Find algebraically the solution set of the equation the absolute value or modulus of 𝑥 plus four equals 𝑥 plus four.
So, in this problem, in fact, what we have are two different situations to consider. So the reason we’re looking at two different situations to consider is because we’ve got the absolute value or modulus of 𝑥 plus four. So actually, what we’re looking at are the positive values of 𝑥 plus four. So first of all, let’s consider 𝑥 plus four is greater than or equal to zero. We can see that this is already going to be positive, so we’re gonna have to do nothing to this. However, if 𝑥 plus four is less than zero, then we’d have to take the negative of that value to make it a positive. And that’s what the absolute value function itself is actually doing.
Now, if we solve for both of our inequalities, on the left-hand side, we’d have 𝑥 is greater than or equal to negative four. And we get that by subtracting four from each side. And on the right-hand side, we’d have 𝑥 is less than negative four. So we can say that if the value of 𝑥 is less than negative four, we come down the right-hand branch. However, if it’s greater than negative four we come down the left-hand branch. So if we’re coming down the left-hand branch, we know that the value of 𝑥 plus four is going to be positive. So we actually don’t need to worry, and what we can write is 𝑥 plus four equals 𝑥 plus four. So what we’re actually saying is that 𝑥 plus four on the left-hand side is equal to 𝑥 plus four on the right-hand side.
So we could just think at this point, ah well, it’s true for all values of 𝑥 obviously. However, what we do need to remember is the range for the values of 𝑥 that we identified. And that is that 𝑥 has got to be greater than or equal to negative four. So therefore, we know to satisfy this equation, we can say that 𝑥 is gonna be all values of 𝑥 that are greater than or equal to negative four. So moving down the right-hand channel, we know that 𝑥 is less than negative four and we also know that 𝑥 plus four is giving us a negative value.
So therefore, what we’re gonna have to do is take a negative of that negative to turn it into a positive. So therefore, what we’re gonna have to work with is negative 𝑥 plus four is equal to 𝑥 plus four. So we’re gonna get negative 𝑥 minus four equals 𝑥 plus four. So then if we add 𝑥 to both sides of the equation, we’re gonna get negative four equals two 𝑥 plus four. And then if we subtract four, we get negative eight equals two 𝑥. And then if I divide both sides by two, I get the answer 𝑥 is equal to negative four.
Well, we might think that 𝑥 equals negative four wouldn’t strictly be speaking correct because if we look it down the right-hand side, we actually said that 𝑥 has got to be less than negative four. However, actually, if we had the negative of zero, it would still be positive, so it’d still be zero. And we actually cover this solution in the 𝑥 is greater than or equal to negative four anyway.
So therefore, we could say that the solution is 𝑥 is greater than or equal to negative four. Or we can actually write this also in interval notation. And if we did that, we’d have left closed interval negative four, comma ∞, right open interval. And what this tells us is that the negative four can be part of our solution set, whereas the ∞ will not be.