Rational Expressions Module

Ms. Lorie Jane L Letada
Rational Expressions is an algebraic expression where both
numerator and denominator are polynomials. During elementary years, you have
learned that a fraction is a ratio of a numerator and denominator where the
denominator is not equal to zero, but in this module, we extend the concept of
fractions to algebraic expressions.
Here is the map of the lessons that will be covered in this module.
Module
2 Rational Expressions
At the end of this module, the learners will:
1. illustrate rational expressions;
2. simplify rational expressions;
3. perform operations on rational algebraic expressions;
4. solve problems involving rational expressions.
Rational Expressions
Simplifying Rational expressions
Operation of Rational Expressions
Adding and Subtracting
Multiplying and Dividing

Module 2: Rational Expressions
Read and analyze each item carefully and encircle the letter of the
best answer.
Pre-test!

A Rational Expressions is reduced to its simplest form if the
numerator and denominator have no common factors. Putting a rational
expression in its simplest form is important as this will make operations
easier.
Simplify the following in its simplest form.
1.
4𝑎
12𝑏
2.
9𝑚
21𝑛
3.
4𝑤2
6𝑤−8𝑤2
Lesson
1
Simplifying Rational
Expressions
Step 1
Factor the numerator and
denominator and get the GCF.
4a = (2) (2) a
12b= (3) (2) (2) b
GCF: (2) (2)
GCF: 4
Thus, the common factor is 4.
Step 2:
Divide out the common factor.
4𝑎
12𝑏
=
4 ( 𝑎 )
4 (3𝑏)
=
𝑎
3𝑏
Thus, the simplify is
𝑎
3𝑏
.
Step 1
9m =
21n =
GCF:
GCF:
Thus, the common factor is __.
Step 2:
Step 1
Get the GCF of each term.
4𝑤2 = (2) (2) (w) (w)
6w = (3) (2) (w)
8𝑤2 = (2)(2) (2)(w)
GCF: (2) (w)
GCF: 2w
Thus, the common factor is 2w.
Step 2:
Factor the common
numerator and
denominator.
4𝑤2
6𝑤−8𝑤2 =
2𝑤 ( 2)
2𝑤 (3−4𝑤 )
Step 3:
Divide out the common
factor.
4𝑤2
6𝑤−8𝑤2 =
2𝑤 ( 2)
2𝑤 (3−4𝑤 )
=
2
3−4𝑤
2
3−4𝑤
.

4.
6𝑝2
2𝑝−4𝑝3
5.
2𝑥−2𝑦
𝑦2− 𝑥2
6.
3𝑥+ 3𝑦
𝑥2− 𝑦2
7.
4𝑝+ 4𝑞
𝑝2− 𝑞2
Step 1
Get the GCF of each term.
6𝑝2 =
2p =
4𝑝3 =
GCF:
GCF:
Step 2:
Factor the common
numerator and
denominator.
6𝑝2
2𝑝−4𝑝3 =
Step 3:
Step 1
Factor the common numerator
and denominator.
2𝑥−2𝑦
𝑦2− 𝑥2 =
2 (𝑦−𝑥)
( 𝑦−𝑥 )(𝑦+ 𝑥)
Step 2:
2𝑥−2𝑦
𝑦2− 𝑥2 =
2 (𝑦−𝑥)
( 𝑦−𝑥 )(𝑦+ 𝑥)
=
2
(𝑥+ 𝑦)
You can only apply the
difference of two squares if:
*The two terms are both
perfect squares.
*The operation is subtraction.
Example:
𝑦2
− 𝑥2
= (y – x ) ( y + x )
Step 1
and denominator.
3𝑥+ 3𝑦
𝑥2− 𝑦2 =
Step 2:
Divide out the common factor. To factor 𝑥2
− 𝑦2
, get
the square root of each term.
First Term: √x2 = x
Second Term: √y2 = y
Using x and y, form the
sum (x + y) & the difference
(x - y).
Thus, x2
- y2
= (x + y) (x – y).
Step 1
and denominator.
4𝑝+ 4𝑞
𝑝2− 𝑞2 =
Step 2:
A variable
is perfect
square if
its
exponent
is even
number.

8.
𝑚2+𝑚−6
𝑚2−7𝑚+10
9.
𝑛2−7𝑛−30
𝑛2−5𝑛−24
10.
𝑏2−49
𝑏2−2𝑏−35
Simplify each rational expression.
1.
5𝑎𝑏2
20𝑎𝑏
= 5.
𝑥−1
5𝑥−5
=
2.
5𝑔ℎ
10𝑔
= 6.
𝑛2+2𝑛−15
𝑛2−𝑛−6
=
3.
15𝑎𝑐2
45𝑎𝑏3 𝑐
= 7.
𝑐2+7𝑐+10
𝑐2+5𝑐
=
4.
𝑥2−9
𝑥2−𝑥−6
= 8.
𝑞2+8𝑞+12
𝑞2+3𝑞−18
=
Skill Booster!
Step 1
Factor the common numerator and
denominator.
𝑚2+𝑚−6
𝑚2−7𝑚+10
=
( 𝑚+3)(𝑚−2 )
(𝑚−5)(𝑚−2)
Step 2:
𝑚2+𝑚−6
𝑚2−7𝑚+10
=
( 𝑚+3)(𝑚−2 )
(𝑚−5)(𝑚−2)
=
(𝑚+3)
(𝑚−5 )
Step 1
denominator.
𝑛2−7𝑛−30
𝑛2−5𝑛−24
=
Step 2:
Step 1
denominator.
𝑏2−49
𝑏2−2𝑏−35
=
(𝑏−7 (𝑏+7 )
(𝑏−7 )(𝑏+5 )
Step 2:
𝑏2−49
𝑏2−2𝑏−35
=
(𝑏−7 (𝑏+7 )
(𝑏−7 )(𝑏+5 )
=
( 𝑏+7 )
(𝑏+5 )
To factor 𝑚2
+ 𝑚 − 6 ,
find any factors of -6
that when you add the
result is 1,
Hence, the correct pair
is 3 and -2.
So, the factors of
𝑚2
+ 𝑚 − 6 = (m +3)
(m -2).
* When you multiply
positive & negative
numbers, the result is
negative.
Example: (5 )(-3) = -15
*When you add positive
& negative numbers, the
sign used in the result
will be based which
absolute num. is greater.
Examples: 5 + -3 = 2
*When you add positive
& negative numbers, the
sign used in the result
will be based which
absolute num. is greater.
Examples: 5 + -3 = 2

