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## College Algebra

### Unit 1: Lesson 3

Multi-step linear inequalities

# 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

• 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!
• 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,
• To whom it may concern,
I hope you and your family is safe especially during this tough pandemic!
• 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. :)
• doesn't the negative and a negative equal to a positive number?
• 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!
• 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 solve an inequality that contains exponents?
Thanks very much!
• What exactly is an inequality?
• It's like and equation, but with the inequality symbols, which are < and >.

An equation uses an = (equal sign).

For example:
3x + 8 = 2x - 4 is an equation.
3x + 8 > 2x - 4 and 3x + 8 < 2x - 4 are inequalities.
• 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.
• 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!