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8.5 Adding and subtracting rational expressions

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<para>This module is from<link document="col10614">Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr.</para><para>A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.</para><para>The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.</para><para>Objectives of this module: be familiar with the basic rule for adding and subtracting rational expressions, be able to add and subtract fractions with the same and with different denominators.</para>

Overview

  • Basic Rule
  • Fractions with the Same Denominator
  • Fractions with Different Denominators

Basic rule

We are now in a position to study the process of adding and subtracting rational expressions. There is a most basic rule to which we must strictly adhere if we wish to conveniently add or subtract rational expressions.

To add or subtract rational expressions conveniently, they should have the same denominators.

Thus, to add or subtract two or more rational expressions conveniently, we must ensure that they all have the same denominator. The denominator that is most convenient is the LCD.

Fractions with the same denominator

The rule for adding and subtracting rational expressions

To add (or subtract) two or more rational expressions with the same denominators, add (or subtract) the numerators and place the result over the LCD. Reduce if necessary. Symbolically,

a c + b c = a + b c
a c − b c = a − b c

Note that we combine only the numerators.

Sample set a

Add or subtract the following rational expressions.

1 6 + 3 6 The denominators are the same . Add the numerators . 1 6 + 3 6 = 1 + 3 6 = 4 6 Reduce . 1 6 + 3 6 = 2 3

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5 x + 8 x The denominators are the same . Add the numerators . 5 x + 8 x = 5 + 8 x = 13 x

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2 a b y 2 w − 5 b y 2 w The denominators are the same . Subtract the numerators . 2 a b y 2 w − 5 b y 2 w = 2 a b − 5 b y 2 w

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3 x 2 + x + 2 x − 7 + x 2 − 4 x + 1 x − 7 The denominators are the same . Add the numerators . 3 x 2 + x + 2 x − 7 + x 2 − 4 x + 1 x − 7 = 3 x 2 + x + 2 + x 2 − 4 x + 1 x − 7 = 4 x 2 − 3 x + 3 x − 7

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5 y + 3 2 y − 5 − 2 y + 4 2 y − 5 The denominators are the same . Subtract the numerators .  But  b e   c a r e f u l  to subtract the  e n t i r e  numerator . Use parentheses! 5 y + 3 2 y − 5 − 2 y + 4 2 y − 5 = 5 y + 3 − ( 2 y + 4 ) 2 y − 5 = 5 y + 3 − 2 y − 4 2 y − 5 = 3 y − 1 2 y − 5 N o t e : 5 y + 3 2 y − 5 − 2 y + 4 2 y − 5 ︸ Observe this part . The term − 2 y + 4 2 y − 5  could be written as + − ( 2 y + 4 ) 2 y − 5 = − 2 y − 4 2 y − 5

A common mistake is to write

− 2 y + 4 2 y − 5  as  − 2 y + 4 2 y − 5
This is not correct, as the negative sign is not being applied to the entire numerator.

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
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Haryormhidey Reply
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Esrael Reply
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Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
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WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
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Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
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Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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