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hey it's professor Dave let's learn
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about absolute values in our haste to
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get through algebra there's one topic we
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didn't mention so before we move on
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let's quickly cover absolute values it's
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important to know what these are
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and how to graph them so let's become
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familiar with this definition the
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absolute value of any real number X
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which is denoted by these little bracket
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type things is the distance from zero to
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that number on a number line
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take the number three this is three
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units away from zero so the absolute
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value of 3 is 3 now look at negative 3
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this is also 3 units away from zero so
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the absolute value of negative 3 is 3
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that brings us to a more practical
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definition of absolute value as being
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the magnitude of a number disregarding
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its sign positive numbers are unchanged
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by taking the absolute value while
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negative numbers simply drop the
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negative sign and become positive this
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applies not just to integers but all
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real numbers whether rational or
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irrational absolute values obey
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properties just like other operations so
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we can see that the absolute value of a
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B can be split up into the product of
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two absolute values and the same goes
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for division however let's note that the
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absolute value of a plus B does not
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equal the absolute value of a plus the
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absolute value of B which we can
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demonstrate with the values 1 and
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negative 1 we clearly don't get the same
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thing when we encounter absolute values
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while solving equations it just adds
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another step if we have the absolute
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value of x equals 2 that means X could
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be 2 or negative 2 so there are two
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solutions to this equation if we try a
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trickier one like the absolute value of
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2x minus 1 equals 5 then again we split
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this up into two equations it could
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either be the case that 2x minus 1
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equals 5 or 2x minus 1 could equal
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negative 5 because once we take the
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absolute value we get 5 either way that
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means we now have to solve both of these
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equations to get the two possible values
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of X for the first one we add 1/2 and x
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equals 3 for the other we do the same
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thing but we get x equals negative 2 so
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those are the two solutions in this case
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the equations can get harder than this
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but the approach will always be the same
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the last thing we want to do is look at
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simple graphs with absolute values we
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know that a graph of y equals x will
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look like this the x and y values and
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every single point are equal to one
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another and it's just a line with a
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slope of 1 and a y-intercept of 0 now
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what if we want to graph y equals the
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absolute value of x well as we said all
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positive values remain unchanged so
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everything from the y axis to the right
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will be just like it is now but negative
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values will have their signs reversed
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when we plug in negative 1 for X we
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don't get negative 1 for y we get
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positive 1 negative 2 gives us positive
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2 and so forth that means that
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everything to the left of the y axis or
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all the negative inputs for X will be
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reflected across the x axis so that we
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get the positive versions of all these Y
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values resulting in this v-shaped graph
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rather than a line we had before this
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can be transformed in all kinds of ways
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that we will learn later for now we just
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want to understand some basics about
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absolute values what they are and what
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they do with that covered let's check
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comprehension
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[Music]
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[Music]
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thanks for watching guys subscribe to my
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channel for more tutorials support me on
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patreon so I can keep making content and
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as always feel free to email me
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professor Dave explains at gmail.com
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[Music]
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you
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[Music]
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