Find algebraically the solution set of the equation seven 𝑥 squared minus four multiplied by the absolute value of 𝑥 equals zero.
When we look at this problem, what we have are two different situations to consider. And the reason that this is the case is because we are looking at the absolute value of 𝑥 as part of our equation, which means that what we’re actually looking at are the positive values of 𝑥. Well, first of all, what we can consider is 𝑥 is greater than or equal to zero. Well, in this case, 𝑥 is already going to be positive, so we won’t need to do anything to it. However, if 𝑥 is less than zero, then what we’re going to do is have to take the negative of that value to make it positive. And that’s what the absolute value function is doing in this question.
Okay, so now, what we need to do is solve considering each of these conditions. So we’re gonna begin by going down the left branch. And as we’ve already said, going down the left branch, we’re looking at values where 𝑥 is greater than or equal to zero. So we don’t need to do anything to the 𝑥. So here, the equation we have to solve is seven 𝑥 squared minus four 𝑥 equals zero. So what we can do here is factor. And we can do that because we can take out 𝑥 as a factor because 𝑥 is in each of our terms. So when we do that, we’re gonna get 𝑥 multiplied by and then, in the parentheses, we have seven 𝑥 minus four. Then this is equal to zero. So therefore, we can say the solutions to this equation are 𝑥 is equal to zero or four-sevenths.
And we get these results because, first of all, if 𝑥 was equal to zero, then we’d have zero multiplied by then zero minus four. Well, zero multiplied by anything is gonna give us zero. And then, to find our other solution, we need to see what would make the expression inside our parentheses equal to zero. So we can do this by setting seven 𝑥 minus five equal to zero. And then adding four to each side of the equation, we get seven 𝑥 is equal to four. And then dividing by seven gives us 𝑥 is equal to four-sevenths, which is the solution that we’re looking for.
Okay, great. So that’s our left-hand branch dealt with. So now, going down the right-hand branch, we have a different scenario because we’re looking at when 𝑥 is less than zero. Well, if we consider when 𝑥 is less than zero, then what we need to do is, in fact, find the negative of this if we’re looking to get the positive that our absolute value is looking for. So therefore, we’re gonna set up the equation seven 𝑥 squared minus four multiplied by negative 𝑥 equals zero. And this, in turn, is going to give us seven 𝑥 squared plus four 𝑥 equals zero. And that’s because if we have negative four multiplied by negative 𝑥, we’re going to get positive four 𝑥.
And once again, we can factor by taking 𝑥 out as a factor to give us 𝑥 multiplied by, and then we’ve got seven 𝑥 plus four in the parentheses. And this is equal to zero, which is gonna give us the solutions 𝑥 equals zero or negative four-sevenths. And this time, it’s negative and that’s because if we look inside our parentheses, we’ve got positive four, whereas we had negative four on the left-hand side. So therefore, if we had negative four-sevenths, seven multiplied by negative four-sevenths plus four would give us the zero that we’re looking for inside our parentheses. So therefore, we can say that the solution set of the equation seven 𝑥 squared minus four multiplied by the absolute value of 𝑥 equals zero is negative four-sevenths, zero, four-sevenths.