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One-step inequality involving addition

How to solve and graph one-step inequalities. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user jyoung310
    What if both sides of the inequality are negative? Does the sign switch?
    (44 votes)
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  • male robot hal style avatar for user agarcia323

    When would would need to flip the direction of the inequality
    (34 votes)
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    • blobby green style avatar for user dannynasir
      When you divide or multiply both sides of the inequality by a negative number. For example: -4x > 9 Here you have to divide both sides by a negative number, negative 4, so you carry out the division just like you would in a regular equality, but the only thing you do differently is you flip the inequality sign. So in this case it would be: x < -9/4. Hope this helps!
      (46 votes)
  • male robot hal style avatar for user Chase K Duncan
    When do you need to flip the inequality?
    (6 votes)
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    • piceratops ultimate style avatar for user lemonpeel50
      You are multiplying or dividing by a negative. For example: the equation 7>1 does not hold when you multiply both sides by negative one, as that would leave you with -7>-1, which is FALSE. Take any two numbers and multiply or divide them by a negative, and see what happens:)

      Hope that helped! :D
      (10 votes)
  • orange juice squid orange style avatar for user Dylan Cappel
    how do you know when the signs switch?
    (1 vote)
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    • male robot donald style avatar for user Jeremy
      Dylan,

      You only need to switch the inequality sign when you multiply or divide both sides of the inequality by a NEGATIVE number. I find this sometimes makes more sense if we take all of the variables away and just look at numbers.

      For example, 3 < 4, right? Well what happens if I multiply both sides by -2, and I DON'T remember to switch the signs? I get -6 < -8… but -6 ISN'T less than -8. -8 is Less than -6. So, since I multiplied by a negative number, if I switch the signs the statement becomes true again: -6 > -8.

      Suppose I divide by a negative number? The same thing happens. Watch:

      Let's say I start with 12 > 9. Clearly that is true, 12 IS greater than 9. But suppose I divide both sides by -3 and forget to switch the signs? Then I get:
      -4 > -3. Well that's not true anymore. -4 ISN'T greater than -3, it's LESS than -3. That's why I have to switch the sign: -4 < -3.

      You'll notice that this only happens when I multiply or divide by a NEGATIVE number. If I multiply or divide by a positive number, I don't need to do anything: 1 < 2 is a true statement. 1 IS less than 2. And if I multiply both sides by POSITIVE 4, I get 4 < 8, which is STILL a true statement.

      I also don't need to switch the signs if I Add or subtract ANY numbers, positive or negative: 6 < 9 is true. And if I subtract 5 from both sides (or add -5) I get: 1 < 4. Still true, and I didn't need to switch the signs.

      It can be harder to see this when there are a bunch of variables in the expression. But if you ever forget, just take the variables out and use numbers, you can check to see when you need to switch the inequality sign all by yourself just like we did above. It shouldn't take you long to prove to yourself that you only switch the signs when you multiply or divide by a negative number.

      Does that help, Dylan?
      (11 votes)
  • male robot hal style avatar for user Harrier
    Can someone help here? When do you flip the sign? I never get these correct!
    (3 votes)
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  • leaf green style avatar for user Alison Cianciolo
    Why do you have to flip the sign when you divide by a negative?
    (2 votes)
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    • leaf red style avatar for user Noble Mushtak
      Multiplying or dividing by a negative is like multiplying -1 and then multiplying or dividing a positive normally. Thus, I'll just explain why multiplying by -1 makes you flip the sign to explain all of it.

      When you multiply by -1, you are flipping the numbers on both sides of the inequality over 0. Remember that -1 is greater than -2 because -1 is less left of 0 on the number line. If you take the negative of each side of the inequality, you have to flip the sign because lesser positive numbers before will become greater than greater positive numbers before. When you flip it over 0, you're switching the numbers places so that one number is left of the other [on the number line] when it was right of it [on the number line] before and vice versa. Let's do some examples to show you this.

      A number with less absolute value will be less when positive, but more when negative.
      1<2-->-1>-2
      Positive numbers are always greater than negative numbers, just as negative numbers are always less than positive numbers
      2>-1-->-2<1
      A number with greater absolute value will be less when negative, but greater when positive.
      -2<-1-->2>1

      I hope this helps!
      (3 votes)
  • piceratops ultimate style avatar for user Neal Menon
    What if I want to write the solution set for this question?
    (2 votes)
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    • leaf green style avatar for user jc
      In the same format as the previous videos, I believe the solution set would be something like this: { x is a real number | x ≤ -2 } which could be read as "x is a real number such that x is less than or equal to negative two."
      (3 votes)
  • leaf green style avatar for user jc
    Can I also express the final answer x≤-2 as (negative infinity, -2] in interval notation?
    (1 vote)
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  • duskpin seedling style avatar for user iwinosa osawe
    can you explain compound inequalities more
    (2 votes)
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  • duskpin seedling style avatar for user iwinosa osawe
    more on multi step inequalities
    (2 votes)
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Video transcript

Solve for x plus 8 is less than or equal to 6 and graph the solution. So we have x plus 8 is less than or equal to 6. So let's solve this inequality for x. And the easiest way to isolate an x on-- let's isolate on the left-hand side since it's already there-- is to just get rid of this 8. And the best way to get rid of this positive 8 is to subtract 8 from both sides So let's subtract 8 from both sides. That won't change the direction of the inequality. So the left-hand side-- x plus 8 minus 8. And you're just left with an x. The right-hand side-- 6 minus 8 is negative 2. And we still have the less than or equal. So we solved the inequality. We have x is less than or equal to negative 2. So let's draw that on a number line. So that's my number line. Let's stick 0 over here. Maybe if we go 1, and then we could go negative 1, negative 2, negative 3. And we could keep going to the left. And we want all of the x's that are less than or equal to negative 2. Since it can be equal to negative 2, we'll put a filled-in line right here at negative 2, and all of the values less than that. If it was just less than, if there wasn't the equal sign, we would have an open dot. But since this is less than or equal to, we've closed this dot. And then we want all of the values below that. And you could just sample a few and verify for yourself that they work. Based on this, negative 3 should work. And if you took negative 3-- negative 3 plus 8 is 5, which is definitely less than 6, so that works. And negative 1 shouldn't work. It's not included in this set over here. So let's try that out. Negative 1 plus 8 is 7, which is definitely not less than 6. So just sampling some points, it seems like we've got the right solution.