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## Algebra (all content)

### Course: Algebra (all content)>Unit 2

Lesson 16: 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 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!
• 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.
• i am in fifth grade being curious and starting to learn some algebra 1
• I am just like you
• why? is it necessary to swap the inequality sign? Is there any way to prove why should we do so and the reasons behind it?
• If you multiply or divide by a negative value, the relationship between the numbers changes. You can confirm this by looking at a simple numeric example:
-2 < 3
Multiply both sides by -5 and you get:
10 > -15
If you didn't reverse the inequality, you would have 10<-15 which is a false statement.

Hope this helps.
• i dont know what i'm doing T^T
• What do you not get?
• why is -3p - p =-4 because we do not know what p is should it not be -3p - p = -3
• -3p-p does *not equal -4, nor -3. It equals -4p. You can't drop the variable.

Remember, -p means -1p
To combine -3p-1p, we add the coefficients.
-3p-1p = (-3-1)p = -4p

Hope this helps.
• 22q+73>52q+63
it says don't round and don't use mix numbers
• The result is a simple proper fraction, so no rounding and no mixed numbers are needed.
Try solving it and see what you 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.

-- 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.