To multiply rational expressions, recall the rules for multiplying
fractions. If the denominators are not equal to zero, then we simply
multiply the numerators and denominators. The same rule applies to
rational expressions.
If a, b, c, and d represent polynomials where b ≠ 0 and d ≠ 0. Then,
Multiply each rational expression and simplify.
1.
𝟓
𝟏𝟐
∗
𝟖
𝟏𝟓
=
2.
𝟔
𝟏𝟎
∗
𝟓
𝟏𝟓
=
3.
𝟏𝟓
𝟐𝒚 𝟐 ∗
𝒚 𝟒
𝟐𝟎𝒚 𝟐 =
Lesson
2
Multipication of Rational
Expressions
𝒂
𝒃
∗
𝒄
𝒅
=
𝒂𝒄
𝒃𝒅
Step 1 Factor each expression.
𝟓
𝟏𝟐
∗
𝟖
𝟏𝟓
=
𝟓
(𝟒)(𝟑)
∗
( 𝟒) (𝟐)
( 𝟑) (𝟓)
Step 2 Cancel all common
factors.
𝟓
(𝟒)(𝟑)
*
(𝟒) (𝟐)
(𝟑) (𝟓)
Step 3: Write what is left.
𝟓
𝟏𝟐
∗
𝟖
𝟏𝟓
=
𝟐
(𝟑) (𝟑)
=
𝟐
𝟗
Step 1 Factor each expression. Step 2 Cancel all common
factors.
Step 1: Factor each expression.
𝟏𝟓
𝟐𝒚 𝟐 ∗
𝒚 𝟒
𝟐𝟎𝒚 𝟐 =
( 𝟓 )( 𝟑)
( 𝟐)(𝒚 𝟐)
∗
(𝒚 𝟐)(𝒚 𝟐)
( 𝟓)( 𝟒)(𝒚 𝟐)
Step 2 Cancel all common factors.
𝟏𝟓
𝟐𝒚 𝟐 ∗
𝒚 𝟒
𝟐𝟎𝒚 𝟐 =
( 𝟓 ) ( 𝟑)
( 𝟐) (𝒚 𝟐)
∗
(𝒚 𝟐) (𝒚 𝟐)
( 𝟓) ( 𝟒) (𝒚 𝟐)
𝟏𝟓
𝟐𝒚 𝟐 ∗
𝒚 𝟒
𝟐𝟎𝒚 𝟐 =
𝟑
(𝟐)(𝟒)
=
𝟑
𝟖

4.
𝟏𝟐
𝟑𝒘 𝟑 ∗
𝟒𝒘 𝟒
𝟏𝟓𝒘 𝟐 =
𝟓.
𝟒𝒙 𝟐−𝟗
𝒙 𝟐−𝟓𝒙+𝟔
∗
𝟑𝒙−𝟔
𝟖𝒙+𝟏𝟐
=
𝟔.
𝒙+𝟓
𝟒
∗
𝟏𝟐𝒙 𝟐
𝒙 𝟐+𝟕𝒙+𝟏𝟎
=
𝟕.
𝟑𝒔
𝟒𝒔+𝟏
∗
𝟐𝒔+𝟏
𝟑𝒔 𝟐 =
𝟖 .
𝟑
𝒙−𝒚
∗
(𝒙−𝒚 ) 𝟐
𝟔
=
Step 1: Factor each
expression.
Step 2 : Cancel all common
factors.
𝟒𝒙 𝟐−𝟗
𝒙 𝟐−𝟓𝒙+𝟔
∗
𝟑𝒙−𝟔
𝟖𝒙+𝟏𝟐
=
(𝟐𝒙−𝟑 )(𝟐𝒙+𝟑)
(𝒙−𝟑)(𝒙−𝟐)
*
𝟑 ( 𝒙−𝟐 )
𝟒(𝟐𝒙+𝟑)
Step 2 : Cancel all common factors.
𝟒𝒙 𝟐−𝟗
𝒙 𝟐−𝟓𝒙+𝟔
∗
𝟑𝒙−𝟔
𝟖𝒙+𝟏𝟐
=
(𝟐𝒙−𝟑 )(𝟐𝒙+𝟑)
(𝒙−𝟑)(𝒙−𝟐)
*
𝟑 ( 𝒙−𝟐 )
𝟒(𝟐𝒙+𝟑)
𝟒𝒙 𝟐−𝟗
𝒙 𝟐−𝟓𝒙+𝟔
∗
𝟑𝒙−𝟔
𝟖𝒙+𝟏𝟐
=
( 𝟐𝒙−𝟑 )
( 𝒙−𝟑)
∗
𝟑
𝟒
=
𝟑 ( 𝟐𝒙−𝟑 )
𝟒 ( 𝒙−𝟑 )
Show your solution here .….
𝟑𝒔
𝟒𝒔+𝟐
∗
𝟐𝒔+𝟏
𝟑𝒔 𝟐 = 𝟑𝒔
𝟐(𝟐𝒔+𝟏)
∗
(𝟐𝒔+𝟏)
𝒔 ( 𝟑𝒔 )
Step 2 : Cancel all common factors.
𝟑𝒔
𝟒𝒔+𝟐
∗
𝟐𝒔+𝟏
𝟑𝒔 𝟐 = 𝟑𝒔
𝟐(𝟐𝒔+𝟏)
∗
(𝟐𝒔+𝟏)
𝒔 ( 𝟑𝒔 )
𝟑𝒔
𝟒𝒔+𝟐
∗
𝟐𝒔+𝟏
𝟑𝒔 𝟐 =
𝟏
𝟐 ( 𝒔)
=
𝟏
𝟐𝒔
Show your solution here .….
Cancelation involving
rational expressions should
be applied carefully. In the
expression
2𝑥−3
𝑥−3
,you cannot
cancel x in the numerator
and denominator even
though
𝑥
𝑥
is equal to 1
because it is not a factor of
2𝑥−3
𝑥−3
. 𝐼𝑛 𝑓𝑎𝑐𝑡,
2𝑥−3
𝑥−3
is already in simplest
form.
Factor, in
mathematics, a
number or
algebraic
expression that
divides another
number or
expression
evenly—i.e., with
no remainder. For
example, 3 and 6
are factors of 12
because 12 ÷ 3 = 4
exactly and 12 ÷ 6
= 2 exactly.

