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## Operations and Algebraic Thinking 229+

### Unit 1: Lesson 5

Multi-step inequalities

# Inequalities with variables on both sides

Sal solves the inequality -3p-7<p+9, draws the solution on a number line and checks a few values to verify the solution. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• i am in fifth grade being curious and starting to learn some algebra 1
• I am just like you
• I am working on a question: -9x-4<-4x+5
the answer i get is x>-1.8 yet my answer key says the answer is x>565 or 11.2
WHAT!? so confused. can someone explain this to me please!
(1 vote)
• Sorry for being late! No, the answer key is not an error and you can solve this equation. All you gotta do is:
Step 1: Add 4x to both sides.
−9x−4+4x<−4x+5+4x
−5x−4<5
Step 2: Add 4 to both sides.
−5x−4+4<5+4
−5x<9
Step 3: Divide both sides by -5.
-5x/-5 < 9/-5
x > -9/5

x > -9/5

Hope that cleared some things up! :)
• At Sal says that when you are dividing a negative number you must switch the inequality. Why is that? What would happen if I didn't? I was just wondering the math behind it and why we have to do that :)
• Dividing or multiplying by a negative reverses the relationship between the numbers which is why you need to reverse the inequality.
For example: 2 < 5
Divide by -1 and you get: -2 < -5 which is false, so reverse the inequality: -2 > -5

Hope this helps.
(1 vote)
• i dont know what i'm doing T^T
• What do you not get?
• Can we reason our way through as to why do we change the inequality sign when we multiply and divide by the negative side?
• Multiplying / dividing by a negative reverses the relation between the sides of the inequality. You can see this is you use an inequality containing only numbers.

For example: Start with 4 < 12
-- Multiply by -2 and you get: -8 < -24 which makes no sense as -8 is now larger than -24.
-- Reverse the inequality: -8 > -24. Now the relationship is correctly stated.

Hope this helps.
• would this have infinite solutions, then?
• Yes there are an infinite number of points less than any given number. In fact, there are an infinite number of points between any two numbers, even as simple as 1 and 2, or .1 and .2, or .001 and .002.
• What if my problem is
1+3x< 2x-x+13
• Simplify the right side by combining like terms: 1+3x<x+13
Your problem is now like the one in the vide0.
-- Move the "x" to the left side using opposite operations
-- Move the 1 to the right side by using opposite operations.

Give it a try. Comment back if you have questions.
• what do you do if there is no variable because I'm confused about that
(1 vote)
• An inequality without a variable is true, or false. The variable is only there to have its value found
• A Khan Academy question on this subject is -42v + 33 > 8v + 91. I noticed that rather than dividing, Khan Academy says to multiply by -1 at the last step. Why is this?
• There are often more than one set of steps that will get you to the correct answer when solving equations and inequalities. Multiplying by -1 and dividing by -1 both create the same result. So, pick the one you want to use.

Personally, I see no need to divide / multiply by -1. The coefficient of "v" is not -1. Just divide by the entire coefficient including the negative:
-- Subtract 8v: -50v + 33 > 91
-- Subtract 33: -50v > 58
-- Divide by -50 and reverse the inequality because of the division by a negative value: v < -58/50
-- Simplify to get: v < -1 4/25

Hope this helps.
• Solve for q.
Reduce any fractions to lowest terms. Don't round your answer, and don't use mixed fractions.

50q+43 > -11q + 70