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# Multi-step inequalities

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

## Want to join the conversation?

• 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....
• 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
• Why does Sal write a negative infinity sign? I don't get what it means.
• Number lines continue forever in 2 directions. We use positive infinity for the rigth side and -infinity for the left side. There is no larger numbers and there is no smallest number. The line extends forever.

Hope that helps,
• Is there a clever way to remember to change the direction of the sign when dividing or multiplying by a negative number?
• 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?

• 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)?
• 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!
• To whom it may concern,
I hope you and your family is safe especially during this tough pandemic!
• we def went through a hard time these past years
(1 vote)
• when would you need to know inequalities?
• 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. :)
• How would you solve an inequality that contains exponents?
Thanks very much!
• doesn't the negative and a negative equal to a positive number?
(1 vote)
• 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!
• I'd love to see a proof showing why we flip the inequality sign when dividing or multiplying by a negative number. I see it in the back of my mind but need help bringing the logic to the surface.
• Look at an example with just numbers.
Consider: -2 < 5
This is currently a true statement.
Now, multiply both sides by -1.
-- If you don't reverse the inequality, you have: 2 < -5, which is a false statement.
-- If you do reverse the inequality, you have 2 > -5, which is a true statement.

Multiplying/dividing by a negative reverses the relationship in the numbers which is why we reverse the inequality.
Hope this helps.
• What exactly is an inequality?