Math Focus
To multiply rational expressions:
• Write each numerator and denominator in factored form.
• Divide out any numerator factor with any matching
denominator factor.
• Multiply numerator by numerator and denominator by
denominator.
• Simplify as needed.
Multiply each rational expression and simplify.
1.
𝟏𝟒
𝟐𝟕
∗
𝟑
𝟕
= 5.
(𝒓 𝟐+𝟑𝒓+𝟐 )
𝒓−𝟏
*
𝒓+𝟑
𝒓 𝟐+𝟓𝒓+𝟔
=
2.
𝟏𝟐𝒏
𝟒𝒎 𝟐 ∗
𝟖𝒎 𝟒
𝟏𝟓𝒏 𝟐 = 6.
(𝒂 𝟐−𝟏 )
𝟏𝟔 𝒂
*
(𝟒𝒂 𝟐
𝟕𝒂+𝟕
=
3.
𝟏𝟐𝒏
𝟒𝒎 𝟐 ∗
𝟖𝒎 𝟒
𝟏𝟓𝒏 𝟐 = 7.
(𝒚 𝟐+𝟒𝒚+𝟒 )
𝒚+𝟑
*
𝟒𝒚+𝟏𝟐
𝒚+𝟐
= =
4.
𝟐𝒎+𝟓
𝟑𝒎−𝟔
*
(𝒎 𝟐+𝒎−𝟔 )
𝟒
=
Skill Booster!

In dividing rational expressions, observe the same rules as when
dividing fractions.
If the denominators are not equal to zero, then
Find the quotient of each rational expression.
1.
𝟐
𝟑
÷
𝟏𝟎
𝟗
2.
𝟓
𝟏𝟐
÷
𝟏𝟖
𝟏𝟓
3.
𝟒𝒙 𝟐 𝒚
𝟏𝟓𝒂 𝟑 𝒃 𝟐 ÷
𝟖𝒙𝒚 𝟐
𝟓𝒂𝒃 𝟑
Lesson
3
Division of Rational
Expressions
𝒂
𝒃
÷
𝒄
𝒅
=
𝒂
𝒃
∗
𝒅
𝒄
=
𝒂𝒅
𝒃𝒄
Step 1 :
Get the reciprocal of
the divisor and proceed
to multiplication.
2
3
÷
10
9
=
2
3
∗
9
10
Step 2 : Factor each expression
and cancel all common factors.
2
3
÷
10
9
=
2
3
∗
9
10
=
2
3
∗ ( 3 )(3)
(5)(2)
Step 3 : Write what is
left
2
3
÷
10
9
=
3
5
Write your solution here…
Step 1 :
Get the reciprocal of the divisor
and proceed to multiplication.
𝟒𝒙 𝟐 𝒚
𝟏𝟓𝒂 𝟑 𝒃 𝟐 ÷
𝟖𝒙𝒚 𝟐
𝟓𝒂𝒃 𝟑 =
𝟒𝒙 𝟐 𝒚
𝟏𝟓𝒂 𝟑 𝒃 𝟐 *
𝟓𝒂𝒃 𝟑
𝟖𝒙𝒚 𝟐
Step 2 : Factor each expression and cancel all
common factors .
𝟒𝒙 𝟐 𝒚
𝟏𝟓𝒂 𝟑 𝒃 𝟐 *
𝟓𝒂𝒃 𝟑
𝟖𝒙𝒚 𝟐 =
(𝟒)(𝒙)(𝒙)(𝒚)
(𝟓)(𝟑)(𝒂)(𝒂)(𝒂)(𝒃)(𝒃)
*
(𝟓)(𝒂)(𝒃)(𝒃)(𝒃)
(𝟒)(𝟐)(𝒙)(𝒚)
Step 3 : Write what is left.
𝒙
𝟑𝒂 𝟐 *
𝒃
𝟐
=
𝒃𝒙
𝟔𝒂 𝟐

