 28 equal to 36? Yes
8 + 28 =36
36 = 36 Correct

Number Problem
Find five consecutive odd integers whose sum is 55.

Solution Let x = 1st odd integer

x+2= 2nd odd integer
x+4= 3rd odd integer
x+6= 4th odd integer
x+8= 5th odd integer

3
 Solve the equation.
x + (x + 2) + (x+4) + (x + 6) + (x + 8) = 55
5x + 20 = 55 Combine like terms.
5x + 20 + (– 20) = 55 + (–20) Addition Property of Equality
5x = 35 Divide both sides by 5
5 5
x=7

x = 7 1st odd integer

x + 2 = 7 + 2 = 9 2nd odd integer
x + 4 = 7 + 4 = 11 3rd odd integer
x + 6 = 7 + 6 = 13 4th odd integer
x + 8 = 7 + 8 = 15 5th odd integer

The five consecutive odd integers are 7, 9, 11, 13, and 15. We can check that the
answers are correct if we observe that the sum of these integers is 55, as required
by the problem.
7 + 9 + 11 + 13 + 15 = 55
55 = 55 correct

Age Problem
Margie is 3 times older than Lilet. In 15 years, the sum of their ages is 38 years.
Find their present ages.

Let Present Age Future age (in 15 years)

Lilet x x + 15
Margie 3x 3x + 15

In 15 years, the sum of their ages is 38 years.

Solve the equation.
(x + 15) + (3x + 15) = 38
4x + 30 = 38 Combine like terms.
4x + 30 + (-30) = 38 + (-30) Subtraction Property of Equality
4x = 8
4 4 Divide both sides 4
x=2
Answer: Lilet’s age now is 2 while, Margie’s age now is 3(2) or 6.

Checking: Margie is 6 which is 3 times older than Lilet who’s only 2 years old. In 15
years, their ages will be 21 and 17. The sum of their ages is 21 + 17 = 38.
Age now In 15 years
Lilet x=2 x + 15 = 2 + 15 = 17
Margie 3x = 3(2) = 6 3x + 15 = 3(2) + 15 = 6 + 15 = 21
Total 17 + 21 = 38

4
 SOLVING PROBLEMS INVOLVING LINEAR INEQUALITIES

Number Relation

The sum of two non-negative integers is 15. Twice the larger integer is greater than
thrice the other integer. Find the least value of the larger integer.

Solution:
Let x be the larger integer and 15 - x be the other integer.
We obtain the inequality 2x > 3(15 - x).

Key word: least value (symbol “ > ” )

Solve the inequality.

2x > 3(15 - x)
2x > 45 - 3x Distribute 3 inside the parenthesis (means multiply
3)
2x + 3x > 45 - 3x + 3x Isolate x to the other side then add 3x to both sides
(Addition Property of Equality)
5x > 45
5 5 Divide both sides by 5
x >9 This means that the larger integer should be
greater than 9.

Therefore, the least value of the larger integer is 10.

Check:
Let us try x = 9 Let us try x = 10
2x > 3(15 - x) 2x > 3(15 - x)
2(9) > 3(15 - 9) 2(10) > 3(15 -10)
18 > 3 (6) 20 > 3(5)
18 > 18 which is not true. 20 > 15 which is true.

Money Problem

Keith has P5 000.00 in a savings account at the beginning of the summer. He wants
to have at least P2 000.00 in the account by the end of the summer. He withdraws
P250.00 each week for food and transportation. How many weeks can Keith withdraw
money from his account?

Solution:

Step 1: Let w be the number of weeks Keith can withdraw money.

5
Step 2:
5000 - 250w ≥ 2000
Amount at the withdraw 250 each At Amount by the end of
beginning week least the summer

Step 3: 5000 – 250w ≥ 2000

5000 + (-5000) – 250w ≥ 2000 + (-5000) Addition Property of Equality
–250 w ≤ - 3000
- 250 - 250 Divide both sides by -250
w ≤ 12

Therefore, Keith can withdraw money from his account not more than 12
weeks. We can check our answer as follows. If Keith withdraws P250 per week 12
times, then the total money withdrawn is P3000. Since he started with P5000, then
he will still have P2000 at the end of the summer.

