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### Course: Pre-algebra > Unit 12

Lesson 2: Equations with parentheses# Equations with parentheses

When an equation such as -9 - (9x - 6) = 3(4x + 6) has parentheses, we can distribute without changing the value of each side. Then combine like terms. Next, we move all the x-terms to one side and the constants to the other. Finally, we solve for x. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- 4r−3=3(3r+4)

I am older than most trying

To learn algebra II

To help my grandkids,

H

Very old school ...

Is there a way to find

Out the value of the letters

In various problems

For example the above

Problem

What value is the r(139 votes)- We can do this in steps:

First, we need to open the parenthesis.

4r-3 = 3(3r+4)

⬇

4r-3 = 9r+12

Then, we can either add 3 to both sides or subtract 12 to get only R's on one side, like this:

4r-3 = 9r+12

⬇

4r = 9r+15**or**4r-15 = 9r

Then, we subtract the extra R's.

4r = 9r+15**or**4r-15 = 9r

⬇

0 = 5r+15**or**-15 = 5r

If you chose the options on the left, you subtract 15 from both sides to get the equation on the right. Then, you divide by 5 to get the value for r:

-15 = 5r

⬇

*-3 = r*

Hope this helps!(33 votes)

- Hi!

What if we were dealing with Inequalities? Would having a negative 9 behind the brackets change the direction of the Inequality?

Here, have a look:

-9 times [9x - 6] > 3 times [4x + 6] » Would that initial negative 9 have influence on the direction the Inequality assumes? Instead of having ''greater than'', would we have ''lesser than''? Or do we turn a blind eye to that initial negative 9 and only change the direction of the Inequality if, later on, we need to multiply or divide?

Hope I explained myself properly.

Cheers!(59 votes)- I really don’t understand what this guy is talking about(21 votes)

- why cant you do -9-1 before you use the distributive property in the beginning of the equation?(22 votes)
- PEMDAS rules always apply when you are simplifying each expression on the sides of the equation.

You can't do the work inside the parentheses because you can't combine unlike terms. But, you have to deal with the parentheses. The only way to remove the parentheses is to use the distributive property.

If you do -9-1 as your 1st step, you are skipping all the PEMDAS rules prior to additionn/subtraction, which can't be done.

Hope this helps.(55 votes)

- what if the the variables are the same value(16 votes)
- They are supposed to be. In this case, Sal is only solving for one variable: x. Even though x appears twice in the equation, x is one number. Just like 9 appears twice, and 6 appears twice, x appears twice but it's one number still. Hopefully that makes sense.(29 votes)

- How do you know if it is negative and not just subtraction? At point 50 seconds in the video he says negative one times negative six. How would you know it's not just subtraction?(12 votes)
- If you mean something like -7(-3), I can answer that. There are three ways to indicate multiplication: an x sign, a dot, or a number in parentheses touching another number. If it is not one of those, and is more like -5-6, then it would be subtraction. Hope this helps!(10 votes)

- Is there a way to memorize all of these steps to the linear equations?(5 votes)
- well use the BODMAS(Bracket, Of, Division, Multiplication, Addition, Substraction) rule. it helps, don't memorize maths.

eg.

solve 18-(20-15/3)

=18-(20-5)

=18-15

=3 (ans)(4 votes)

- what I don't like about Kahn Academy is that they show you a video of a simple problem, then expect you to understand it right away, then go on to give problems that are way harder then the example.(14 votes)
- Let all dat stress go😫from work, them meetings Let all dat stress go(5 votes)

- what does parenthesis mean?(4 votes)
- parenthesis are '(' and ')' . Calculations between () or parenthesis have the highest priority when it comes to sequencing.(16 votes)

- Hi I am going to 7th grade and going to be learning algebra 1 what are some tips I can use to help me understand this topic more.(3 votes)
- Great question, and smart of you to prepare!

★**Algebra**, and both are used in Geometry.*uses and builds upon*all past Arithmetic math skills and concepts

It's very useful in life and is applied in many different situations, so we can find the answers we need.

★**Algebra**is**a method to solve for unknown values**, using logic, balance, and Arithmetic.

so…

•**Review Arithmetic**, (algebra is full of fractions, exponents, signed integers, etc), and…**Take advantage of, and learn from**Khan Academy's Algebra prep classes.

•**Pre-Algebra**

•**Get Ready for Algebra 1***Continued in Comments*(10 votes)

- Hi, Sal. Thanks for taking the time to explain this. You created this distribution property, right?

Why did you add an invisible 1 to the first parentheses i.e. ( -9- 1(9x -6)) you didn't add to the second parentheses i.e. (3(4x + 6)). I would appreciate it if you can provide an in-depth explanation for this.

