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## Algebra basics

### Course: Algebra basics>Unit 3

Lesson 5: Two-step inequalities

# Two-step inequalities

Two-step inequalities are slightly more complicated than one-step inequalities (duh!). This is a worked example of solving ⅔>-4y-8⅓. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• why is it so hard to learn?
• because it helps our brains grow and keeps us on top of becoming smarter.
• How would you then answer a question like this type?
4x − 3/2− 3x − 3/3< 3
I cant find any references to this type of equation, since i don't think it is a multi-step inequality
• a calculator would probably help
• I was trying to solve a multi-step inequality but I don't know how to. The inequality is:
-1/4(8n-12) ≥ −2n−1 When I try to solve it the n variable no longer exists and I don't understand how I would continue to solve. When I looked online they said the answer would be all real numbers are solutions but I still do not understand. Please help.
• 1) Distribute the -1/4 on the left side
-1/4(8n-12) ≥ −2n−1
-2n+3 ≥ −2n−1
2) Add 2n to both sides
3 ≥ −1
So, as you have found already, the variable disappears. The inequality that remains is True. 3 is greater than -1. This is called an identify. It tells us that any value of "n" that you use in the inequality will be a solution. So, the solution is all real numbers. You can see this for yourself by picking any value you like for "n" and checking to see if it is a solution. All values will work.

Here are a couple of other scenarios that you might run into:
If your inequality had become 3 ≥ 3, this is also true (an identity). So, again the solution is all real numbers.

If your inequality becomes something like 3 < 1, this is False ( 3 is never less than 1). This is called a contradiction. It means that the inequality has no solution.
• does it matter which side the variable terms are
• If you mean "Does it matter-left or right side of the equals sign?" then no.
However, it is important to get the variable on one side, either the left or the right.
• Why do you have to swap the inequality so that it is less than?
• *When you multiply both sides by a negative value you make the side that is greater have a "bigger" negative number, which actually means it is now less than the other side!* This is why you must flip the sign whenever you multiply by a negative number.
• This is harder than rocket science!
• Actually, if you can solve a 2-step equation, you can solve 2-step inequalities. The same techniques are used. The only thing that is different is that if you multiply/divide the inequality by a negative value, then you reverse the inequality.

And, by the way, multi-step inequalities and many more advanced math topics are used in rocket science. So, a 2-step inequality is not harder than rocket science.
• Where did you get the 25/3 from?
• Sal converted the mixed number 8 1/3 to an improper fraction which is equal to 25/3. Hope this helped!
• At Sal divides -12y by -12 but when I divided by a negative number in the problem: 16x -7 < or = -71 (I divided - -64 by -16) the answer was wrong. I don't understand why it was wrong, please explain why.
-- Did you reverse the inequality since you divided by a negative?
-- How did you get 2 "-" in front of the 64? -71+7 = -64
-- Did you finish the math? -64/(-16) = 4

So the final result should be: x >= 4

Hope this helps.
Note: If this if in a practice problem, use the Hints to figure out what you did incorrect. This allows you to learn from your mistake(s).
• At , Sal simplifies 2/3 by multiplying it by three(because he's doing that to the other side of the inequality sign). He then crosses out both threes and leaves the two. Why? How does that work? It's probably really obvious, but can someone answer this?
• Sure!, A fraction is just a number divided by another number, therefore to undo the division you multiply. Sal crossed them out because they cancel out each other.
For example,
if you have 6/3 that will equal 2. then multiply by three to get 6. But you should just multiply by the denominator and cancel out everything at the beginning to get 6!
Hope this helps!