![Ms. Lorie Jane L Letada
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,](https://image.slidesharecdn.com/rationalexpressionsmodule-200715042836/75/Rational-Expressions-Module-20-2048.jpg)
![Module 2: Rational Expressions
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
𝒄𝒅−𝟐𝒄−𝟑𝒅+𝟔
=
𝒄 𝟐− 𝒅 𝟐−𝟐𝒄 +𝟐𝒅−𝟓
𝒄𝒅−𝟐𝒄−𝟑𝒅+𝟔
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](https://image.slidesharecdn.com/rationalexpressionsmodule-200715042836/75/Rational-Expressions-Module-21-2048.jpg)
![Ms. Lorie Jane L Letada
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.](https://image.slidesharecdn.com/rationalexpressionsmodule-200715042836/75/Rational-Expressions-Module-24-2048.jpg)
![Module 2: Rational Expressions
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](https://image.slidesharecdn.com/rationalexpressionsmodule-200715042836/75/Rational-Expressions-Module-25-2048.jpg)
![Ms. Lorie Jane L Letada
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.](https://image.slidesharecdn.com/rationalexpressionsmodule-200715042836/75/Rational-Expressions-Module-26-2048.jpg)
Uploaded byLorie Jane Letada
Rational Expressions Module
This document provides instruction on rational expressions. It begins with a definition of rational expressions as algebraic expressions where both the numerator and denominator are polynomials. The document then outlines the key lessons to be covered: illustrating, simplifying, and performing operations on rational expressions. Examples are provided of simplifying rational expressions by factoring and canceling common factors between the numerator and denominator. The document also demonstrates multiplying rational expressions using the same rules as multiplying fractions, as well as canceling common factors.
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Rational Expressions Module
- 2. Ms. Lorie JaneL 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 Rational Expressions Multiplying and Dividing Rational Expressions
- 3. Module 2: RationalExpressions Read and analyze each item carefully and encircle the letter of the best answer. Pre-test!
- 4. Ms. Lorie JaneL Letada
- 5. Module 2: RationalExpressions
- 6. Ms. Lorie JaneL Letada 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 Factor the numerator and denominator and get the GCF. 9m = 21n = GCF: GCF: Thus, the common factor is __. Step 2: Divide out the common factor. 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 Thus, the simplify is 2 3−4𝑤 .
- 7. Module 2: RationalExpressions 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: Divide out the common factor. Step 1 Factor the common numerator and denominator. 2𝑥−2𝑦 𝑦2− 𝑥2 = 2 (𝑦−𝑥) ( 𝑦−𝑥 )(𝑦+ 𝑥) Step 2: Divide out the common factor. 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 Factor the common numerator 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 Factor the common numerator and denominator. 4𝑝+ 4𝑞 𝑝2− 𝑞2 = Step 2: Divide out the common factor. A variable is perfect square if its exponent is even number.
- 8. Ms. Lorie JaneL Letada 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: Divide out the common factor. 𝑚2+𝑚−6 𝑚2−7𝑚+10 = ( 𝑚+3)(𝑚−2 ) (𝑚−5)(𝑚−2) = (𝑚+3) (𝑚−5 ) Step 1 Factor the common numerator and denominator. 𝑛2−7𝑛−30 𝑛2−5𝑛−24 = Step 2: Divide out the common factor. Step 1 Factor the common numerator and denominator. 𝑏2−49 𝑏2−2𝑏−35 = (𝑏−7 (𝑏+7 ) (𝑏−7 )(𝑏+5 ) Step 2: Divide out the common factor. 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
- 9. Module 2: RationalExpressions 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 3: Write what is left. 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) Step 3: Write what is left. 𝟏𝟓 𝟐𝒚 𝟐 ∗ 𝒚 𝟒 𝟐𝟎𝒚 𝟐 = 𝟑 (𝟐)(𝟒) = 𝟑 𝟖
- 10. Ms. Lorie JaneL Letada 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. Step 3: Write what is left. Step 1: Factor each expression. 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) Step 3: Write what is left. 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 .…. Step 1: Factor each expression. 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.
- 11. Module 2: RationalExpressions 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!
- 12. Ms. Lorie JaneL Letada 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
- 13. Module 2: RationalExpressions 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 Write your solution here… 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 Write your solution here… 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
- 14. Ms. Lorie JaneL Letada 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!
- 15. Module 2: RationalExpressions 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:
- 16. Ms. Lorie JaneL Letada 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.
- 17. Module 2: RationalExpressions 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!
- 18. Ms. Lorie JaneL Letada 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.
- 19. Module 2: RationalExpressions 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
- 20. Ms. Lorie JaneL Letada 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,
- 21. Module 2: RationalExpressions 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 𝒄𝒅−𝟐𝒄−𝟑𝒅+𝟔 = 𝒄 𝟐− 𝒅 𝟐−𝟐𝒄 +𝟐𝒅−𝟓 𝒄𝒅−𝟐𝒄−𝟑𝒅+𝟔 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
- 22. Ms. Lorie JaneL Letada 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
- 23. Module 2: RationalExpressions 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 Rational Expressions
- 24. Ms. Lorie JaneL Letada 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.
- 25. Module 2: RationalExpressions 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
- 26. Ms. Lorie JaneL Letada 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.
- 27. Module 2: RationalExpressions 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!
- 28. Ms. Lorie JaneL Letada 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 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 . 𝒂 𝒃 ∗ 𝒄 𝒅 = 𝒂𝒄 𝒃𝒅 𝒂 𝒃 ÷ 𝒄 𝒅 = 𝒂 𝒃 ∗ 𝒅 𝒄 = 𝒂𝒅 𝒃𝒄
- 29. Module 2: RationalExpressions 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
- 30. Ms. Lorie JaneL Letada 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!
- 31. Module 2: RationalExpressions 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.
- 32. Ms. Lorie JaneL Letada 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.
- 33. Module 2: RationalExpressions 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
- 34. Ms. Lorie JaneL Letada 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!
- 35. Module 2: RationalExpressions
- 36. Ms. Lorie JaneL Letada 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