When you combine absolute values and inequalities, things can get complicated very quickly. So, I want to look at just a couple of basic examples of some inequalities that contain absolute values.
So, perhaps, the most basic you can find is just an absolute value of x that is less than a constant. We’ll say 3.
The following transcript is provided for your convenience.
So, if the absolute value of x is less than 3, that means that we need to solve it two different ways. First, with x as a positive number, which gives us x is less than 3, and then secondly, with x as a negative number, which gives us -x is less than 3, or multiplying both sides by -1 and flipping the sign, x is greater than -3.
And so, we have these two solutions, and they both must be satisfied, so we have a range of values for x when we say that x is greater than -3, but less than 3, or x is between -3 and 3. And so, this range of values for x is the solution to that inequality.
And, it’s bounded, because the absolute value is given an upper maximum. We’re going to look in this example here, at a situation where the absolute value of x is greater than a constant. And so, if the absolute value is greater than a constant that means that x can be either positive or negative, and extend to infinity in either direction, and still satisfy this equation.
So, whereas in this one, we had 2 finite bounds, here we’re going to have 2 infinite bounds. So, we’ll go ahead and solve this with a positive x first, we get x is greater than 3, and so, that’s one option for a solution, and then, assuming x is negative, we can have -x is greater than 3, or multiplying by -1 and flipping the sign, x is less than -3.
And so, now, we have two ranges, bounded on one side by 3 or -3, and bounded on the other side by an infinity. So, x can be between 3 and infinity, or between negative infinity and -3.
And so, these are the two ranges of x that satisfy this inequality. Now, there’s one more example I want to look at real briefly, and that is the example of a no solution problem.
So, suppose we have the absolute value of x plus 3 is less than 2.
Now, if we try to solve this, we’ll subtract 3 from both sides, and that’ll give us absolute value of x is less than -1.
And so, we can look at this, and we can see that there’s going to be no solution because the absolute value function tells us that this side of the equation is going to positive. And so, if we have a positive number here that has to be less than -1, that just doesn’t work. There’s no value for x that’ll satisfy this equation. And so, when you absolute values and inequalities, these are three of the things you can encounter.