4.
𝟏𝟐𝒎 𝟐
𝟓𝒏 𝟐 ÷
𝟏𝟐𝒎 𝟑
𝟐𝟓𝒏 𝟑
5.
𝒓−𝒔
𝒓 𝟐− 𝒔 𝟐 ÷
𝒓+𝒔
𝒓 𝟐+ 𝟐𝒓𝒔+ 𝒔 𝟐
6.
𝒂+𝒃
𝒂−𝒃
÷
𝒂 𝟐+𝒂𝒃
𝒂 𝟐−𝒃 𝟐
7.
𝟗𝒅 𝟐−𝟔𝒅𝒆+ 𝒆 𝟐
𝟐𝒅𝒆 𝟐 ÷
𝟑𝒅−𝒆
𝟔𝒅 𝟑 𝒆
Step 1 :
Get the reciprocal of the divisor and
proceed to multiplication.
𝒓−𝒔
𝒓 𝟐− 𝒔 𝟐
÷
𝒓+𝒔
𝒓 𝟐+ 𝟐𝒓𝒔+ 𝒔 𝟐
=
𝒓−𝒔
𝒓 𝟐− 𝒔 𝟐
∗
𝒓 𝟐+ 𝟐𝒓𝒔+ 𝒔 𝟐
𝒓+𝒔
Step 2 and 3 :
Factor each expression and cancel all
common factors. Write, what is left.
𝒓−𝒔
𝒓 𝟐− 𝒔 𝟐
÷
𝒓+𝒔
𝒓 𝟐+ 𝟐𝒓𝒔+ 𝒔 𝟐
=
𝒓−𝒔
(𝒓−𝒔)(𝒓+𝒔 )
∗
(𝒓+𝒔)(𝒓+𝒔)
𝒓+𝒔
𝒓−𝒔
𝒓 𝟐− 𝒔 𝟐
÷
𝒓+𝒔
𝒓 𝟐+ 𝟐𝒓𝒔+ 𝒔 𝟐
= 1
Solution:
𝟗𝒅 𝟐−𝟔𝒅𝒆+ 𝒆 𝟐
𝟐𝒅𝒆 𝟐
÷
𝟑𝒅−𝒆
𝟔𝒅 𝟑 𝒆
=
𝟗𝒅 𝟐−𝟔𝒅𝒆+ 𝒆 𝟐
𝟐𝒅𝒆 𝟐
*
𝟔𝒅 𝟑 𝒆
𝟑𝒅 −𝒆
=
(𝟑𝒅−𝒆)(𝟑𝒅−𝒆)
𝟐(𝒅)(𝒆)
*
(𝟑)(𝟐)(𝒅)(𝒅)(𝒅)(𝒆)
(𝟑𝒅−𝒆)
=
(𝟑𝒅−𝒆)
𝟏
*
𝟑 (𝒅)(𝒅)
𝟏
=3𝒅 𝟐 (𝟑𝒅 − 𝒆)
Any number or
variables divided
by itself is always
equal to 1.
Example:
(𝒓−𝒔)(𝒓+𝒔)
(𝒓−𝒔)(𝒓+𝒔 )
= 1
Any number or
variables divided
by 1 is always
equal to itself.
Examples:
(𝟑𝒅−𝒆)
𝟏
= (3d-e)
𝟑 (𝒅)(𝒅)
𝟏
= 3𝒅 𝟐

Find the quotient and simplify.
1.
𝟏𝟓𝒂 𝟐 𝒃 𝟑
𝟏𝟔𝒙𝒚
÷
𝟑𝟎𝒃 𝟐
𝟖𝒙𝒚
= 4.
𝒙 𝟐−𝒙−𝟐
𝒙+𝟑
÷
𝟐𝒙+𝟐
𝟑𝒙+𝟗
=
2.
𝟕
𝟏𝟖
÷
𝟔
𝟗𝒂
= 5.
𝟑𝒘+𝟏𝟓
𝟏𝟎
÷
𝟏𝟐
𝟒𝒘+𝟐𝟎
=
3.
𝟗
𝒎−𝟑
÷
𝟏𝟐
𝒎 𝟐−𝟗
= 6.
𝒙 𝟐−𝟑𝒙−𝟏𝟖
𝒙 𝟐+𝟓𝒙+𝟔
÷
𝒙 𝟐−𝟖𝒙+𝟏𝟐
𝒙 𝟐−𝟒
=
In your lower grade math, you usually learn how to add and subtraction
fraction before you are taught multiplication and division. However,
with rational expressions, multiplication and division are sometimes taught
first because these operations are easier to perform than addition and
subtraction. Addition and subtraction of rational expressions are not as easy to
perform as multiplication because, as with numeric fractions, the process
involves finding common denominators. By working carefully and writing down
the steps along the way, you can keep track of all of the numbers and variables
and perform the operations accurately.
Skill Booster!

In adding and subtracting rational expressions with same
denominators, perform the operation in the numerator and keep the
denominator. The resulting sum and difference must be reduced to
lowest terms whenever possible.
Provided the denominator is not equal to zero, then
Perform the indicated operations.
1.
𝟑
𝟓
+
𝟏
𝟓
2.
𝟕
𝟗
+
𝟏
𝟗
2.
𝟑𝒙+𝟐
𝒙+ 𝟓
+
𝟒𝒙
𝒙+ 𝟓
3. .
𝟒𝒎+𝟐
𝒏+ 𝟓
+
𝟗𝒎+𝟐
𝒏+ 𝟓
Lesson
4
Addition and Subtraction of Rational
Expressions
(Same Denominators )
𝒂
𝒃
+
𝒄
𝒃
=
𝒂 + 𝒄
𝒃
𝒂
𝒃
−
𝒄
𝒃
=
𝒂 − 𝒄
𝒃
Step 1 :
Since the denominator is the same, add
the numerator and copy the denominator.
𝟑
𝟓
+
𝟏
𝟓
=
𝟑+𝟏
𝟓
=
𝟒
𝟓
Step 1 :
Step 1 :
Since the denominator is the same, combine
the numerator and keep the denominator.
𝟑𝒙+𝟐
𝒙+ 𝟓
+
𝟒𝒙
𝒙+ 𝟓
=
𝟑𝒙+𝟐+𝟒𝒙
𝒙+ 𝟓
Step 2 :
Combine like terms.
𝟑𝒙+𝟐
𝒙+ 𝟓
+
𝟒𝒙
𝒙+ 𝟓
=
𝟑𝒙+𝟒𝒙+𝟐
𝒙+ 𝟓
=
𝟕𝒙+𝟐
𝒙+ 𝟓
Solution:

