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### Course: Algebra 1>Unit 2

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?

• 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 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! :)
• 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.
• Erm what the flip ;-;
• I felt that.
(1 vote)
• i dont know what i'm doing T^T
• What do you not get?
• i am sooo lost
• By this point in math, you would have learned how to solve equations where the variable is on both sides. Use the same techniques here to solve the inequality.
-- Get the variable on only one side by using opposite operations to add/subtract one of the terms containing the variable.
-- Next, If there is a constant on the same side as the variable, again, move it using opposite operations.
-- Last, divide by the coefficient in front of the variable to finish isolating the variable.

The only things that are different are:
1) If you multiply / divide both sides of the equation by a negative value, you need to reverse the inequality.
2) Equations create 1 solution. With inequalities, you will have a large number of solutions. For example: x>1 has a solution set of all real numbers larger than 1.

Hope this helps.
• Why do we have to find P?
• 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.
• why is -3p - p =-4 because we do not know what p is should it not be -3p - p = -3