Let Us Practice

Direction: Solve the following problems on linear equations in one variable.

1. One number is 5 less than 4 times another. If the sum of the two numbers is
11, find the numbers.

2. Find five consecutive even integers whose sum is 130.

3. Jim is 4 years less than twice David’s age. The sum of their ages is 23. Find
their ages.

4. Maria has scores of 93, 86, and 89 on three tests. What must her average
score on the next two tests be in order for her to have an average of 90?
5. Alex is paid P 2,800 a week plus a commission of P 500 on each television set
he sells. How many sets must he sell to make P 5,300 in a week?

Let Us Practice More

Problem Solving. Solve the following problems on linear inequalities in one variable.
Show your step-by-step solution.

1. The capacity of an elevator is at most 750 kg. Six passengers weighing a total
of 425 kg are already inside the elevator. What is the maximum additional
weight that the elevator can carry?

2. The sum of two non-negative integers is 20. Twice the larger integer is greater
than thrice the other integer. Find the least value of the larger integer.

6
 3. Jacob wants to buy some pencils at a price of P4.50 each. He has no more
than P55.00. What is the greatest number of pencils can Jacob buy?
4. Maiah won 40 lollipops playing basketball at the school fair. She gave two to
every student in her math class. She has at least 7 lollipops left. Find the
maximum number of students in her class.
5. Isaiah has P3 000.00 in a savings account at the beginning of the summer.
He wants to have at least P1 000.00 in the account by the end of the summer.
He withdraws P100.00 each week for food and transportation. How many
weeks can Isaiah withdraw money from his account?

Let Us Remember

Solving word problems involving linear equations and inequalities in one

variable often creates difficulties for some students. It is therefore a need that one
already has a knowledge in translating verbal phrases/sentences to algebraic
expressions/equations/inequalities, and the different properties of linear
equations/inequalities in one variable.

There are problems in real life that require several answers. Those problems
use the concept of equation and inequality. Here are some points to remember when
solving word problems that use linear equations and inequalities in one variable.

Steps in Solving Problems

1. Read and understand the problem carefully.
2. Represent the unknowns using variables.
3. Formulate an equation or inequality.
4. Solve the formulated equation/inequality.
5. Check or justify your answer.

Let Us Assess

Direction: Read the following questions carefully and write the letter of the
correct answer on a separate sheet of paper.

1. A man’s age is 36 and that of his daughter is 3 years. In how many years will
the man be 4 times as old as his daughter?
a. 8 b. 10 c. 12 d. 14

2. A number less than 13 equals 73. What is the number?

a. 86 b. 60 c. - 86 d. - 60

7
3. Belle bought a T.V. Set for Php 8249.36 and a watch for Php 1249.36. How
much did she spend in all?
a. Php6498.72 c. Php9498.72
b. Php8498.72 d.Php 7498.72

4. The sum of three consecutive integers is 306. What is the largest integer?
a. 103 b. 105 c. 107 d. 109

5. If a number is increased by 15, the result is 42. What is the number?

a. 57 b. 27 c. 630 d. 42/15

6. The sum of two numbers is – 15 . If one number is 5, what is the other

number?
a. – 20 b. – 15 c. 10 d. 20

7. In 12 years, Kenneth will be 52. How old is he now?

a. 40 b. 64 c. 70 d. 84

8. If Ramon can do twice as many push-ups as Nilo, who does 75, how many
push-ups can he do?
a. 37.5 b. 73 c. 150 d. 77

9. If the weight of a package is multiplied by 1/3, the result is 15 kg. What is

the package’s weight?
a. 5 kg b. 15 1/3 kg c. 14 2/3 kg d. 45 kg

10. The difference between two numbers is less than 96. If the larger number is
245, what is the possible smaller number?
a. 96 b. 129 c. 149 d. 194

11. There are four more boys than girls in a class of less than fifty-two students.
How many boys could possibly be there in a class?
a. 13 b. 24 c. 25 d. 28

12. Vivian is three years younger than twice Maricel’s age. The sum of their ages
is at most 42. What are their oldest possible ages?
a. Maricel is 27 and Vivian is 15. c. Maricel is 30 and Vivian is 12.
b. Maricel is 12 and Vivian is 30. d. Maricel is 15 and Vivian is 27.

