Video Transcript
In this video, we’re gonna look at
how to solve equations that involve an unknown in two places, one requiring an
absolute value to be taken and one not. Now remember, taking the absolute
value just means taking the positive version of a number, so ignoring the negative
sign. We’ll look at two ways to solve the
problems: one purely algebraically considering different possible values and the
other by looking at the graphs and considering the effect of taking the absolute
value. Okay, let’s look at our first
example.
Well we’ve got to solve the
absolute value of 𝑥 plus two is equal to 𝑥 plus two. And we’ve got two different
situations to consider: if 𝑥 plus two is greater than or equal to zero, then
it’s already positive so we don’t need to do anything fancy; if 𝑥 plus two is
less than zero, then we need to take the negative of that value in order to turn
it into a positive value. That’s what the absolute value
function is doing. So the values of 𝑥 down each
of those branches are, well subtracting two from either side of the inequality
on the left there, is 𝑥 is greater than or equal to negative two, and doing the
same for the other inequality, 𝑥 is less than negative two. So if the value of 𝑥 is less
than negative two, we’re gonna come down the right hand-branch; and if the value
of 𝑥 is greater than or equal to negative two, we’re gonna come down to the
left-hand branch. And if we come down the
left-hand branch, that means that the value of 𝑥 plus two is positive so we
don’t need to worry. So we’ll just use the
expression 𝑥 plus two. So what we’re saying is 𝑥 plus
two on the left-hand side is equal to 𝑥 plus two on the right-hand side, and
that’s true for all values of 𝑥. But remember, although it’s all
values of 𝑥, we were stretching ourselves to only values of 𝑥 which were
greater than or equal to negative two in order to get in here. So it’s all values of 𝑥 that
are greater than or equal to negative two. Now going down the right
channel, we know that 𝑥 is less than negative two, and we know that 𝑥 plus two
is giving us an answer which is negative, so we’ve gotta take the negative of
that negative to turn it into a positive. So that means the negative of
𝑥 plus two is equal to 𝑥 plus two or negative 𝑥 take away two is equal to 𝑥
plus two. Well if I add 𝑥 to both sides
of that equation, I get negative two is equal to two 𝑥 plus two; then if I
subtract two from both sides, I get negative four is equal to two 𝑥; and then
if I divide both sides by two, I get the answer 𝑥 is equal to negative two. Now if we’re being really
strict about that, we said that 𝑥 was less than negative two to come down this
route in the first place, so that is not a valid solution. But in fact, strictly speaking,
𝑥 could be equal to negative two to come down this route because the negative
of zero is still zero. So taking the absolute value of
that would in fact work. In fact, it doesn’t matter in
either case because we have already covered that solution on the other side over
here. So the answer is 𝑥 is greater
than or equal to negative two, which we can also write in interval notation like
this. negative two is included, so we’ve got the square bracket and it goes all
the way up to positive infinity, which is- always has the parentheses or the
round bracket.
Now we also promised to have a
look at this graphically. So if I draw the graph of 𝑦
equals 𝑥 plus two, straight away from the equation 𝑦 equals 𝑥 plus two, we
can see the slope is one. So every time I increase my
𝑥-coordinate by one my 𝑦-coordinate also goes up by one, and the 𝑦-intercept
is two. And it cuts the 𝑥-axis when 𝑦
is equal to zero, so zero is equal to 𝑥 plus two. Subtracting two from both
sides, 𝑥 is equal to negative two. So it cuts the 𝑥-axis, 𝑥 is
negative two, cuts the 𝑦-axis at two, and has a slope of one. Now let’s think about the graph
of 𝑦 equals the absolute value of 𝑥 plus two. Well, this is gonna be just
like the graph of 𝑦 equals 𝑥 plus two when it happens to be above the 𝑥-axis,
when the 𝑦-coordinates are positive. But when the 𝑦-coordinates go
negative, we’re reflecting that in the 𝑥-axis so that all the negative
𝑦-coordinates map onto their positive equivalents. So the orange line there is the
line 𝑦 equals the absolute value of 𝑥 plus two. And to the right of where the
𝑦-coordinate is zero at 𝑥 equals negative two and in fact even including when
𝑥 equals negative two. So I put a little solid dot in
here. That region there, we’re just
using the equation 𝑦 equals 𝑥 plus two to generate the 𝑦-coordinates for that
orange line from the 𝑥-coordinates. Now to the left of that, that
equation would generate negative 𝑦-coordinates, so we’re taking the negative of
the negative 𝑦-coordinates to generate these positive coordinates up here. So to the left of 𝑥 equals
negative two we’re using the equation that the 𝑦-coordinate is equal to the
negative of the coordinate generated by 𝑥 plus two. In other words, 𝑦 equals
negative 𝑥 take away two So the question was, when are those two things
equal? When does the line 𝑦 equals
the absolute value of 𝑥 plus two intersect with the line 𝑦 equals 𝑥 plus
two? And it’s all of these values
here, including when 𝑥 is equal to negative two, and all of these off into
infinity in that direction here. So the set of 𝑥-values that
match that is when 𝑥 is equal to negative two and everything to the right of
that. So we’ve got the same answers,
by either doing it algebraically or using the graph, but using the graph gives
you a bit more insight into why that’s the answer and what’s really going
on.
Let’s look at number two then.
