So we're given the following equation, that 5 times the
absolute value of x plus 3 minus 3 is equal to 7.
It's always a little daunting to see an equation with an
absolute value sign.
And when you only have one of them, like this, what I like
to do is isolate it, and just kind of think it through from
that point.
So let's try to isolate the absolute
value of x plus 3 here.
Let's try to isolate that part of our equation.
So the first thing we might want to do is add 3 to both
sides of the equation.
That'll get rid of this minus 3, or this negative 3, on the
left-hand side.
So let's add 3 to both sides of this equation.
And this'll turn the equation into-- do it in that same pink
color --the left-hand side will still be 5 times the
absolute value of x plus 3.
The minus 3, or the negative 3 plus the 3,
those will cancel out.
That'll just be 0.
And then that will be equal to 7 plus 3, which is 10.
Now, we have 5 times the thing we want to isolate.
The best way to isolate it completely is to divide both
sides of this equation by 5.
So if you divide that side by 5, and then the
right-hand side by 5.
We divide it by 5 so that these guys will cancel out.
5 times something divided by 5 is just that something.
So these cancel out.
That's 2.
So we're left with the absolute value of x plus 3 is
equal to 10 divided by 5, which is 2.
So we've simplified the equation a good bit, now we
just have to put our thinking caps on a little bit.
If the absolute value of something is 2, what does it
mean that that something is?
What are the two numbers that, if I were take it's absolute
value, I could get 2.
I'll do a little thing on the side here.
We know that the absolute value of 2 is equal to 2.
So maybe this thing was equal to 2.
So maybe x plus 3 is equal to 2.
If x plus 3 is equal to 2, and you take its absolute value,
you're going to get 2 again.
But we also know that the absolute value of negative 2
is also equal to 2.
So maybe x plus 3 is equal to negative 2.
Because if x plus 3 is equal to negative 2, and we take its
absolute value, then we're going to get 2 again.
So we could also write or x plus 3 could be equal to
negative 2.
That's what I mean about thinking
about it a little bit.
Another way to think about it-- I've said this in other
videos --is absolute value means distance from 0.
So if we were to draw a number line here, that is 0, this is
saying that, whatever this quantity is inside the
absolute value sign, its distance from 0 is 2.
So what numbers are 2 away from 0?
Well, you have positive 2.
I'll write a positive there explicitly.
And you also have a negative 2.
So this thing here could be a positive 2, or it could be a
negative 2.
Either way you take the absolute value of a positive 2
or a negative 2, you're going to get a 2.
So let's solve these.
So over here we can subtract 3 from both
sides of this equation.
So if you subtract 3 from both sides, you get-- The left-hand
side, you're just left with an x.
These 3's cancel out.
That's the whole point of subtracting the 3.
x will be equal to 2 minus 3 is negative 1.
So that is one solution to our absolute value equation.
And what's our other solution?
Well here, once again, let's subtract 3 from both sides.
So you subtract 3 from both sides.
The left-hand side just becomes an x.
The right-hand side, negative 2 minus 3 is negative 5.
So this is our other solution.
And let's verify that they both work.
So if x is equal to negative 1, what does
this equation become?
We have 5 times the absolute value of negative 1
plus 3 minus 3.
And if this really is a solution, then this should be
equal to positive 7.
So let's see.
This is 5 times negative 1 plus 3 is 2.
So it's 5 times the absolute value of 2 minus 3.
The absolute value of 2 is just 2.
So this is 5 times 2 minus 3.
5 times 2 is 10 minus 3, which is, indeed, equal to 7.
So this is definitely a solution.
Let's try the other one out. x is equal to negative 5.
Running out of some real estate, but I could clear some
up right here.
If x is equal to negative 5, same drill.
5 times the absolute value of negative 5 plus 3 minus 3.
This again should be equal to 7.
Negative 5 plus 3 is negative 2.
Since there's 5 times the absolute value of negative 2
minus 3, the absolute value of negative 2 is positive 2.
So this is 5 times 2 minus 3, which is 10 minus 3, which is
again equal to 7.
And notice, in this situation, the thing in the absolute
value sign became a negative 2, because we said, oh, x plus
3 could be equal to negative 2.
In the magenta situation, the thing in the absolute value
sign was positive 2, because this was a situation where we
assumed that the thing in the absolute value sign was a
positive 2.
Now if we were to graph these solutions on the number line,
if we were to graph the two solutions on a number line--
I'll draw a new number line here.
Let me draw a new number line, just like that.
Both of these are less than 0, so I'll put 0 all
the way over here.
Maybe 0, 1, 2.
Let's go negative, negative 1, negative 2, negative 3,
negative 4, negative 5, negative 6.
This solution right here-- x is equal to negative 1 --we
could plot on the number line right over there.
And then the other solution-- x is equal to negative 5 --we
can plot it right over there.
Hopefully you found that enjoyable.