Adding and Subtracting Rational Expressions with Like Denominators


   

Return to the Purplemath home page

Try a MathHelp.com demo lesson
Join MathHelp.com
Login to MathHelp.com
align=”left” valign=”top”>

 

Index of lessons
Print this page (print-friendly version) | Find local tutors

 

Adding
and Subtracting Rational Expressions:
    Examples
(page
2 of 3)



As you follow along in
these examples, note how I do everything neatly and orderly. You should
model your homework after these exercises, to help minimize errors.

  • Simplify the following:
    • (5x - 1)/(x + 8) - (3x + 4)/(x + 8)

    These fractions already
    have a common denominator, so I can just add. But I’ll use parentheses
    on the numerators, to make sure I carry the "minus" through
    the second parentheses. (A common mistake would be to take the "minus"
    sign only onto the "3x"
    and not onto the "
    4".)

      (5x - 1)/(x + 8) - (3x + 4)/(x + 8) = (2x - 5)/(x + 8)

    Then the answer is:

      (2x - 5)/(x + 8)

  • Simplify the following:
    • 3x/(x^2 + 3x - 10) - 6/(x^2 + 3x - 10)

    Again, these already
    have a common denominator, so I can just combine them as they are. But
    the denominator is a quadratic, so I’ll want to factor the numerator
    when I’m done, to check and see if anything cancels out.

      3x/(x^2 + 3x - 10) - 6/(x^2 + 3x - 10) = [3(x - 2)]/[(x + 5)(x - 2)] = 3/(x + 5)

    As you can see, something
    did cancel. You always need to remember this step: factor the denominator
    and numerator (if possible) and check for common factors. By the way,
    since I was able to cancel off the "x – 2" factor,
    this eliminated a zero from the denominator. Depending on your book
    and on your instructor, you may (or may not) need to account for this
    change in the domain of the fraction.

    Copyright
    © Elizabeth Stapel 2003-2011 All Rights Reserved

    Advertisement

    The complete answer is:

      3/(x + 5) for x not equal to 2

Depending on your instructor,
you might not need the "for x not equal to 2"
part. If you’re not sure, ask now, before the test.

  • Simplify the following:
    • (x + 4)/2x - (x - 1)/x^2

    First I have to convert
    these fractions to the common denominator of 2x2.
    (If you’re not sure about the common denominator, do the factor table,
    as shown in the second example on the
    previous
    page
    , to check.)
    Then I’ll add and, if possible, cancel off common factors.

      (x + 4)/2x - (x - 1)/x^2 = (x^2 + 4x)/2x^2 - (2x - 2)/2x^2 = (x^2 + 2x + 2)/2x^2

    Note how I used parentheses
    to keep my subtraction straight. I wanted to be sure to carry the "minus"
    through properly, and the extra step with the parentheses is very helpful
    for this. Nothing cancelled in this case, so the answer is:

      x^2 + 2x + 2)/2x^2

It isn’t common that you
will be able to simplify a rational addition or subtraction problem, but
you should get in the habit of checking. I would bet good money that you’ll
have a problem that simplifies on the test.

  • Simplify the following:
    • 4x/(2x - 1) - 5/(x - 6)

    The two denominators
    have no common factors, so the common denominator will be
    (2x – 1)(x – 6).

      4x/(2x - 1) - 5/(x - 6) = (4x^2 - 34x + 5)/[(2x - 1)(x - 6)]

    The numerator doesn’t
    factor, so there is no chance of anything cancelling off. It is customary
    to leave the denominator factored like this, so, unless your instructor
    says otherwise, don’t bother multiplying the denominator out. The answer
    is:

      (4x^2 - 34x + 5)/[(2x - 1)(x - 6)]

  • Simplify the following:
    • 3/(x + 2) + 2

    Don’t let this one throw
    you. The denominator of the "2"
    is just "
    1",
    so the common denominator will be the only other denominator of interest:
    "
    x + 2".

      3/(x + 2) + 2 = 3/(x + 2) + (2x + 4)/(x + 2) = (2x + 7)/(x + 2)

    Nothing cancels, so the
    answer is:

      (2x + 7)/(x + 2)

<< Previous   Top   |   1 | 2 | 3  |   Return
to Index    Next >>

Cite this article
as:

Stapel, Elizabeth.
"Adding and Subtracting Rational Expressions: Examples." Purplemath. Available from https://www.purplemath.com/modules/rtnladd2.htm.
Accessed
 

 

MathHelp.com Courses

K12 Math

5th Grade Math
6th Grade Math
Pre-Algebra
Algebra 1
Geometry
Algebra 2

College Math

College Pre-Algebra
Introductory Algebra
Intermediate Algebra
College Algebra

Standardized Test Prep

ACCUPLACER Math
ACT Math
ASVAB Math
CBEST Math
CHSPE Math
CLEP Math
COMPASS Math
FTCE Math
GED Math
GMAT Math
GRE Math
MTEL Math
NES Math
PERT Math
PRAXIS Math
SAT Math
TABE Math
TEAS Math
TSI Math

more tests…

  Copyright � 2003-2014   Elizabeth
Stapel   |    About
  |    Terms of Use
  |    Linking   |    Site Licensing

 

  Feedback
  |    Error?   

 

 

 

Found

The document has moved here .


Apache Server at www.mesacc.edu Port 80