13. If the perimeter of a square is 12 units, what is the area?

a. 9 sq.u. b. 12 sq.u. c. 16 sq.u. d. 36 sq.u.

14. Tanya is 28 years older than Marcus. In 6 years, Tanya will be three times
as old as Marcus. How old is Tanya now?
a. 34 b. 36 c. 38 d. 40
b.

8
 15. Thrice a number subtracted from 15 is at most -12. Which of the following
best describes the number?
a. The number is at least 9. c. The number is less than 9.
b. The number is at most 9. d. The number is more than 9.

Let Us Enhance

With the learnings you have about solving problems involving linear equation
and inequality in one variable, do this activity in a separate sheet of paper.
Create 1 real-life word problem, it could be age, number, money or number
relation problem.

Do the following:

1. Based from the problem given, represent the unknowns using

variables.

Answer:_____________________________________________________________

2. Formulate an equation or inequality.

Answer:_____________________________________________________________

3. Solve the formulated equation/inequality

Answer:_____________________________________________________________

4. Checking:
Answer:_____________________________________________________________

RUBRICS
Points Descriptor
15 Has answered all questions with correct solution
10 Has answered all questions but has an error in the solution
5 Has answered two questions only

9
 Let Us Reflect

Give a short reflection about the topic discussed in the module.

10
 11
Let Us Practice More
1.) Let x be the additional weight that the elevator can carry
425 + x ≤ 750
425 + (-425) + x ≤ 750 + (-425)
x ≤ 325
Therefore, the elevator can still carry a maximum additional weight of 325 kg.
2.) Let x be the larger integer
20 – x = the other integer
2x > 3 (20 – x)
2x > 60 – 3x
2x + 3x > 60 – 3x + 3x
5x > 60
5x > 60
5 5
x > 12
Therefore, the least value of the larger integer is 13.
3.) Let x be the number of pencils
4.50x ≤ 55
4.50x ≤ 55
4.50 4.50
x ≤ 12.22
Therefore, the greatest number of pencils that Jacob can buy is 12 pencils.
Let Us Practice: Let Us Try:
16 39 1. c 6. b 11. c
1.) 3.2 and 7.8 or and
5 5 2. b 7. a 12. a
3. b 8. d 13. a
2.) 22, 24, 26, 28, 30 4. d 9. c 14. d
5. c 10. b 15. c
3.) David is 9 yrs old and Jim is 14
4.) 92
5.) 5 television sets
Answer Key
 12
Let Us Assess:
1. a 6. a 11. a
2. d 7. a 12. d
3. c 8. c 13. a
4. a 9. d 14. b
5. b 10. d 15. a
Let Us Practice More
4) Let x be the number of students
40 – 2x ≥ 7
40 + (-40) – 2x ≥ 7 + (-40)
– 2x ≥ – 33
– 2x ≤ – 33
–2 –2
x ≤ 16.5
Therefore, the maximum number of students in Maiah’s Math class is 16
students.
5) Let x be the number of weeks Isaiah can withdraw money from his account
3000 – 100x ≥ 1000
3000 + (-3000) – 100x ≥ 1000 + (-3000)
– 100x ≥ - 2000
– 100x ≤ - 2000
– 100 - 100
x ≤ 20
Therefore, Isaiah can withdraw money from his account in 20 or less weeks.
 References

Crisostomo, R., de Sagun, P.,& Padua,A.(2013). Our world of Math 7. Quezon

City:Vibal Publishing.
Oronce, Orlando A. and Mendoza, Marilyn O., Worktext in Mathematics, E-
math: Manila: Rex Book Store, Inc., 2012:233-336
https://resources.saylor.org/wwwresources/archived/site/wp-
content/uploads/2012/07/1.9-Age-Practice.pdf
https://jqurtas.wordpress.com/2018/03/22/number-relation-problems-
with-solution/
https://www.google.com/search?q=money+problems+worksheet+with+answ
ers&biw=1242&bih=597&e
https://www.google.com/search?q=age+word+problems+with+solution&biw =1242
&bih=597&ei

13
For inquiries or feedback, please write or call:

Department of Education – Region XI

F. Torres St., Davao City

Telefax:

Email Address: lrms.regionxi@deped.gov.ph

14