4.
𝟓𝒓 +𝟐
𝒓+ 𝟒
−
𝟏𝟎𝒓
𝒓+ 𝟒
5.
𝒑 −𝟐
𝒒− 𝟐
−
𝟑𝒑
𝒒− 𝟐
6.
𝟐𝒘+ 𝟒
𝒘 𝟐−𝟔𝒘+𝟗
−
𝒘−𝟏
𝒘 𝟐−𝟔𝒘+𝟗
7.
𝒆 + 𝟓
𝒇 𝟐+𝟗
−
𝟔𝒆+𝟐
𝒇 𝟐+𝟗
Step 1 :
Since the denominator is the same, combine
the numerator and keep the denominator.
𝟓𝒓 +𝟐
𝒓+ 𝟒
−
𝟏𝟎𝒓
𝒓+ 𝟒
=
𝟓𝒓 +𝟐 −𝟏𝟎𝒓
𝒓+ 𝟒
Step 2 :
Combine like terms.
𝟓𝒓 +𝟐
𝒓+ 𝟒
−
𝟏𝟎𝒓
𝒓+ 𝟒
=
𝟓𝒓 −𝟏𝟎𝒓 + 𝟐
𝒓+ 𝟒
=
−𝟓𝒓+𝟐
𝒓+ 𝟒
Solution:
When you subtract
two numbers, just follow
the KCC Rules ;
Example: 5r - 10r
*Keep the Subtrahend 5r
*Change the operation
from minus to add (5r +
* Change the sign of the
minuend to its opposite
sign from +10r to -10r
5r + -10r, and proceed to
the rules of addition.
5r + -10r = -5r
Solution:
𝟐𝒘+ 𝟒
𝒘 𝟐−𝟔𝒘+𝟗
−
𝒘−𝟏
𝒘 𝟐−𝟔𝒘+𝟗
=
𝟐𝒘+ 𝟒 − (𝒘−𝟏)
𝒘 𝟐−𝟔𝒘+𝟗
=
𝟐𝒘+ 𝟒−𝒘+𝟏
𝒘 𝟐−𝟔𝒘+𝟗
=
𝒘+ 𝟑
𝒘 𝟐−𝟔𝒘+𝟗
Multiply negative
to each term
Combine like terms
Solution:
𝒆 + 𝟓
𝒇 𝟐+𝟗
−
𝟔𝒆+𝟐
𝒇 𝟐+𝟗
=
𝒆 + 𝟓 − (𝟔𝒆 + 𝟐)
𝒇
𝟐
+𝟗
= ----------------
= ----------------
Multiply negative
to each term
Combine like terms
* When you multiply
unlike terms, the result is
negative.
Example: (-1 )(w) = -w
*When you multiply like
terms, the result is
positive.
Example: (-1)(-1) = 1
*When you add unlike
terms, the sign used in
the result will be based
which absolute num. is
greater.
Examples: e + -6e = -5e
We use -5e since -6e
has the greater absolute
value and its sign is
negative.

Perform the indicated operation for rational expressions
with like denominators.
1.
𝟓𝒌
𝟏𝟎𝒍
−
𝟐𝒌
𝟏𝟎𝒍
5.
𝟓𝒛− 𝟒
𝒛 𝟐− 𝟏
−
𝟒𝒛−𝟑
𝒛 𝟐− 𝟏
2.
𝟏𝟓𝒕
𝟐𝒖 𝟐 +
𝟗
𝟐𝒖 𝟐 6.
𝟐𝒗+𝟏
(𝒗+𝟏)
+
𝒗 𝟐
(𝒗+𝟏)
3.
𝒑 𝟐
𝒑 + 𝟐
−
𝟖 +𝟐𝒑
𝒑+𝟐
7.
𝒂 𝟐
𝒂+𝒃
+
𝟐𝒂𝒃+𝒃 𝟐
𝒂+𝒃
4.
𝒆 + 𝟏𝟐
𝟗
+
𝟐𝒆−𝟑
𝟗
𝟖.
𝟒𝒒+ 𝟑
𝒒 𝟐− 𝟐𝟓
+
𝟐−𝟑𝒒
𝒒 𝟐− 𝟐𝟓
Skill Booster!

When the denominators of the two rational algebraic expressions
are different, we need to find the least common denominator (LCD)
of the given expressions.
The least common denominator must contain all prime factor of
each denominator raised to the highest power. It is the common
multiple of the denominator.
Provided that the denominators are not equal to zero, then,
Perform the indicated operation and simplify if possible.
1.
𝟏
𝟐𝒓
+
𝟐
𝟑𝒔
Lesson
5
Addition and Subtraction of Rational
Expressions
(Different Denominators )
𝒂
𝒃
+
𝒄
𝒅
=
𝒂𝒅 + 𝒄𝒅
𝒃𝒅
𝒂
𝒃
−
𝒄
𝒅
=
𝒂𝒅 − 𝒄𝒅
𝒃𝒅
Step 1 : Find the least
common denominator (LCD).
𝟐𝒓 = ( 2 ) ( r )
3s = (3) (s)
LCD = (2) (r ) (3)(s)
LCD = (2) (3) ( r ) (s)
LCD = 6rs
Step 2 : Express each fraction with
the LCD as the denominator.
𝟏
𝟐𝒓
+
𝟐
𝟑𝒔
=
𝟏 (𝟑𝒔)+𝟐 (𝟐𝒓)
𝟔𝒓𝒔
𝟔𝒓𝒔
𝟐𝒓
= 3s
𝟔𝒓𝒔
𝟐𝒓
= 3s
𝟔𝒓𝒔
𝟑𝒔
= 2r
Step 3 : Add the numerator
and simplify when possible.
𝟏
𝟐𝒓
+
𝟐
𝟑𝒔
=
𝟏 (𝟑𝒔)+𝟐 (𝟐𝒓)
𝟔𝒓𝒔
=
𝟑𝒔+ 𝟒𝒓
𝟔𝒓𝒔
Observe how to add
dissimilar fraction.

2.
𝟒
𝟑𝒙
+
𝟐
𝟐𝒚
3.
𝟐
𝒓+𝟏
+
𝒙
𝒓−𝟏
Step 1 .
Combine the denominator.
𝟐
𝒓+𝟏
+
𝒙
𝒓−𝟏
=
(𝒓+𝟏) ( 𝒓−𝟏)
Step 2 .
Cross Multiply.
𝟐
𝒓+𝟏
+
𝒙
𝒓−𝟏
=
𝟐 ( 𝒓−𝟏 ) +𝒙 (𝒓+𝟏 )
(𝒓+𝟏) ( 𝒓−𝟏)
Step 1 : Find the least
common denominator (LCD).
𝟑𝒙 =
2y =
LCD =
LCD =
LCD =
Step 2 : Express each fraction with
the LCD as the denominator.
Step 3 : Add the numerator
and simplify when possible.
Step 3 .
Multiply the numerator.
𝟐
𝒓+𝟏
+
𝒙
𝒓−𝟏
=
𝟐 ( 𝒓−𝟏 ) +𝒙 (𝒓+𝟏 )
(𝒓+𝟏) ( 𝒓−𝟏)
=
𝟐 𝒓−𝟏 + 𝒙𝒓 +𝒙
(𝒓+𝟏) ( 𝒓−𝟏)
=
𝟐 𝒓+𝒙+𝒙𝒓−𝟏
(𝒓+𝟏) ( 𝒓−𝟏)
You can only add or
subtract if they have the
same variables and
exponents regardless of its
numerical coefficients
Example: 𝟐𝒙 𝟑
and 𝟑𝒙 𝟑

