How to Solve an Absolute Value Equation

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  • 0:10 Review of Absolute Values
  • 0:51 Solving Basic…
  • 2:21 Splitting an Equation into Two
  • 5:02 Lesson Summary
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Lesson Transcript
Instructor: Zach Pino
Once you get familiar with any new operation, the next step in any algebra class is to learn how to solve equations with that operation in them. Absolute values are no different. Solve absolute value equations here!

Review of Absolute Values

Absolute value involves the magnitude of the specified numbers
Absolute Value Example

Alright, so you're a pro at taking absolute values. |10| = ? 10! |-10| = ? Still 10! Negative numbers turn positive again because all an absolute value cares about is how 'large' the number is or what the magnitude is, not what sign it is. You even know that absolute values can be thought about as how far away that number is away from zero. But all that has just been to get you ready for this: solving absolute value equations.

Now this is a really common theme in algebra classes; simply learning a new skill isn't usually good enough and you're almost always going to end up having to learn how to apply that skill in order to solve an equation.

Solving Basic Equations with Absolute Values

So what are we talking about here? How about solve the |x| = 15. Well, any time we are asked to solve, it's our job to find a value for the variable that makes the equation true. So what number can I substitute in for x that will make the absolute value of it 15? Or you could think about this as what number is 15 units away from zero. 15 seems like the obvious choice, but we can't forget that -15 is also 15 units away from zero, which means that we actually get two answers here. x could be 15 or -15. This makes sense, right? If I just plug 15 into the absolute value, I just get 15 back out. Or if I plug -15 into the absolute value, I, again, just get 15 back out.

Absolute value equations result in two answers
Absolute Value Equations Two Answers

This brings up a really important point: absolute value equations give us two answers. |x| = 20? x is 20 or -20. |x| = 1,000? x is 1,000 or -1,000. |x| = -5,000? x equals 5,000 or negative - wait! Absolute values can't be negative numbers, so no matter what I plug in for x, there's no way I'm going to get -5,000. So anytime an absolute value is set equal to a negative number, there is no solution.

Splitting an Equation into Two

Let's take the difficulty up a bit now. What if it asks you to solve |x-5| = 20? Now it can be really tempting to simply undo the subtraction with addition because they're inverse operations. So we add 5 to both sides and we get |x| = 25 and then we say, 'hey, two answers! 25 and -25 and we're done.' But you can't forget that absolute value bars kind of act like parentheses, which means you have to do everything on the inside before you take the absolute value. So when you first add 5 to both sides like that, you're kind of breaking that rule, which means doing that is a big no-no. You can't do that.

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