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One-step inequalities: -5c ≤ 15

In addition to solving the inequality, we'll graph the solution. Remember to swap if you mutiply both sides of the inequality by a negative number. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user Roy
    Why does the inequality get flipped when you multiply both sides by a negative?
    (65 votes)
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    • blobby green style avatar for user bobwburns
      It is the rule.
      When you mult by - the value goes opposite so the sign that says what is opposite must change

      x < 5......... lets say x is 4
      -x > -5 so that means -4 is greater than -5
      Remember the number line and see that as you go to the right things get bigger.
      That is the point of using the number line.
      (94 votes)
  • starky tree style avatar for user -Forests Of Faith-
    couldn't you just divide both by -5 and get the same answer?
    (26 votes)
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  • leaf grey style avatar for user TR
    How could this be used in everyday life?
    (20 votes)
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  • primosaur ultimate style avatar for user Nancy
    So why exactly must we flip the signs?
    (6 votes)
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    • mr pink green style avatar for user David Severin
      Think of it this way: if I have -3 x< 9, suppose I do not like to flip inequality signs by dividing by a negative Starting with the equation -3x<9
      My other choice is to move my x by opposites
      -3x + 3x < 9 + 3x
      0< 9 +3x I still do not have x isolated, so I subtract 9
      0-9<9-9+3x
      -9 < 3x divide by 3
      -9/3 < 3/3x
      -3 < x if -3 is less than x, then when I flip this equation
      x > -3 x must be greater than -3

      Starting from beginning -3x < 9 divide by -3 and flip inequality sign
      -3/-3x > 9/-3
      x > -3

      It is easier to remember to flip the inequality sign than to go through the whole process of moving my x to make it positive which makes me move everything else also - the difference between one step and multiple steps

      Hope that helps and you understand the inequality flip now
      (16 votes)
  • male robot donald style avatar for user Omri Rabon
    how did sal get 15 times -1/5 = -3 im confused
    (5 votes)
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    • piceratops ultimate style avatar for user Just Keith
      Multiplying by a fraction is the same as dividing by the reciprocal of the fraction. (A reciprocal is the number you get if you switch the numerator and denominator. So, the reciprocal of 2/3 is 3/2 and the reciprocal of 1/5 is 5.)

      So, 15 × (-⅕) = 15 ÷ (-5) = -3
      (15 votes)
  • leaf green style avatar for user branspammck
    can't you just divide instead of using the inverse I've tested this on multiple problems and it seems to work every time? just wondering now instead of figuring out I can't do this down the road once problems become more complicated. Thanks.
    (3 votes)
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  • aqualine seed style avatar for user Adil
    can't we divide both sides by 5
    (2 votes)
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    • hopper jumping style avatar for user Michael Laman
      You can, but you're left with "-c", not "c". You're solving for "c", not "-c".

      Dividing both sides of "-5c ≤ 15" by 5 gives "-c ≤ 3". Perhaps you're uncomfortable with "flipping" the equality.

      If you're still uncomfortable with flipping the side then a way to see why.

      -c ≤ 3 [ The "-c" expression ]
      0 ≤ 3 + c [ Add "c" to both sides ]
      -3 ≤ c [ Subtract "-3" from both sides ]

      Now the "c" is on the greater than side of the expression.

      Flipping the sign is the correct procedure.

      I hope this helps.
      (5 votes)
  • leaf blue style avatar for user batman
    What if you were to add a negative number to both sides? Would you flip the sign then?
    Like in x+(-8)>6
    (3 votes)
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  • piceratops tree style avatar for user Abhimanyu Panda
    in where does the 1/5 comes from
    (3 votes)
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  • spunky sam red style avatar for user Mahek Sureka
    what if you multiply a negative number? does the sign flip
    (2 votes)
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    • duskpin ultimate style avatar for user Dominic Nguyen
      Yes, adding to Allison's answer, because you would technically have to divide by a negative in a case like that. For example in -x/8>10 (I'm going to go step by step -not going straight to multiply each side by -8) Multiply each side by 8 to get -x>80, to isolate x from here, you have to divide by a negative 1, so the sign does switch. Hope this clears up any confusion you have on this
      (3 votes)

Video transcript

Solve for c and graph the solution. We have negative 5c is less than or equal to 15. So negative 5c is less than or equal to 15. I just rewrote it a little bit bigger. So if we want to solve for c, we just want to isolate the c right over here, maybe on the left-hand side. It's right now being multiplied by negative 5. So the best way to just have a c on the left-hand side is we can multiply both sides of this inequality by the inverse of negative 5, or by negative 1/5. So we want to multiply negative 1/5 times negative 5c. And we also want to multiply 15 times negative 1/5. I'm just multiplying both sides of the inequality by the inverse of negative 5, because this will cancel out with the negative 5 and leave me just with c. Now I didn't draw the inequality here, because we have to remember, if we multiply or divide both sides of an inequality by a negative number, you have to flip the inequality. And we are doing that. We are multiplying both sides by negative 1/5, which is the equivalent of dividing both sides by negative 5. So we need to turn this from a less than or equal to a greater than or equal. And now we can proceed solving for c. So negative 1/5 times negative 5 is 1. So the left-hand side is just going to be c is greater than or equal to 15 times negative 1/5. That's the same thing as 15 divided by negative 5. And so that is negative 3. So our solution is c is greater than or equal to negative 3. And let's graph it. So that is my number line. Let's say that is 0, negative 1, negative 2, negative 3. And then I could go above, 1, 2. And so c is greater than or equal to negative 3. So it can be equal to negative 3. So I'll fill that in right over there. Let me do it in a different color. So I'll fill it in right over there. And then it's greater than as well. So it's all of these values I am filling in in green. And you can verify that it works in the original inequality. Pick something that should work. Well, 0 should work. 0 is one of the numbers that we filled in. Negative 5 times 0 is 0, which is less than or equal to 15. It's less than 15. Now let's try a number that's outside of it. And I haven't drawn it here. I could continue with the number line in this direction. We would have a negative 4 here. Negative 4 should not be included. And let's verify that negative 4 doesn't work. Negative 4 times negative 5 is positive 20. And positive 20 is not less than 15, so it's good that we did not include negative 4. So this is our solution. And this is that solution graphed. And I wanted to do that in that other green color. Here you go. That's what it looks like.