Find all possible values of 𝑥 given that the absolute value of 𝑥 minus five-sixths is equal to five-sixths.
So in this problem, what we actually have are in fact two different situations to consider. And the reason that we actually have two different situations to consider is because we’re looking at the absolute value of 𝑥 minus five-sixths. So what we’re actually looking for are the positive values of 𝑥 minus five-sixths.
Well, first of all, let’s consider 𝑥 minus five-sixths is greater than or equal to zero. Well, as this is already greater than or equal to zero, and so it’s going to be positive, it means that we don’t actually have to do anything to it. However, if 𝑥 minus five-sixths is less than zero, then what we’re going to do is we’re gonna have to take the negative of that value to make it positive. And that’s what the absolute value function itself is actually doing.
Well now if we look to solve those inequalities, on the left-hand side, we’d have 𝑥 is greater than or equal to five-sixths. And we’d get that by adding five-sixths to each side of the inequality. And on the right-hand side, what we get is 𝑥 is less than five-sixths. And we get that by again adding five-sixths to each side of the inequality. So therefore, what we can say is that if the value of 𝑥 is less than five-sixths, we come down the right-hand branch. However, if it’s greater than or equal to five-sixths, we’d come down the left-hand branch.
Well, coming down the left-hand branch, we know that 𝑥 minus five-sixths is going to be greater than zero. So therefore, we don’t have to worry. So what we can write is 𝑥 minus five-sixths is equal to five-sixths. So now what we can do is solve to find 𝑥. And we can do that by adding five-sixths to both side of the equation. So then we get 𝑥 equal to ten-sixths, which simplifies to 𝑥 is equal to five-thirds.
Okay, great. That’s the left-hand side dealt with. Now let’s go down the right-hand side. Well, moving down the right-hand channel, we know that 𝑥 is less than five-sixths and 𝑥 minus five-sixths is less than zero. So therefore, to turn this into a positive, what we need to do is take the negative of this. So therefore, what we get is negative 𝑥 minus five-sixths is equal to five-sixths. So therefore, multiplying the parentheses through by negative one, we get negative 𝑥 plus five-sixths equals five-sixths. So then, if we subtract five-sixths from both sides of the equation, we get negative 𝑥 equals zero. Then, dividing through by negative one, we get 𝑥 is equal to zero.
So therefore, we can say that the possible values of 𝑥 given that the absolute value of 𝑥 minus five-sixths equals five-sixths are five-thirds or zero.