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Simplifying with Parentheses (page 1 of 3)


This topic is really part of studying the Order of Operations, but simplifying with parentheses is probably the sub-topic that causes students the most difficulty, so this lesson is meant to provide a little extra help in this area.

When simplifying expressions with parentheses, you will be applying the Distributive Property. That is, you will be distributing over (multiplying through) a set of parentheses in order to simplify a given expression. I will walk you through some examples of increasing difficulty, and you should note, as this lesson progresses, the importance of simplifying as you go and of doing each step neatly, completely, and exactly.

  • Simplify 3(x + 4).

    To "simplify" this, I have to get rid of the parentheses. The Distributive Property says to multiply the 3 onto everything inside the parentheses. I sometimes draw arrows to emphasize this:

      3(x + 4), drawn with arrows

    Then I multiply the 3 onto the x and onto the 4:

     

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      3(x + 4), drawn with arrows

      3(x) + 3(4)

      3x + 12

Written all in one line, this would look like: Copyright Elizabeth Stapel 2003-2011 All Rights Reserved

    3(x + 4) = 3(x) + 3(4) = 3x + 12

The most common error at this stage is to take the 3 through the parentheses but only onto the x, forgetting to carry it through onto the 4 as well. If you need to draw little arrows to help you remember to carry the multiplier through onto everything inside the parentheses, then use them!

  • Simplify 2(x 4)

    I have to take the 2 through the parentheses. This gives me:

      2(x 4)
      2(x) 2(4)

      2x + 8

The common mistake students make with this type of problem is to lose a "minus" sign somewhere, such as doing "2(x 4) = 2(x) 2(4) = 2x 8". Did you notice how the "4" somehow turned into a "4" when the 2 went through the parentheses? That's why the answer ended up being wrong. Be careful with the "minus" signs! Until you are confident in your skills, take the time to write out the distribution, complete with the signs, as I did.

    2(x 4)
    2(x) 2(4)

    2x + 8

If you have difficulty with the subtraction, try converting it to addition of a negative:

    2(x 4)
    2(x + [4])
    2(x) + (2)(4)
    2x + 8

Do as many steps as you need to, in order to consistently get the correct answer.

  • Simplify (x 3)

    I have to take the "minus" through the parentheses. Many students find it helpful to write in the little understood "1" before the parentheses:

      1(x 3)

    I need to take a 1 through the parentheses:

      (x 3)
      1(x 3)

      1(x) 1(3)

      1x + 3

      x + 3

Note that "1x + 3" and "x + 3" are technically the same thing; in my classes, either would be a perfectly acceptable answer. However, some teachers will accept only "x + 3" and would count
"
1x + 3" as not fully simplified. It would be wise to check with your instructor, especially if you find it helpful to write in that understood "1".

  • Simplify 2 + 4(x 1)

    The order of operations tells me that multiplication comes before addition. I can't do the "2 + " until I have taken the 4 through the parentheses.

      2 + 4(x 1)
      2 + 4(x) + 4(1)

      2 + 4x + (4)

      2 4 + 4x

      2 + 4x

      4x 2

I would accept either of "4x 2" and "2 + 4x" as a valid answer. However, most texts expect the answer to be written in "descending order" (with the variable term first, and then the plain number). You should know that the two expressions of the answer are the same, but that some instructors insist that the answer be written in descending order. It would probably be best to get in the habit now of writing your answers in descending order.

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Cite this article as:

Stapel, Elizabeth. "Simplifying with Parentheses." Purplemath. Available from
    http://www.purplemath.com/modules/simparen.htm. Accessed
 

 



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