4.
𝟐𝒙
𝒙 +𝟐
-
𝟓
𝒙−𝟐
5.
𝒙+𝟏
𝒙 +𝟐
-
𝒙−𝟏
𝒙−𝟐
n
Put your solution here…..
Step 1 .
Combine the denominator.
𝒙+𝟏
𝒙 +𝟐
-
𝒙−𝟏
𝒙−𝟐
=
( 𝒙+𝟏)−(𝒙−𝟏)
( 𝒙 +𝟐) (𝒙−𝟐 )
Step 2 .
Cross Multiply.
𝒙+𝟏
𝒙 +𝟐
-
𝒙−𝟏
𝒙−𝟐
=
[( 𝒙+𝟏)( 𝒙−𝟐 ) ]−[ ( 𝒙−𝟏)( 𝒙+𝟐 )]
( 𝒙 +𝟐)( 𝒙−𝟐)
Step 3 .
Multiply the numerator.
𝒙+𝟏
𝒙 +𝟐
-
𝒙−𝟏
𝒙−𝟐
=
[( 𝒙+𝟏)( 𝒙−𝟐 ) ]−[ ( 𝒙−𝟏)( 𝒙+𝟐 )]
( 𝒙 +𝟐)( 𝒙−𝟐)
𝒙+𝟏
𝒙 +𝟐
-
𝒙−𝟏
𝒙−𝟐
=
(𝒙 𝟐−𝟐𝒙−𝟐) −( 𝒙 𝟐−𝟐𝒙−𝟐 )
𝒙 𝟐−𝟒
Use FOIL method to
simplify these the
numerator.
Use FOIL method to simplify
these the denominator.
Step 4.
Simplify the numerator.
𝒙+𝟏
𝒙 +𝟐
-
𝒙−𝟏
𝒙−𝟐
=
(𝒙 𝟐−𝟐𝒙−𝟐) −( 𝒙 𝟐−𝟐𝒙−𝟐 )
𝒙 𝟐−𝟒
𝒙+𝟏
𝒙 +𝟐
-
𝒙−𝟏
𝒙−𝟐
=
𝒙 𝟐−𝟐𝒙−𝟐 − 𝒙 𝟐+ 𝟐𝒙 +𝟐
𝒙 𝟐−𝟒
𝒙+𝟏
𝒙 +𝟐
-
𝒙−𝟏
𝒙−𝟐
=
𝟎
𝒙 𝟐−𝟒
= 0
Multiply negative to each terms.
Combine like terms,

6.
𝒎+𝟑
𝒎 + 𝟒
-
𝒏 −𝟏
𝒏− 𝟐
7.
𝒄 +𝟏
𝒅− 𝟐
-
𝒅 −𝟏
𝒄 − 𝟑
8.
𝒘 +𝟑
𝒒− 𝟏
+
𝒒 −𝟏
𝒘+ 𝟑
Put your solution here….
Solution:
𝒄 +𝟏
𝒅− 𝟐
-
𝒅 −𝟏
𝒄 − 𝟑
=
[( 𝒄 +𝟏)( 𝒄−𝟑)]−[( 𝒅−𝟏)(𝒅−𝟐)
( 𝒅− 𝟐)(𝒄−𝟑 )
=
(𝒄 𝟐−𝟐𝒄−𝟑 ) −(𝒅 𝟐−𝟐𝒅+𝟐)
𝒄𝒅−𝟐𝒄−𝟑𝒅+𝟔
=
𝒄 𝟐−𝟐𝒄−𝟑 − 𝒅 𝟐+𝟐𝒅−𝟐
=
𝒄 𝟐− 𝒅 𝟐−𝟐𝒄 +𝟐𝒅−𝟓
Cross multiply
Get the LCD
Use FOIL Method
Combine like terms
Solution:
Like Terms - both
terms have the
same variable
group abc; ex. abc
and 22abc
Unlike Terms are
terms that are not
in group ;example
−4𝑦3
and 5𝑥3
because the
variables are
different

Perform the indicated operation and simplify.
𝟏.
𝟏
𝟓𝒑
+
𝟐
𝟐𝒒
4.
𝟐𝒔
𝒓 +𝟓
-
𝟓𝒓
𝒔 −𝟓
𝟐.
𝒙
𝟒𝒉
+
𝟐
𝟑𝒊
5.
𝒂 +𝟑
𝒃− 𝟏
+
𝒃 −𝟏
𝒂+ 𝟑
𝟑.
𝟑𝒑
𝒑 +𝟐
-
𝟐
𝒑−𝟑
6.
𝒇 +𝟏
𝒈− 𝟐
+
𝟐𝒇
𝒈 + 𝟑
Skill Booster!
Math Guro
Math Guro
Rational Expressions – nothing more than a fraction in which the numerator and/or the
denominator are polynomials. Here are some examples of rational expressions.
Fractions –represents a part of a whole or, more generally, any number of equal parts.
Numerator –the number above the line in a common fraction showing how many of the
parts indicated by the denominator are taken, for example, 2 in 2/3.
Denominator –the number below the line in a common fraction; a divisor.
Like Terms – Like terms - both terms have the same variable group abc; ex. abc and 22abc
Unlike Terms – examples −4𝑦3 and 5𝑥3 because the variables are different
Similar Fraction – fractions that have the same denominator; Examples:
𝟐
𝟑
,
𝟏
𝟑
and
4
3
Dissimilar Fraction – fractions that have different denominators; Examples:
𝟐
𝟓
,
𝟏
𝟔
and
𝟒
𝟕
Variables – A symbol for a number we don't know yet. It is usually a letter like x or y.
Example: in x + 2 = 6, x is the variable.
Literal Coefficients – literal coefficient is a variable used to represent a number

