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Multi-step inequalities

Sal solves several multi-step linear inequalities. Created by Sal Khan.

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  • aqualine ultimate style avatar for user Blue Phoenix
    At , Sal says that you swap the inequality sign when you divide by a negative number. But I'm pretty sure my teacher taught me that when you divide by a negative, you would change > to a less than OR EQUAL TO symbol, not just to a <. So confused...who is right....
    (16 votes)
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    • old spice man blue style avatar for user Owen
      I am sorry, but your Math teacher must have misspoke. When solving inequalities, like, say, this one:

      -2x+5<25

      You would cancel out the +5 with -5 and subtract 25 by 5, so you're left with this:

      -2x<20.

      But now, since you're dividing by -2 (remember that multiplying or dividing by a negative number will reverse the sign) it will no longer be less than, it will be greater than:

      -2x/-2>20/-2

      x>-10.

      So, therefore, you cannot go from < or > to an "or equal to" sign just by dividing or multiplying by a negative number.

      Behold. Math.

      Hope this helps :D
      (40 votes)
  • male robot hal style avatar for user Seth

    How would you do it if you had to go backwards (You were given the solution and asked to find the inequality that has that solution)?
    (14 votes)
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    • leaf red style avatar for user dryhter
      Just like in simple math. If I said " add two numbers together that equal six
      2+4=6, and we are done. as Sal likes to say. So ... ..
      Pic a number -1 make an expression where X = -1
      4x+3<-1. You see it worked and just like the addition there are only a couple of possibilities compared to all the possibilities that could work. Just check your work!
      (17 votes)
  • aqualine ultimate style avatar for user Govind_Beemisetty
    Why does Sal write a negative infinity sign? I don't get what it means.
    (13 votes)
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  • marcimus purple style avatar for user Ariana Kohen
    To whom it may concern,
    I hope you and your family is safe especially during this tough pandemic!
    (13 votes)
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  • blobby green style avatar for user Kole_Rollins
    when would you need to know inequalities?
    (8 votes)
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    • piceratops tree style avatar for user Theodore Stillson
      It is helpful to know inequalities in the future: say you are baking something, for example a cake, and you can't remember how much sugar you needed. You knew that it was more than how much flour you needed, multiplied by two. This could be expressed as S< 2F.
      You may not see inequalities pop out at you as: "Oh. That's an inequality!", but they are there. They are there everyday. It could be in homework or cooking or practically anything, but they are there. :)
      (7 votes)
  • starky tree style avatar for user No lian
    doesn't the negative and a negative equal to a positive number?
    (2 votes)
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    • leafers ultimate style avatar for user Christelle Winter
      A negative number multiplied by a negative number gives a positive result,
      but a negative number added to a negative number gives a negative result.
      Imagine it's -2 degrees outside and the temperature drops another 5 degrees, then it is now -7 degrees. Basically it is (-2)+(-5) = (-7)
      Hope this helps!
      (11 votes)
  • purple pi purple style avatar for user Madeleine
    Is there a clever way to remember to change the direction of the sign when dividing or multiplying by a negative number?
    (6 votes)
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    • leaf yellow style avatar for user Madialyn Neyohaven
      Think of the negative sign as a bad thing, or losing something.
      Think of the positive sign as a good thing, or gaining something.
      ( - )( + ) = losing something good = negative
      ( - )( - ) = losing something bad = positive
      ( + )( + ) = gaining something good = positive
      ( + )( - ) = gaining something bad = negative
      Did that help?
      (2 votes)
  • aqualine seed style avatar for user Peeta mellark
    How would you solve an inequality that contains exponents?
    Can anyone answer me?
    Thanks very much!
    (5 votes)
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  • ohnoes default style avatar for user TheBubblegumShark
    What exactly is an inequality?
    (4 votes)
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  • blobby green style avatar for user Gabriela Perez
    at do you have to always subtract the largest number or the smallest number because when I always do the problems I'm getting the correct answer but the sign is always the opposite I just had to subtract the other way wrong which is so confusing which is not mentioned. I've been trying to do this for 3 hours now and I can't get how it's always wrong.
    (3 votes)
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    • leaf orange style avatar for user Hayden
      You want all x's on one side of the equation. If that means subtracting a larger number of x's from a smaller number of x's, that's ok. However, if you end up with a negative number of x's, you need to divide both sides by that same negative number so that you end up with x = (some number). If you do have to divide by a negative number, make sure you flip the inequality (< or >). For example:

      5x - 4 > 2x + 2
      subtract 5x from both sides
      -4 > -3x + 2
      subtract 2 from both sides
      -6 > -3x
      divide both sides by -3
      remember the inequality flips because we divide by negative three
      2 < x

      Hope this helps!
      (4 votes)

