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### Course: Algebra (all content)>Unit 8

Lesson 2: Solving absolute value equations

# Worked example: absolute value equations with one solution

To solve absolute value problems with one solution, identify expression in absolute value bars. Recall absolute value is zero only if expression is zero. Make expression equal right-hand side. Use algebra to find x values that satisfy equation. Check solutions by plugging in. Graph solutions on number line. Mark points. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• why isn't there a number line?
• What effect does the absolute value bring into an equation?
• it means that if the the equation equals an integer greater or less than 0 it will have 2 answers, which correlate to the graph later on in algebra. as you can see with this video, when an absolute value equals 0, it is just 0. a special exception.
• Why is this video on the website twice in a row? The other video is just called absolute value equations and also shows the number line in the solution as well.
• Kyle, the difference in this video is that we are checking the solution. We did not do this in the last video.
• My algebra teacher is working on these type of equations with us in class, but she taught us to switch the signs in the middle. In this case we would of switched the negative to a positive. How come you do it different?
• Alyssa,

I think the reason that the method is a little bit different here than what you are used to is because in this particular problem there was a 0 on the right hand side of the equation.

The equation Sal was solving was |3x-9|=0, right? Well, normally, you solve equations with an absolute value in them by considering two cases. In the first case, you assume that the stuff inside the parentheses is already a positive number, so you can just get rid of the absolute value signs and solve the problem as though they aren't even there. Then, once you've finished doing the problem that way, you have to come back and solve the second case. In the second case, you assume that the stuff inside the absolute value signs was negative, and taking the absolute value of it made it positive. Therefore, the absolute value symbols changed the sign of the stuff inside of them, which is the same thing as changing their signs. So you get rid of the absolute value signs and change the sign of everything inside the absolute value, and then you solve the whole problem a second time. When you do this method of accounting for case 1 and then case 2, you usually find that you get two different answers.

But for THIS problem, we didn't need to do any of that, because the problem was equal to 0, which means that the stuff inside the absolute value sign also had to be equal to 0. We didn't need to think about the two different possibilities of the stuff inside the absolute value signs, because we knew that it had to be equal to 0.
• Okay, so this is a math question relating to a homework question! So I understand everything in the video, but how would you solve a absolute value equation that looked like: I X+5 I= 3x - 7? Would you move the X over to the 3X and get I 4X I = -12? and then solve? Any help is greatly appreciated :)
• Can you explain to me why, for example |3x-9|=0, that the -9 does not change to positive even though it has the absolute value bars? thanks :)
• You can't take the absolute value of the individual parts.
We follow PEMDAS rules. The absolute value is a special case of parentheses. We do the work inside the parentheses 1st. The absolute value gets applied to the result, not the parts.
Consider this numeric example: |3(5)-9|
1) Using PEMDAS: Mulitply, substract, then do absolute value once there is one number.
|3(5)-9| = |15 -9| = |6| = 6

2) If you do absolute value 1st (make everything positie), do you still get 6?
|3(5)-9| becomes 3(5)+9 = 15 + 9 = 24.

We now have 2 different answers for the same problems. Math creates consistent answers if you follow the rules. The fact that these are not the same tells use one of them is the wrong approach.
The 1st version follows the correct set of step.

Hopefully, this makes it easier to see why we can't just go in and change all the signs to positive. We would get the wrong answers.
• WHAT to do is the equation is set to y=l3x-9l
• Zmomin,
You have an equation with two variables. It is already solved for y. It has many solutions. You can graph the solution set. The graph would look like this.

I hope that helps.
• Isn't this the same video as the last one?
• It's not exactly the same video, but it uses the same equation to explain the same concept.