The only application of
rational equations that we will cover
in this class is shared work
problems, which means two or more
people (or objects) working together
on a job to share the work.
▪ Person A does a job in 2 hours, so
their rate is
1
2
the job per hour
▪ Person B does the same job in 3
hours, so their rate is
1
3
of the job per
hour
▪ Persons A and B do the job together in 𝑥 hours, so their rate
together is
1
𝑥
of the job per hour
1
2
+
1
3
=
1
𝑥
These are the steps on how to solve work problems.
1. Make sure the units of time are consistent.
For example, if one person does a job in 45 minutes, and
another person does the same job in 2 hours, convert both rates to
either minutes or hours. (60 mins. x 2 hrs. = 120 mins.)
In terms of minutes, the equation would be
1
45
+
1
120
=
1
𝑥
2. Make sure your answer makes sense .
If one person does a job in 45 minutes, and another person does
the same job in 2 hours, it shouldn’t take both people working
together more than 45 minutes to do the job.
6
Lesson Problem Solving Involving

1. It takes a boy 90 minutes to mow the lawn, but his sister can mow
it in an hour. How long (in minutes) would it take them to mow the
lawn if they worked together using two lawn mowers?
Let us look at the following problems.
Step 1:
Number of minutes that the
boy and his sister work
together
Step 2:
Let
1
𝑥
= number of mins.
work together
𝟏
𝟗𝟎
+
𝟏
𝟔𝟎
=
𝟏
𝒙
Step 3:
𝟏
𝟗𝟎
+
𝟏
𝟔𝟎
=
𝟏
𝒙
5400𝒙 [
𝟏
𝟗𝟎
+
𝟏
𝟔𝟎
=
𝟏
𝒙
] 5400x
60x + 90x = 5400
150x = 5400
𝟏𝟓𝟎
𝟏𝟓𝟎
𝒙 =
𝟓𝟒𝟎𝟎
𝟏𝟓𝟎
X = 36
Step 4:
Therefore, there are 36
minutes to mow the lawn.

2. Two pipes can be used to fill a swimming pool. When the first pipe
is closed, the second pipe can fill the pool in 9 hours. When the
second pipe is closed, the first pipe can fill the pool in 7 hours. How
long (in hours) will it take to fill the pool if both pipes are open?
Round your answer to two decimal places.
Step 1:
Number of hours that both
pipes will fill the pool.
Step 2:
Let
1
𝑥
= number of hours
that the pipes will fill
together the pool
𝟏
𝟗
+
𝟏
𝟕
=
𝟏
𝒙
Step 3:
𝟏
𝟗
+
𝟏
𝟕
=
𝟏
𝒙
63𝒙 [
𝟏
𝟗
+
𝟏
𝟕
=
𝟏
𝒙
] 63x
7x + 9x = 63
16x = 63
𝟏𝟔
𝟏𝟔
𝒙 =
𝟔𝟑
𝟏𝟔
X = 3.94
X = 3 hrs. , 56
mins (0.94 x 60) and 15 sec
(.25 x 60)
Step 4:
Therefore, there are 3.94
hours( to fill the pool
together.
LCD (9,7, X)
63x ÷ 9
63𝑥
𝑥
LCD (9,7, X)
When you want to get
the LCD in a rational
equation and perform
the operation, make
sure to divide both side
with the LCD.
Example:
63𝒙 [
𝟏
𝟗
+
𝟏
𝟕
=
𝟏
𝒙
] 63x

3. Walter and Helen are asked to paint a house. Walter can paint
the house by himself in 12 hours and Helen can paint the house by
herself in 16 hours. How long would it take to paint the house if
they worked together?
4. On average, Mary can do her homework assignments in 100
minutes. It takes Frank about 2 hours to complete a given
assignment. How long (in minutes) will it take the two of them
working together to complete an assignment?
Step 1:
Step 2:
Step 3:
Step 4:
Step 1:
Number of minutes that both of
them will finish the assignment
together.
Step 2:
Let x = num. of minutes
Finish the task together
1
100
+
1
120
=
1
𝑥
Step 3:
𝟏
𝟏𝟎𝟎
+
𝟏
𝟏𝟐𝟎
=
𝟏
𝒙
12000𝒙 [
𝟏
𝟏𝟎𝟎
+
𝟏
𝟏𝟐𝟎
=
𝟏
𝒙
] 12000𝒙
120x + 100x = 12000
220x = 12000
𝟐𝟐𝟎
𝟐𝟐𝟎
𝒙 =
𝟏𝟐𝟎𝟎𝟎
𝟐𝟐𝟎
x = 54.55
x = 54.55 mins
Step 4:
Therefore, there are 54.55 minutes
to finish the assignment together.

Read and analyze each item carefully and solve the
problem.
1. Working together, Bill and Tom painted a fence in 4 hours. If
Tom has painted the same fence before by himself in 7 hours,
how long (in hours) would it take Bill on his own? Round your
answer to one decimal place.
2. Tom and Jerry have to stuff and mail 1000 envelopes for a new
marketing campaign. Jerry can do the job alone in 6 hours. If
Tom helps, they can get the job done in 4 hours. How long
would it take Tom to do the job by himself?
3. One pipe can fill a swimming pool in 10 hours, while another
pipe can empty the pool in 15 hours. How long would it take to
fill the pool if both pipes were accidentally left open?
4. One roofer can put a new roof on a house three times faster
than another. Working together they can roof a house in 5 days.
How long would it take the faster roofer working alone?
5. Triplets, Justin, Jason, and Jacob are working on a school
project. Justin can complete the project by himself in 6 hours,
Jason can complete the project by himself in 9 hours, and Jacob
can complete the project by himself in 8 hours. How long would
it take the triplets to complete the project if they work
together?
Skill Booster!

