The rule for rewriting an absolute value equation without abso...

Question

The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84. $∣ x^{2} - 6 ∣ = ∣ 5 x ∣$

Accepted Answer

Welcome back. I'm so glad you're here. We've got this absolute value equation, we have the absolute value of X squared plus equals the absolute value of nine X. And wow, we've got an absolute value equals an absolute value. How do we solve for that? We take a step back and think for a moment what is absolute value Anyway? An absolute value is the distance of value is from zero. So in this case you could choose either one, but I'm going to choose X squared plus 14. How far is that from zero? Well that is nine X units away from zero but it's not only nine X units in the positive direction, it's nine X units in the negative direction as well. So we can set X squared plus 14 equal to negative nine X and X squared plus 14 equal to positive nine X. And you might be asking yourself well what about the other way around? We've got absolute value on both sides. What about is nine X equal to X squared plus 14 units away from zero in the positive direction and is nine X equal to X squared plus 14 units from zero in the negative direction as well. And yeah, if you were to go through and check those will equal our same equations that we've set up here. Once you do a little bit of simplifying. So I'm just going to go ahead with the two equations, I've set up here in black but you could have done it in the red equations as well. It'll be the same answer. So now looking at our X squared plus 14 equals negative nine X. I am going to bring the nine X over to the left so that we can set up a quadratic equation and hopefully factor so we've got X squared plus nine X plus 14 equals zero. And then we'll set up our parentheses and what times what gives us X squared. That's X times X. And what times what Multiplies to Equal 14. That adds up to equal a positive nine. That's going to be a plus two and a plus seven. And if you need to you can pause the video and foil that to double check. But continuing on here we're going to set each of those equal to zero each of those two factors. So we have X plus two equals zero. We'll subtract two from both sides and one of our solutions is X equals negative two. And now for the other one, X plus seven equals zero. Will subtract seven from both sides. And that will give us a solution of X equals negative seven. While there's two of them. Now let's take a look at our X squared plus 14 equals nine X solution our equation. Excuse me. So we'll subtract nine X from both sides. That will give us X squared minus nine X plus 14 equals zero. Again we can factor this. We ask ourselves what times what equals X. That's X times X. What times what equals 14 a positive 14 but adds up to a negative nine. That's going to be a minus two and a minus seven. And again if you want to check that with foil you can pause and go ahead and do that. But now I'll continue on and set each of those factors equal to zero, X minus two equals zero and x minus seven equals zero and I will add two to both sides for x minus two equals zero and we get one solution of x equals two. And for our x minus seven we'll add seven to both sides. So we'll now have X equals seven and we've got four answers here and we always want to pause and ask ourselves does this make sense? Does it make sense that I've got four solutions? And in this case yes it does. Because we have absolute, that's going to give us two solutions. And because the degree of our equation was too because it was x squared. Well that means we're going to have to x solutions for that. And then because it's absolute value have to double however many solutions the degree indicates you should have so yes, four solutions does make sense for this equation. We look at our answer choices and our solution matches answer choice B Well done. We'll catch on the next one

Key Concepts

Absolute Value Definition and Properties

Solving Equations Involving Absolute Values

Quadratic Equations and Factoring

Watch next