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Two-step inequalities

Two-step inequalities are slightly more complicated than one-step inequalities (duh!). This is a worked example of solving ⅔>-4y-8⅓. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • winston default style avatar for user Daniel Hernandez
    why is it so hard to learn?
    (44 votes)
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  • blobby green style avatar for user Samanthac.phelan
    How would you then answer a question like this type?
    4x − 3/2− 3x − 3/3< 3
    I cant find any references to this type of equation, since i don't think it is a multi-step inequality
    (19 votes)
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  • blobby green style avatar for user jonnaanisa
    I was trying to solve a multi-step inequality but I don't know how to. The inequality is:
    -1/4(8n-12) ≥ −2n−1 When I try to solve it the n variable no longer exists and I don't understand how I would continue to solve. When I looked online they said the answer would be all real numbers are solutions but I still do not understand. Please help.
    (10 votes)
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    • stelly blue style avatar for user Kim Seidel
      1) Distribute the -1/4 on the left side
      -1/4(8n-12) ≥ −2n−1
      -2n+3 ≥ −2n−1
      2) Add 2n to both sides
      3 ≥ −1
      So, as you have found already, the variable disappears. The inequality that remains is True. 3 is greater than -1. This is called an identify. It tells us that any value of "n" that you use in the inequality will be a solution. So, the solution is all real numbers. You can see this for yourself by picking any value you like for "n" and checking to see if it is a solution. All values will work.

      Here are a couple of other scenarios that you might run into:
      If your inequality had become 3 ≥ 3, this is also true (an identity). So, again the solution is all real numbers.

      If your inequality becomes something like 3 < 1, this is False ( 3 is never less than 1). This is called a contradiction. It means that the inequality has no solution.
      (17 votes)
  • ohnoes default style avatar for user Rachel Chan
    does it matter which side the variable terms are
    (6 votes)
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  • duskpin ultimate style avatar for user McBoi
    Why do you have to swap the inequality so that it is less than?
    (7 votes)
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  • female robot grace style avatar for user CatFantasy
    At , Sal simplifies 2/3 by multiplying it by three(because he's doing that to the other side of the inequality sign). He then crosses out both threes and leaves the two. Why? How does that work? It's probably really obvious, but can someone answer this?
    (3 votes)
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    • piceratops sapling style avatar for user rubixspeller
      Sure!, A fraction is just a number divided by another number, therefore to undo the division you multiply. Sal crossed them out because they cancel out each other.
      For example,
      if you have 6/3 that will equal 2. then multiply by three to get 6. But you should just multiply by the denominator and cancel out everything at the beginning to get 6!
      Hope this helps!
      (3 votes)
  • aqualine ultimate style avatar for user Milky Way
    Please, at , why did he say that 8x1/3 is equal to 25? Don't we have to do 8 times 1 and as the numerator is one, we have to keep the 3? So why is it not equal to 8/3?
    (2 votes)
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  • primosaur ultimate style avatar for user Lily Harden
    Where did you get the 25/3 from?
    (3 votes)
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  • blobby green style avatar for user Nylisha Russell
    Math is so hard it used to be so easy in 5th grade
    (3 votes)
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  • aqualine sapling style avatar for user cole_scafferi
    Yes, I get it but I needed to know what y is. So in this problem, y=? How do I get that?
    (3 votes)
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Video transcript

We have the inequality 2/3 is greater than negative 4y minus 8 and 1/3. Now, the first thing I want to do here, just because mixed numbers bother me-- they're actually hard to deal with mathematically. They're easy to think about-- oh, it's a little bit more than 8. Let's convert this to an improper fraction. So 8 and 1/3 is equal to-- the denominator's going to be 3. 3 times 8 is 24, plus 1 is 25. So this thing over here is the same thing as 25 over 3. Let me just rewrite the whole thing. So it's 2/3 is greater than negative 4y minus 25 over 3. Now, the next thing I want to do, just because dealing with fractions are a bit of a pain, is multiply both sides of this inequality by some quantity that'll eliminate the fractions. And the easiest one I can think of is multiply both sides by 3. That'll get rid of the 3's in the denominator. So let's multiply both sides of this equation by 3. That's the left-hand side. And then I'm going to multiply the right-hand side. 3, I'll put it in parentheses like that. Well, one point that I want to point out is that I did not have to swap the inequality sign, because I multiplied both sides by a positive number. If the 3 was a negative number, if I multiplied both sides by negative 3, or negative 1, or negative whatever, I would have had to swap the inequality sign. Anyway, let's simplify this. So the left-hand side, we have 3 times 2/3, which is just 2. 2 is greater than. And then we can distribute this 3. 3 times negative 4y is negative 12y. And then 3 times negative 25 over 3 is just negative 25. Now, we want to get all of our constant terms on one side of the inequality and all of our variable terms-- the only variable here is y on the other side-- the y is already sitting here, so let's just get this 25 on the other side of the inequality. And we can do that by adding 25 to both sides of this equation. So let's add 25 to both sides of this equation. And with the left-hand side, 2 plus 25 five is 27 and we're going to get 27 is greater than. The right-hand side of the inequality is negative 12y. And then negative 25 plus 25, those cancel out, that was the whole point, so we're left with 27 is greater than negative 12y. Now, to isolate the y, you can either multiply both sides by negative 1/12 or you could say let's just divide both sides by negative 12. Now, because I'm multiplying or dividing by a negative number here, I'm going to need to swap the inequality. So let me write this. If I divide both sides of this equation by negative 12, then it becomes 27 over negative 12 is less than-- I'm swapping the inequality, let me do this in a different color-- is less than negative 12y over negative 12. Notice, when I divide both sides of the inequality by a negative number, I swap the inequality, the greater than becomes a less than. When it was positive, I didn't have to swap it. So 27 divided by negative 12, well, they're both divisible by 3. So we're going to get, if we divide the numerator and the denominator by 3, we get negative 9 over 4 is less than-- these cancel out-- y. So y is greater than negative 9/4, or negative 9/4 is less than y. And if you wanted to write that-- just let me write this-- our answer is y is greater than negative 9/4. I just swapped the order, you could say negative 9/4 is less than y. Or if you want to visualize that a little bit better, 9/4 is 2 and 1/4, so we could also say y is greater than negative 2 and 1/4 if we want to put it as a mixed number. And if we wanted to graph it on the number line-- let me draw a number line right here, a real simple one. Maybe this is 0. Negative 2 is right over, let's say negative 1, negative 2, then say negative 3 is right there. Negative 2 and 1/4 is going to be right here, and it's greater than, so we're not going to include that in the solution set. So we're going to make an open circle right there. And everything larger than that is a valid y, is a y that will satisfy the inequality.