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## Algebra (all content)

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

Lesson 2: Solving absolute value equations- Intro to absolute value equations and graphs
- Worked example: absolute value equation with two solutions
- Worked example: absolute value equations with one solution
- Worked example: absolute value equations with no solution
- Solve absolute value equations

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# 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?(38 votes)
- Noah,

Very observant of you.

The orginal problem did say "and graph the solution on a number line".

Here is the graph just for you. http://www.khanacademy.org/cs/graph-of-3x-90/1523111424(42 votes)

- What effect does the absolute value bring into an equation?(14 votes)
- 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.(10 votes)

- 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.(8 votes)
- Kyle, the difference in this video is that we are checking the solution. We did not do this in the last video.(3 votes)

- 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?(4 votes)
- 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.(3 votes)

- 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 :)(4 votes)
- 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 :)(2 votes)
- 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.(4 votes)

- WHAT to do is the equation is set to y=l3x-9l(2 votes)
- 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.

https://www.khanacademy.org/cs/y3x-9/5442811621015552

I hope that helps.(3 votes)

- Isn't this the same video as the last one?(2 votes)
- It's not exactly the same video, but it uses the same equation to explain the same concept.(2 votes)

- what does it mean to check for extraneous solutions?(2 votes)
- Hi Benjamin,

An extraneous solution is a solution that, when plugged back in to the original equation, does not work. Extraneous solutions can occur when dealing with absolute values or quadratics. In certain situations, even though you may get two solutions, when the solutions are plugged back in to the original equation, one of them may not work. The solution that does not work would then be considered extraneous.

Hope that helps :-)(2 votes)

- didn't you do this in you previous video? like the one before it?(2 votes)

## Video transcript

We're told, solve the absolute
value of 3x minus 9 is equal to 0, and graph the solution
on a number line. So let's just rewrite the
absolute value equation. They told us that the absolute
value of 3x minus 9 is equal to 0. So we're told that the
absolute value of the something-- in this case the
something is 3x minus 9-- is equal to 0. If I told you that the absolute
value of something is equal to 0, I'm telling you
that something has to be exactly 0 away from 0, or 0 away
from the origin on the number line. So the only thing that that
something could be is 0. If I told you that the absolute
value of x is equal to 0, you know that x has
to be equal to 0. That's the only value whose
absolute value is 0. So if I told you that the
absolute value of 3x minus 9 is 0, than we know that 3x minus
9 has to be equal to 0, and that's kind of unique about
the 0 is that, it's the only number the has a unique,
that's only the absolute value of 0. If you had, say, a 1 here, you
could say, oh well, then this thing could be a 1
or a negative 1. But here, if you have a 0,
this thing can only be 0. So solving this equation is
fairly straightforward. If we want to isolate the 3x,
get rid of the negative 9 on the left-hand side, we add 9 to
both sides of the equation. Add 9 to both sides
of the equation, these 9's cancel out. That's the whole point. On the left-hand side, you're
just left with 3x, and on the right-hand side, you are
just left with 9. Now we want to solve for x,
so we have 3 times x. Let's divide it by 3, because 3
times x divided by 3 is just going to be x. But if we divide the left side
by 3, we have to divide the right side by 3. So we are left with-- these
guys cancel out. x is equal to 9 over
3, which is 3. And that's our solution. Now let's try out. Let's make sure that this
actually works. Let's substitute it back into
our original equation. So we have the absolute
value of 3 times x. Instead of x, I'll just put in
our actual answer that we got, 3 times 3 minus 9 has got
to be equal to 0. So what's this going
to be equal to? 3 times 3 is 9. So it's the absolute value of
9 minus 9, which is the absolute value of 0, which
is, indeed, 0. So it does, indeed, equal
0, and we are done.