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. $∣ 2 x^{2} - 4 ∣ = ∣ 2 x^{2} ∣$

Accepted Answer

Welcome back. I'm so glad you're here we are. Given this absolute value equation, the absolute value of eight X squared minus equals the absolute value of eight X squared. Well this is a really unusual equation. It's got absolute value on both sides and it's got this eight X squared on both sides. But on the left the eight X squared is minus 64. And if we just had eight X squared minus 64 equals eight X squared. Well that would be nonsense. That would be no solution. But those absolute value bars give us an opportunity for a solution. So let's go ahead and see if we can find one. Let's recall for a moment what the absolute value is. The absolute value is. The distance of value is from zero. So we could take either side in this example and set it and ask this question. We ask what is the distance of? Eight X squared minus 64 from zero. And the distance is eight X squared units. But it's not eight X squared units only in the positive direction, it's also eight X squared units in the negative direction as well. So eight X squared minus 64 is equal to a negative eight X squared and eight X squared minus is equal to a positive eight X squared. And you could have absolutely set it up where you took eight X squared and said that was eight X squared minus 64 units in the positive direction from zero and eight x squared was negative eight X squared minus or eight X squared minus 64 units in the negative direction. You could absolutely be solving that one, those two equations as well. It'll give you the same answer as what we're doing. So I'm just going to go ahead with what we've set up and now to solve for zero, let's get all our eight X squared on the same side of the equation. So I'll subtract eight X squared from both sides, giving us negative 64 equals a negative 16 X squared. We'll divide both sides by negative 16. And so on the left hand side negative 64 divided by negative 16 is four. On the right hand side we have X squared or x squared equals four will take the square root of both sides and the square root of four is plus or minus two. So X equals plus or minus two are two solutions for X. Now let's do the other equation Again. We want to get those eight x squared on the same side of the equal sign. So that will give us a negative 64 equals well, eight X squared minus eight X squared is zero. And this is exactly where if there were no absolute values yet our solution or that equation would not have made any sense. And this is the example of where it makes no sense. So we disregard that part and now we're just going to look at the X equals two and the X equals negative two. And I want to go back and test those out to show you how this does in fact work. So we go back to our original equation. The absolute value of X squared minus equals the absolute value of X squared. And let's start off with a positive two. So we've got the absolute value of eight times two squared minus 64 equals the absolute value of eight times two squared. And that's the absolute value of eight times two squared is four minus 64 equals the absolute value of eight times two squared is four. So eight times four is 32. So we've got the absolute value of 32 minus 64 equals the absolute value of eight times four is 32. And see how this works out. We've got on the left, the absolute value of 32 minus 64. Well that's the absolute value of negative 32 Equals the absolute value of 32. And yeah, the absolute value of -32 is 32 and that does equal the absolute value of 32 which is also 32. So too does work. Now let's show you how negative two works. So we take our original equation the absolute value of eight times negative two squared and for that x minus 64 equals the absolute value of eight times now negative two squared. So the absolute value of eight times negative two squared is four minus 64 equals eight. Or the absolute value of eight times negative two squared is still four and eight times four. Again that's 32 minus 64 equals the absolute value of eight times for again, 32. So again we've got 32 minus 64 so again we've got the absolute value of negative 32 equals the absolute value of 32. And again that's 32 equals 32. So both two and negative two. Work. First solutions of X. We look at our answer choices and that matches with answer choice C. Well done, we'll catch on the next one.

Key Concepts

Absolute Value Definition and Properties

Equations Involving Two Absolute Values

Solving Quadratic Equations

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