Absolute Value Equations

Absolute Value Equations

Let's do some equations that deal with absolute values.

And just as a bit of a review,

when you take the absolute value of a number.

Let's say I take the absolute value of -1.

What you're really doing is

you're saying, how far is that number from 0?

And in the case of -1, if we draw a number line right there

-- that's a very badly
drawn number line.

If we draw a number line right there, that's 0.

You have a -1 right there.

Well, it's 1 away from 0.

So the absolute value of -1 is 1.

And the absolute value of 1 is also 1 away from 0.

It's also equal to 1.

So on some level, absolute value is the distance from 0.

But another, I guess simpler way to think of it,

it always results in the positive version of the number.

The absolute value of -7,346 is equal to 7,346.

So with that in mind, let's try to

solve some equations with absolute values in them.

So let's say I have the equation

the absolute value of x -5 is equal to 10.

And one way you can interpret this,

and I want you to think about this, this is actually saying

that the distance between x and 5 is equal to 10.

So how many numbers that are exactly 10 away from 5?

And you can already think of the solution to this equation,

but I'll show you how to solve it systematically.

Now this is going to be true in two situations.

Either x -5 is equal to +10.

If this evaluates out to +10,

then when you take the absolute value of it,

you're going to get +10.

Or x - 5 might evaluate to -10.

If x - 5 evaluated to -10, when you take the absolute value of it,

you would get 10 again.

So x - 5 could also be equal to -10.

Both of these would satisfy this equation.

Now, to solve this one,

add 5 to both sides of this equation.

You get x is equal to 15.

To solve this one, add 5 to both sides of this equation.

x is equal to -5.

So our solution,

there's two x's that satisfy this equation.

x could be 15.

15 - 5 is 10, take the absolute value,

you're going to get 10, or x could be -5.

- 5 minus 5 is -10.

Take the absolute value, you get 10.

And notice, both of these numbers

are exactly 10 away from the number 5.

Let's do another one of these.

Let's do another one.

Let's say we have

the absolute value of x + 2 is equal to 6.

So what does that tell us?

That tells us that either x + 2,

that the thing inside the absolute value sign, is equal to 6.

Or the thing inside of the absolute value sign,

the x + 2, could also be -6.

If this whole thing evaluated to -6,

you take the absolute value, you'd get 6.

So, or x + 2 could equal -6.

And then if you subtract 2 from both sides of this equation,

you get x could be equal to 4.

If you subtract 2 from both sides of this equation,

you get x could be equal to -8.

So these are the two solutions to the equation.

And just to kind of have it gel in your mind,

that absolute value, you can kind of view it as a distance,

you could rewrite this problem

as the absolute value of x minus -2 is equal to 6.

And so this is asking me,

what are the x's that are exactly 6 away from -2?

Remember, up here we said,

what are the x's that are exactly 10 away from +5?

Whatever number you're subtracting from +5,

these are both 10 away from +5.

This is asking,

what is exactly 6 away from -2?

And it's going to be 4, or -8.

You could try those numbers out for yourself.

Let's do another one of these.

Let's do another one, and we'll do it in purple.

Let's say we have the absolute value of 4x.

I'm going to change this problem up a little bit.

4x -1.

The absolute value of 4x -1, is equal to--

actually, I'll just keep it-- is equal to 19.

So, just like the last few problems,

4x -1 could be equal to 19.

Or 4x -1 might evaluate to -19.

Because then when you take the absolute value,

you're going to get 19 again.

Or 4x -1 could be equal to -19.

Then you just solve these two equations.

Add 1 to both sides of this equation--

we could do them simultaneous, even.

Add 1 to both sides of this, you get 4x is equal to 20.

Add 1 to both sides of this equation,

you get 4x is equal to -18.

Divide both sides of this by 4, you get x is equal to 5.

Divide both sides of this by 4, you get x is equal to -18/4,

which is equal to -9/2.

So both of these x values satisfy the equation.

Try it out.

-9/2 x 4.

This will become a -18.

-18 minus 1 is -19.

Take the absolute value, you get 19.

You put a 5 here, 4 x 5 is 20.

Minus 1 is +19.

So you take the absolute value.

Once again, you'll get a 19.

Let's try to graph one of these, just for fun.

So let's say

I have y is equal to the absolute value of x +3.

So this is a function, or a graph,

with an absolute value in it.

So let's think about two scenarios.

There's one scenario

where the thing inside of the absolute value is positive.

So you have the scenario where x + 3

I'll write it over here: x + 3 is > 0.

And then you have the scenario where x +3 is < 0.

When x +3 is > 0,

this graph, or this line--or I guess we can't call it a line--

this function, is the same thing as y is equal to x +3.

If this thing over here is > 0,

then the absolute value sign is irrelevant.

So then this thing is the same thing

as y is equal to x +3.

But when is x +3 > 0?

Well, if you subtract 3 from both sides,

you get x is > -3.

So when x is > -3,

this graph is going to look just like y is equal to x +3.

Now, when x +3 is < 0.

When the situation where this--

the inside of our absolute value sign--is negative,

in that situation this equation is going to be

y is equal to the negative of x +3.

How can I say that?

Well, look, if this is going to
be a negative number, if x

plus 3 is going to be a negative
number-- that's what

we're assuming here-- if it's
going to be a negative number,

then when you take the absolute
value of a negative

number, you're going to
make it positive.

That's just like multiplying
it by negative 1.

If you know you're taking the
absolute value of a negative

number, it's just like
multiplying it by negative 1,

because you're going to
make it positive.

And this is going to
be the situation.

x plus 3 is less than 0.

If we subtract 3 from both
sides, when x is less than

negative 3.

So when x is less than negative
3, the graph will

look like this.

When x is greater than negative
3, the graph will

look like that.

So let's see what that
would make the

entire graph look like.

Let me draw my axes.

That's my x-axis, that's
my y-axis.

So just let me multiply this
out, just so we have it in mx

plus b form.

So this is equal to negative
x minus 3.

So let's just figure out
what this graph would

look like in general.

Negative x minus 3.

The y-intercept is negative
3, so 1, 2, 3.

And negative x means it
slopes downward, has a

downward slope of 1.

So it would look like this.

The x-intercept would be
at x is equal to--.

So if you say y is equal to 0,
that would happen when x is

equal to negative 3.

So it's going to go
through that line,

that point right there.

And the graph, if we didn't
have this constraint right

here, would look something
like this.

That's if we didn't constrain
it to a certain interval on

the x-axis.

Now this graph, what
does it look like?

Let's see.

It has its y-intercept
at positive 3.

Just like that.

And where's its x-intecept?

When y is equal to 0,
x is negative 3.

So it also goes through that
point right there, and it has

a slope of 1.

So it would look something
like this.

That's what this graph
looks like.

Now, what we figured out is
that this absolute value

function, it looks like this
purple graph when x is less

than negative 3.

So when x is less than negative
3-- that's x is equal

to negative 3 right there-- when
x is less than negative

3, it looks like this
purple graph.

Right there.

So that's when x is less
than negative 3.

But when x is greater than
negative 3, it looks like the

green graph.

It looks like that.

So this graph looks like
this strange v.

When x is greater than negative
3, this is positive.

So we have the graph of-- we
have a positive slope.

But then when x is less than
negative 3, we're essentially

taking the negative of the
function, if you want to view

it that way, and so we have
this negative slope.

So you kind of have this
v-shaped function, this

v-shaped graph, which is
indicative of an absolute

value function.