Lesson 1: Simplifying Rational Expression.
A Rational Expressions is reduced to its simplest form if the
numerator and denominator have no common factors.
Lesson 2: Multiplying Rational Expressions
To multiply rational expressions, recall the rules for multiplying
fractions. If the denominators are not equal to zero, then we simply
multiply the numerators and denominators. The same rule applies to
rational expressions.
If a, b, c, and d represent polynomials where b ≠ 0 and d ≠ 0. Then,
Lesson 3: Dividing Rational Expressions
In dividing rational expressions, observe the same rules as when dividing
fractions.
If the denominators are not equal to zero, then
Sum It Up !
Step 1
4a = (2) (2) a
12b= (3) (2) (2) b
GCF: (2) (2)
GCF: 4
Thus, the common factor is 4.
Step 2:
4𝑎
12𝑏
=
4 ( 𝑎 )
4 (3𝑏)
=
𝑎
3𝑏
𝑎
3𝑏
.
𝒂
𝒃
∗
𝒄
𝒅
=
𝒂𝒄
𝒃𝒅
𝒂
𝒃
÷
𝒄
𝒅
=
𝒂
𝒃
∗
𝒅
𝒄
=
𝒂𝒅
𝒃𝒄

Lesson 4: Adding and Subtracting Rational with Same
Denominator
In adding and subtracting rational expressions with same
denominators, perform the operation in the numerator and keep the
denominator. The resulting sum and difference must be reduced to
lowest terms whenever possible.
Provided the denominator is not equal to zero, then
Lesson 5: Adding and Subtracting Rational with
Different Denominator
When the denominators of the two rational algebraic expressions
are different, we need to find the least common denominator (LCD)
of the given expressions.
Provided that the denominators are not equal to zero, then,
Lesson 6: Work Problems Involving Rational Equations
Shared work problems mean two or more people (or objects)
working together on a job to share the work.
▪ Persons A and B do the job together in 𝑥 hours, so their rate
together is
1
𝑥
of the job per hour
1
2
+
1
3
=
1
𝑥
𝒂
𝒃
+
𝒄
𝒃
=
𝒂 + 𝒄
𝒃
𝒂
𝒃
−
𝒄
𝒃
=
𝒂 − 𝒄
𝒃
𝒂
𝒃
+
𝒄
𝒅
=
𝒂𝒅 + 𝒄𝒅
𝒃𝒅
𝒂
𝒃
−
𝒄
𝒅
=
𝒂𝒅 − 𝒄𝒅
𝒃𝒅

A. Direction: Read and analyze each questions carefully and encircle
the letter of the correct answer.
1. Which of the following expressions is a rational algebraic expression?
2. What is the value of a non-zero polynomial raised to 0?
a. constant c. undefined
b. zero d. cannot be determined
3. What rational algebraic expression is the same as
𝑥2−1
𝑥−1
?
a. x + 1 c. 1
b. x – 1 d. -1
4. You own 3 hectares of land and you want to mow it for farming. What will
you do to finish it at the very least time?
a. Rent a small mower. c. Do kaingin.
b. Hire 3 efficient laborers. d. Use germicide.
6. Angelo can complete his school project in x hours. What part of the job
can be completed by Angelo after 3 hours?
a. x + 3 b. x – 3 c.
𝑥
3
d.
3
𝑥
Post-Test!

9. Your father, a tricycle driver, asked you regarding the best motorcycle
to buy. What will you do to help your father?
a. Look for the fastest motorcycle.
b. Canvass for the cheapest motorcycle.
c. Find an imitated brand of motorcycle.
d. Search for fuel-efficient type of motorcycle.
10. Your friend asked you to make a floor plan. As an engineer, what
aspects should you consider in doing the plan?
a. Precision c. Appropriateness
b. Layout and cost d. Feasibility
11. As a contractor in number 19, what is the best action to do in order to
complete the project on or before the deadline but still on the budget plan?
a. All laborers must be trained workers.
b. Rent more equipment and machines.
c. Add more equipment and machines that are cheap.
d. There must be equal number of trained and amateur workers.

15. Your SK Chairman planned to construct a basketball court. As a
contractor, what will you do to realize the project?
a. Show a budget proposal.
b. Make a budget plan.
c. Present a feasibility study.
d. Give a financial statement.

B. Direction: Classify the following expressions if it either Rational
Expression or not. Put each expression on the correct box.
Rational Algebraic
Expressions
Not Rational
Algebraic
Expressions

C.Direction: Match each rational algebraic expression to its
equivalent simplified expression from choices A to E. Write the rational
expression in the appropriate column. If the equivalent is not among
the choices, write it in column F.
D. Hours and Prints!

References
Diaz, Z., Mojica M. (2013) . Next Century Mathematics 8; Quezon City ; Phoenix Publishing House , Inc
Escaner, J., Sepida, M., Catalla, D. (2013)., Spiral Math 8; Quezon Ciity; Trinita Publishing , Inc
Mathematics 8 Learner’s Module K-12; DepEd K-12 Modified Curriculum Guide and Teacher’s Guide for
Mathematics
http://www.mesacc.edu/~scotz47781/mat120/notes/work/work_probs.htmlhttps://web.ics.purdue.edu/
~pdevlin/Traditional%20Class/Lesson%2011/Applications%20of%20Rational%20Equations.pdf
https://www.google.com/search?safe=active&sxsrf=ALeKk02mur0lCuIoGdtBBGuCWadLdxUynw%3A1594786462276&e
i=noIOX7ivEJTdmAW-iY-
IBQ&q=adding+and+subtracting+rational+expressions&oq=ADDING+AND+SUBTR&gs_lcp=CgZwc3ktYWIQARgAMgQIAB
BDMgQIABBDMgQIABBDMgQIABBDMgQIABBDMgQIABBDMgQIABBDMgQIABBDMgQIABBDMgQIABBDOgQIIxAnOgUIA
BCRAjoHCAAQsQMQQzoFCAAQsQM6CggAELEDEIMBEEM6AggAOgQIABAKUOM6WIVxYN58aANwAHgAgAGvA4gBsyaSA
QkwLjkuOC4zLjGYAQCgAQGqAQdnd3Mtd2l6&sclient=psy-ab
https://www.youtube.com/watch?v=uVpsz-xpnPo
https://www.youtube.com/watch?v=1KGVwaUhR-s
https://www.youtube.com/watch?v=RROSgr4oXjU
https://www.youtube.com/watch?v=WbvLjmK4Kmc

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