What is the solution set of the equation three modulus 𝑥 minus 66 is equal to zero?
The first thing we could have drawn our attention to is these two vertical lines because what these actually mean is the modulus or absolute value of 𝑥. And what the modulus or the absolute value actually are, is it means that we’re interested in only the positive results or nonnegative results. So we’re now gonna use that to actually solve our equation. So we’ve got three multiplied by the modulus of 𝑥 minus 66 is equal to zero.
Well, the first step we’re gonna do is actually we’re gonna have three modulus 𝑥 equal to something. So we’re gonna add 66 onto each side. So we can now say that three multiplied by the modulus of 𝑥 is equal to 66. Well, we’re now gonna actually break it down into two equations. And the reason we’re gonna do this is kind of shown in this little illustration here. Because actually, if we had the modulus of 66, well the answer that would be 66, but also if we had the modulus of negative 66, then that would actually also be 66 is the answer. So because of this, we’re gonna setup and solve two equations: one where three 𝑥 is equal to 66 and the other where three 𝑥 is equal to negative 66. As the answer to both of these will actually give us the same answer of 66 when the modulus or absolute value is included.
Okay, so let’s solve these. So by solving the left-hand side, we’ve divided both sides by three which gives us that 𝑥 is equal to 22. So this is our first possible solution. And then again, if we divide both sides of the equation on the right-hand side by three, we actually get our second solution. So we get 𝑥 is equal to negative 22. So therefore, we can say that our solution set is that 𝑥 is equal to negative 22 or 22.
Well, we can actually double check that by substituting our values for 𝑥 from our set back into the original equation. So here we go, substitute in negative 22, so I’ve got three multiplied by the modulus of negative 22 minus 66. And this is gonna give us 66 minus 66. And that’s 66 because it’s three multiplied by, and as we said, the modulus or absolute value of negative 22. So we’re only interested in the actual positive value. So it’s the same as three multiplied by 22 which is 66. And that gives us zero. So great, that actually meets our original equation.
So now we can try the second value. And this time we substitute in 𝑥 equals 22. So we get three multiplied by the modulus of 22 minus 66, which again gives us 66 minus 66 which again leads us to the answer we want which is zero. So fantastic! We’ve checked our answer. And we know that yes, it’s the correct solution, that the solution set is: 𝑥 is equal to negative 22 or 22.