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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.

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  • blobby green style avatar for user joe.dalessio47
    I am older than most trying
    To learn algebra II
    To help my grandkids,
    Very old school ...
    Is there a way to find
    Out the value of the letters
    In various problems
    For example the above
    What value is the r
    (157 votes)
    Default Khan Academy avatar avatar for user
    • leafers ultimate style avatar for user SJTheOne
      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!
      (52 votes)
  • old spice man green style avatar for user Pedro

    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.

    (62 votes)
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  • piceratops tree style avatar for user Harpreet Lota
    why cant you do -9-1 before you use the distributive property in the beginning of the equation?
    (25 votes)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (58 votes)
  • blobby green style avatar for user wgualtiere
    what if the the variables are the same value
    (17 votes)
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    • orange juice squid orange style avatar for user Mary Goss
      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.
      (32 votes)
  • leaf green style avatar for user walter hartwell white
    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.
    (18 votes)
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  • piceratops sapling style avatar for user Jacquelyn  Caldwell
    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?
    (14 votes)
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    • starky ultimate style avatar for user Patrick
      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!
      (12 votes)
  • male robot hal style avatar for user Andrew St. Clair
    Is there a way to memorize all of these steps to the linear equations?
    (6 votes)
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  • area 52 purple style avatar for user KNIGHTSTER
    what does parenthesis mean?
    (3 votes)
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  • blobby green style avatar for user 0070194758
    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.
    (2 votes)
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    • hopper cool style avatar for user Seed Something
      Great question, and smart of you to prepare!

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

      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.

      Review Arithmetic, (algebra is full of fractions, exponents, signed integers, etc), and…

      Take advantage of, and learn from Khan Academy's Algebra prep classes.
      Get Ready for Algebra 1

      Continued in Comments
      (11 votes)
  • blobby green style avatar for user underrated programmer
    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.
    (5 votes)
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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.