Hey it’s Professor Dave; let’s learn about absolute values.
In our haste to get through algebra, there’s one topic we didn’t mention, so before we
move on let’s quickly cover absolute values.
It’s important to know what these are and how to graph them, so let’s become familiar
with this definition.
The absolute value of any real number X, which is denoted by these little bracket type things,
is the distance from zero to that number on a number line.
Take the number three.
This is three units away from zero, so the absolute value of three is three.
Now look at negative three.
This is also three units away from zero, so the absolute value of negative three is three.
That brings us to a more practical definition of absolute value, as being the magnitude
of a number disregarding its sign.
Positive numbers are unchanged by taking the absolute value, while negative numbers simple
drop the negative sign and become positive.
This applies not just to integers, but all real numbers, whether rational or irrational.
Absolute values obey properties just like other operations, so we can see that the absolute
value of AB can be split up into the product of two absolute values, and the same goes
for division.
However, let’s note that the absolute value of A plus B does not equal the absolute value
of A plus the absolute value of B, which we can demonstrate with the values one and negative
one.
We clearly don’t get the same thing.
When we encounter absolute values while solving equations, it just adds another step.
If we have the absolute value of X equals two, that means X could be two or negative
two, so there are two solutions to this equation.
If we try a trickier one, like the absolute value of two X minus one equals five, then
again, we split this up into two equations.
It could either be the case that two X minus one equals five, or two X minus one could
equal negative five, because once we take the absolute value, we get five either way.
That means we now have to solve both of these equations to get the two possible values of
X.
For the first one, we add one, divide by two, and X equals three.
For the other, we do the same thing, but we get X equals negative two.
So those are the two solutions in this case.
The equations can get harder than this, but the approach will always be the same.
The last thing we want to do is look at simple graphs with absolute values.
We know that a graph of Y equals X will look like this.
The X and Y values at every single point are equal to one another, and it’s just a line
with a slope of one and a Y-intercept of zero.
Now what if we want to graph Y equals the absolute value of X?
Well, as we said, all positive values remain unchanged, so everything from the Y axis to
the right will be just like it is now.
But negative values will have their signs reversed.
When we plug in negative one for X, we don’t get negative one for Y, we get positive one.
Negative two gives us positive two, and so forth.
That means that everything to the left of the Y axis, or all the negative inputs for
X, will be reflected across the X axis, so that we get the positive versions of all these
Y values, resulting in this V-shaped graph rather than the line we had before.
This can be transformed in all kinds of ways that we will learn later, for now, we just
want to understand some basics about absolute values, what they are, and what they do.
With that covered, let’s check comprehension.