Video transcript

Let's do a few more problems that bring together the concepts that we learned in the last two videos. So let's say we have the inequality 4x plus 3 is less than negative 1. So let's find all of the x's that satisfy this. So the first thing I'd like to do is get rid of this 3. So let's subtract 3 from both sides of this equation. So the left-hand side is just going to end up being 4x. These 3's cancel out. That just ends up with a zero. No reason to change the inequality just yet. We're just adding and subtracting from both sides, in this case, subtracting. That doesn't change the inequality as long as we're subtracting the same value. We have negative 1 minus 3. That is negative 4. Negative 1 minus 3 is negative 4. And then we'll want to-- let's see, we can divide both sides of this equation by 4. Once again, when you multiply or divide both sides of an inequality by a positive number, it doesn't change the inequality. So the left-hand side is just x. x is less than negative 4 divided by 4 is negative 1. x is less than negative 1. Or we could write this in interval notation. All of the x's from negative infinity to negative 1, but not including negative 1, so we put a parenthesis right there. Let's do a slightly harder one. Let's say we have 5x is greater than 8x plus 27. So let's get all our x's on the left-hand side, and the best way to do that is subtract 8x from both sides. So you subtract 8x from both sides. The left-hand side becomes 5x minus 8x. That's negative 3x. We still have a greater than sign. We're just adding or subtracting the same quantities on both sides. These 8x's cancel out and you're just left with a 27. So you have negative 3x is greater than 27. Now, to just turn this into an x, we want to divide both sides by negative 3. But remember, when you multiply or divide both sides of an inequality by a negative number, you swap the inequality. So if we divide both sides of this by negative 3, we have to swap this inequality. It will go from being a greater than sign to a less than sign. And just as a bit of a way that I remember greater than is that the left-hand side just looks bigger. This is greater than. If you just imagine this height, that height is greater than that height right there, which is just a point. I don't know if that confuses you or not. This is less than. This little point is less than the distance of that big opening. That's how I remember it. But anyway, 3x over negative 3. So now that we divided both sides by a negative number, by negative 3, we swapped the inequality from greater than to less than. And the left-hand side, the negative 3's cancel out. You get x is less than 27 over negative 3, which is negative 9. Or in interval notation, it would be everything from negative infinity to negative 9, not including negative 9. If you wanted to do it as a number line, it would look like this. This would be negative 9, maybe this would be negative 8, maybe this would be negative 10. You would start at negative 9, not included, because we don't have an equal sign here, and you go everything less than that, all the way down, as we see, to negative infinity. Let's do a nice, hairy problem. So let's say we have 8x minus 5 times 4x plus 1 is greater than or equal to negative 1 plus 2 times 4x minus 3. Now, this might seem very daunting, but if we just simplify it step by step, you'll see it's no harder than any of the other problems we've tackled. So let's just simplify this. You get 8x minus-- let's distribute this negative 5. So let me say 8x, and then distribute the negative 5. Negative 5 times 4x is negative 20x. Negative 5-- when I say negative 5, I'm talking about this whole thing. Negative 5 times 1 is negative 5, and then that's going to be greater than or equal to negative 1 plus 2 times 4x is 8x. 2 times negative 3 is negative 6. And now we can merge these two terms. 8x minus 20x is negative 12x minus 5 is greater than or equal to-- we can merge these constant terms. Negative 1 minus 6, that's negative 7, and then we have this plus 8x left over. Now, I like to get all my x terms on the left-hand side, so let's subtract 8x from both sides of this equation. I'm subtracting 8x. This left-hand side, negative 12 minus 8, that's negative 20. Negative 20x minus 5. Once again, no reason to change the inequality just yet. All we're doing is simplifying the sides, or adding and subtracting from them. The right-hand side becomes-- this thing cancels out, 8x minus 8x, that's 0. So you're just left with a negative 7. And now I want to get rid of this negative 5. So let's add 5 to both sides of this equation. The left-hand side, you're just left with a negative 20x. These 5's cancel out. No reason to change the inequality just yet. Negative 7 plus 5, that's negative 2. Now, we're at an interesting point. We have negative 20x is greater than or equal to negative 2. If this was an equation, or really any type of an inequality, we want to divide both sides by negative 20. But we have to remember, when you multiply or divide both sides of an inequality by a negative number, you have to swap the inequality. So let's remember that. So if we divide this side by negative 20 and we divide this side by negative 20, all I did is took both of these sides divided by negative 20, we have to swap the inequality. The greater than or equal to has to become a less than or equal sign. And, of course, these cancel out, and you get x is less than or equal to-- the negatives cancel out-- 2/20 is 1/10. If we were writing it in interval notation, the upper bound would be 1/10. Notice, we're including it, because we have an equal sign, less than or equal, so we're including 1/10, and we're going to go all the way down to negative infinity, everything less than or equal to 1/10. This is just another way of writing that. And just for fun, let's draw the number line. Let's draw the number line right here. This is maybe 0, that is 1. 1/10 might be over here. Everything less than or equal to 1/10. So we're going to include the 1/10 and everything less than that is included in the solution set. And you could try out any value less than 1/10 and verify that it will satisfy this inequality.