Question Video: Solving Absolute Value Equations

What is the solution set of the equation (13𝑥/8) − 1 = (1/4)|9𝑥 − 6|?

05:19

Video Transcript

What is the solution set of the equation 13𝑥 over eight minus one equals a quarter multiplied by the absolute value of nine 𝑥 minus six?

So in this problem, what we have are in fact two different situations to consider. And the reason that we have two different scenarios to consider is because the equation involves the absolute value of nine 𝑥 minus six. And that’s because the absolute value means that what we want to consider are the positive values of nine 𝑥 minus six.

Well, first of all, what we can do is consider that nine 𝑥 minus six is greater than or equal to zero. Well, as this is already greater than zero, we know that it’s going to be positive. So we don’t have to do anything more to it. However, if nine 𝑥 minus six is less than zero, then what we’re going to have to do is take the negative of that value to give us the positive that we need. So now that’s dealing with our absolute value function.

So next, what we’re gonna do is look to solve those inequalities. Well, if we did that, on the left-hand side, what we’d have is 𝑥 is greater than or equal to six-ninths. And that’s because what we did is added on six and then divided by nine. Well, in fact, this would simplify because we could divide the numerator and denominator both by three to give us 𝑥 is greater than or equal to two-thirds. And then on the right-hand side, once again, we’d have 𝑥 is less than this time than six-ninths. Again, dividing through by three gives us 𝑥 is less than two-thirds.

So what this tells us is that if the value of 𝑥 is greater than or equal to two-thirds, we go down the left-hand branch. However, if it was less than two-thirds, we go down the right-hand branch. We’re going down the left-hand branch. We know that nine 𝑥 minus six is greater than or equal to zero. So therefore, we don’t have to change anything.

So now we can rewrite our equation without the absolute value sign. So what we have is 13𝑥 over eight minus one equals a quarter multiplied by nine 𝑥 minus six. And then what we can do to remove the fraction in front of our parentheses is multiply through by four. And when we do that, we’re gonna get 13𝑥 over two minus four equals nine 𝑥 minus six. Then, adding four to each side of the equation gives us 13𝑥 over two equals nine 𝑥 minus two. And then multiplying through by two gives us 13𝑥 is equal to 18𝑥 minus four. And then what we’re gonna do is subtract 13𝑥 and add four to each side of the equation. And this gives us four is equal to five 𝑥. So therefore, we can say that 𝑥 is gonna be equal to four-fifths.

Okay, great, so what we’ve done now is been down the left-hand branch ’cause we’ve looked at the scenario where nine 𝑥 minus six is greater than or equal to zero. Well, now what we’re going to do is go down the right-hand branch, where nine 𝑥 minus six is less than zero. Well, because it’s less than zero, we therefore need the negative of it to give us the positive we require because of our absolute value. So therefore, what we’re gonna have is 13𝑥 over eight minus one equals a quarter multiplied by negative nine 𝑥 minus six. And to tidy up, we can rewrite the right-hand side as negative a quarter multiplied by nine 𝑥 minus six.

This time, what we can do is multiply through by eight just to show a different method that we could’ve used even for the left-hand side. And when we multiply each term through by eight, we’re gonna get 13𝑥 minus eight equals negative two multiplied by nine 𝑥 minus six, remembering that if we’re multiplying through by eight, it needs to be every term. A common mistake here would be to forget to multiply the negative one on the left-hand side. Now, distributing across the parentheses on the right-hand side gives us 13𝑥 minus eight equals negative 18𝑥 plus 12. Well, now what we can do is add 18𝑥 and add eight to each side of the equation. And doing so gives us 31𝑥 equals 20. And then dividing through by 31 gives us 20 over 31.

So therefore, we can say that the solution set of the equation 13𝑥 over eight minus one equals a quarter multiplied by the absolute value of nine 𝑥 minus six is 20 over 31 and four over five.

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