Thanks in advance.(6 votes)- Just ignore this. I have finally figured this out myself using another method.(5 votes)

## Video transcript

We have the equation negative 9
minus this whole expression, 9x minus 6-- this whole thing
is being subtracted from negative 9-- is equal to
3 times this whole expression, 4x plus 6. Now, a good place to start is
to just get rid of these parentheses. And the best way to get rid of
these parentheses is to kind of multiply them out. This has a negative 1-- you
just see a minus here, but it's just really the same thing
as having a negative 1-- times this quantity. And here you have a 3
times this quantity. So let's multiply it out using
the distributive property. So the left-hand side of
our equation, we have our negative 9. And then we want to multiply the
negative 1 times each of these terms. So negative 1 times
9x is negative 9x, and then negative 1 times negative
6 is plus 6, or positive 6. And then that is going to be
equal to-- let's distribute the 3-- 3 times 4x is 12x. And then 3 times 6 is 18. Now what we want to do, let's
combine our constant terms, if we can. We have a negative 9 and a 6
here, on this side, we've combined all of our like terms.
We can't combine a 12x and an 18, so let's
combine this. So let's combine the negative 9
and the 6, our two constant terms on the left-hand
side of the equation. So we're going to have
this negative 9x. So we're going to have negative
9x plus-- let's see, we have a negative 9 and then
plus 6-- so negative 9 plus 6 is negative 3. So we're going to have a
negative 9x, and then we have a negative 3, so minus
3 right here. That's the negative 9 plus
the 6, and that is equal to 12x plus 18. Now, we want to group all the
x terms on one side of the equation, and all of the
constant terms-- the negative 3 and the positive 18 on the
other side-- I like to always have my x terms on the left-hand
side, if I can. You don't have to have them on
the left, so let's do that. So if I want all my x terms on
the left, I have to get rid of this 12x from the right. And the best way to do that is
to subtract 12x from both sides of the equation. So let me subtract 12x from the
right, and subtract 12x from the left. Now, on the left-hand side, I
have negative 9x minus 12x. So negative 9 minus 12,
that's negative 21. Negative 21x minus 3 is equal
to-- 12x minus 12x, well, that's just nothing. That's 0. So I could write a 0
here, but I don't have to write anything. That was the whole point of
subtracting the 12x from the left-hand side. And that is going to be equal
to-- so on the right-hand side, we just are
left with an 18. We are just left with
that 18 here. These guys canceled out. Now, let's get rid of
this negative 3 from the left-hand side. So on the left-hand side, we
only have x terms, and on the right-hand side, we only have
constant terms. So the best way to cancel out a negative
3 is to add 3. So it cancels out to 0. So we're going to add
3 to the left, let's add 3 to the right. And we get-- the left-hand side
of the equation, we have the negative 21x, no other x
term to add or subtract here, so we have negative 21x. The negative 3 and the plus 3,
or the positive 3, cancel out-- that was the whole
point-- equals-- what's 18 plus 3? 18 plus 3 is 21. So now we have negative
21x is equal to 21. And we want to solve for x. So if you have something times
x, and you just want it to be an x, let's divide by
that something. And in this case, that something
is negative 21. So let's divide both sides of
this equation by negative 21. Divide both sides
by negative 21. The left-hand side, negative
21 divided by negative 21, you're just left with an x. That was the whole point behind dividing by negative 21. And we get x is equal
to-- what's 21 divided by negative 21? Well, that's just negative 1. Right? You have the positive version
divided by the negative version of itself, so it's
just negative 1. So that is our answer. Now let's verify that this
actually works for that original equation. So let's substitute negative 1
into that original equation. So we have negative 9-- I'll do
it over here; I'll do it in a different color than we've
been using-- we have negative 9 minus-- that 1 wasn't there
originally, it's there implicitly-- minus 9
times negative 1. 9 times-- I'll put negative 1
in parentheses-- minus 6 is equal to-- well, actually,
let me just solve for the left-hand side when I substitute
a negative 1 there. So the left-hand side becomes
negative 9, minus 9 times negative 1 is negative
9, minus 6. And so this is negative 9
minus-- in parentheses-- negative 9 minus 6
is negative 15. So this is equal
to negative 15. And so we get negative 9-- let
me make sure I did that-- negative 9 minus 6,
yep, negative 15. So negative 9 minus negative 15,
that's the same thing as negative 9 plus 15,
which is 6. So that's what we get on the
left-hand side of the equation when we substitute x is
equal to negative 1. We get that it equals 6. So let's see what happens when
we substitute negative 1 to the right-hand side
of the equation. I'll do it in green. We get 3 times 4 times
negative 1 plus 6. So that is 3 times negative
4 plus 6. Negative 4 plus 6 is 2. So it's 3 times 2,
which is also 6. So when x is equal to negative
1, you substitute here, the left-hand side becomes 6, and
the right-hand side becomes 6. So this definitely works out.