Simplifying with Parentheses

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:

Then I multiply the 3 onto the x and onto the 4:
3(x) + 3(4)

3x + 12

Written all in one line, this would look like:

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.

Parentheses inside of parentheses are called "nested" parentheses. The process of simplification works the same way as in the simpler examples on the previous page, but you do need to be a little more careful as you work your way through the grouping symbols.

Simplify 4[x + 3(2x + 1)]

With nested parentheses like this, the safest plan is to work from the inside out. So I'll take the 3 through the inner parentheses first, before I even think about dealing with the 4 and the square brackets. I'll also simplify as much as I can as I go along. Note that I write each step out completely as I go:

4[x + 3(2x + 1)]
4[x + 3(2x) + 3(1)]
4[x + 6x + 3]
4[7x + 3]
4[7x] + 4[3]
28x + 12

FYI: The traditional sequence of grouping symbols, working from the inside out, is "parentheses", then "brackets", and then "braces"; then you repeat the sequence, as necessary. But this is not, to my knowledge, a rule; it's just a common convention.

Simplify 9 – 3[x – (3x + 2)] + 4

I won't do anything with the "9 –" or the "+ 4" until I simplify inside the brackets and parentheses. I'll work from the inside out:

9 – 3[x – (3x + 2)] + 4
9 – 3[x – 1(3x + 2)] + 4
9 – 3[x – 1(3x) – 1(2)] + 4
9 – 3[x – 3x – 2] + 4
9 – 3[–2x – 2] + 4
9 – 3[–2x] – 3[–2] + 4
9 + 6x + 6 + 4
6x + 19

It is not required that you write out this many (or this few) steps. You should be careful to do one step at a time, though, writing things out completely and simplifying as you go. You should do as many steps as you need in order to consistently arrive at the correct answer.

Simplify 5 + 2{ [3 + (2x – 1) + x] – 2}
I'll work carefully from the inside out:

5 + 2{ [3 + (2x – 1) + x] – 2}
5 + 2{ [3 + 2x – 1 + x] – 2}
5 + 2{ [2x + x + 3 – 1] – 2}
5 + 2{ [3x + 2] – 2}
5 + 2{3x + 2 – 2}
5 + 2{3x}
5 + 6x
6x + 5

Reference:
http://www.purplemath.com/modules/simparen2.htm