Solve the absolute value of 𝑥
plus five is equal to negative five 𝑥 plus two. So again we’ve got an absolute
value, so there are two different scenarios. Either the value inside the
absolute function there 𝑥 plus five is already greater than or equal to zero or
it’s negative and it needs changing into a positive value. So just thinking about the
𝑥-values that correspond to that, if 𝑥 plus five is greater than or equal to
zero, that happens when 𝑥 is greater than or equal to negative five; if 𝑥 plus
five is less than zero, then that happens when 𝑥 is less than negative
five. So going down the left-hand
branch first, if the contents of the absolute value were already positive, then
we can just take 𝑥 plus five because it’s already positive and we can solve
that equation. So adding five 𝑥 to both sides
gives us six 𝑥 plus five equals two. And then subtracting five from
both sides gives us a six 𝑥 is equal to negative three. And then dividing both sides by
six gives us 𝑥 is equal to negative three over six which is negative a
half. So just before we consider that
a proper solution, we do need to think, “Is that in the region that would have
got us here anyway?” In order to get down this
branch, 𝑥 had to be greater than or equal to negative five, and negative a half
is greater than or equal to negative five, so we’re fine. That is a good solution. So let’s look at the other
side.
On this side, remember 𝑥 plus
five gave us a negative answer so we need to convert that to its positive
equivalent. So instead of taking 𝑥 plus
five, we’re taking the negative of 𝑥 plus five to turn it into a positive
coordinate, and that’s equal to negative five 𝑥 plus two. So distributing the negative
over the parentheses there, we’ve got negative 𝑥 take away five is equal to
negative five 𝑥 plus two, and so adding five 𝑥 to both sides now gives us four
𝑥 minus five on the left-hand side and just two on the right-hand side. so if I
now add five to both sides, I’ve got four 𝑥 equals seven. And now finally, just divide
both sides by four and I get 𝑥 is equal to seven over four. But wait! We said to get down this
branch, 𝑥 needed to be less than negative five; but in doing that, we’ve come
up with an answer 𝑥 is equal to positive seven over four. So this is a spurious nonsense
solution, so that’s not a correct solution here. So the only answer we’ve got
that’s valid is 𝑥 equal to negative a half. Let’s have a look at the graph
of that and just see why that’s the case.
So let’s first consider the
graph of 𝑦 equals negative five 𝑥 plus two that has a slope of negative five
and it cuts the 𝑦-axis at two. So negative five every time I
increase my 𝑥-coordinate by one, my 𝑦 coordinate is gonna go down by five. So we’re looking at a very
sharp downhill line, and it cuts the 𝑥-axis when 𝑦 equals zero. So putting the 𝑦-coordinate
equal to zero, we’ve got zero is equal to negative five 𝑥 plus two, adding five
𝑥 to both sides gives me five 𝑥 is equal to two, and then dividing both sides
by five gives me 𝑥 is equal to two-fifths. So it cuts the 𝑥-axis when 𝑥
is two-fifths. So that’s roughly what the line
𝑦 equals minus five 𝑥 plus two looks like. Now let’s consider the line of
the absolute value of 𝑥 plus five. Well that’s gonna be just the
same as the line 𝑦 equals 𝑥 plus five unless 𝑦-coordinate would’ve been
negative, in which case it becomes positive. So 𝑦 equals 𝑥 plus five has a
slope of one. It cuts the 𝑦-axis at 𝑦
equals five and it cuts the 𝑥-axis at 𝑥 equals minus five. So that’s the line 𝑦 equals 𝑥
plus five. But remember, we said the
absolute value of 𝑥 plus five wherever we’ve got negative 𝑦 coordinates
they’re gonna get reflected to their sort of positive equivalents up here. So this part of the line to the
left of 𝑥 equals negative five is actually gonna bounce up like this in that
direction. Now those two lines cross over
in just one place; and in the region where they cross over, we look at the
intersection of this line 𝑦 equals negative five 𝑥 plus two with this line 𝑦
equals just 𝑥 plus five. And when we solve that
equation, we get an 𝑥-coordinate of negative a half, which is exactly what we
got when we just did it algebraically. So why did our algebraic
approach give us this false answer of 𝑥 is equal to seven over four? Well what it did was it
extended this part of the line here off in this direction; and then way off down
here with this line and this line intersected, it calculated the corresponding
𝑥-coordinate up here of seven over four. Now that isn’t a valid
solution, so you do have to be very very careful when using the algebraic
approach. Using these graphs helps you
really to visualise which are valid solutions and which are not valid
solutions.
So let’s just summarise the main
points of what we’ve learned. The first thing we have to consider
is when the absolute value kicks in. And by that, we just mean when do
we have to change the answers that we get from our original function. So for example, if we’re looking at
the absolute value of 𝑥 plus five, if the value of 𝑥 plus five is greater than or
equal to zero anyway, we can just take that value; it’s already positive. But if the value of 𝑥 plus five
was less than zero, we’d have to take that negative answer and turn it into a
positive answer. And the way that we do that is we
take the negative of that negative answer. And remember, when 𝑥 plus five is
less than zero, if I take away five from each side, that happens when 𝑥 is less
than negative five. So if 𝑥 is less than negative
five, I’m going to take the negative of 𝑥 plus five to turn it into a positive
value; but if 𝑥 is greater than or equal to negative five, then I can just use the
𝑥 plus five value because it’s already greater than zero or equal to zero. Then we have to do two sets of
calculations, one in this situation and one in this situation, and we end up with
two different solution sets. And thirdly, you have to check if
the solutions that you get match the criteria above them. So for example over here, we know
that 𝑥 has got to be less than negative five. So if this comes up with an answer
of 𝑥 equals four, then we know that’s not valid because it’s not less than negative
five; but if it came up with an answer of negative ten, then that is in the valid
region; it is a valid answer. And finally, consider using drawing
a graph. The graphical approach is a visual
approach which helps you to identify valid solutions, perhaps a bit more easily more
obviously. Good luck then with solving
equations involving absolute and nonabsolute values of an unknown.
