Absolute Value: Solve Algebraically
Objective
Solving absolute value functions through graphing
Now that we know exactly how to graph the absolute value function
And we can see how the function changes, depending on how we stretch, move, or reflect the function
It’s time to use this to solve some absolute value functions
Now there are two ways to do this, and this PowerPoint will go over both ways.
Let me be clear:
EITHER WAY IS OKAY
My job, is to show you how to do it with as many ways as I can think of.
Your job is to pick one and solve it that way.
Solve by graphing (no calculations)
So to solve by graphing, we are going to take the equation we are given.
And we’re going to break it up into 2 equations, then graph those two equations
Now, this idea is sort of weird, but it makes logical sense, so before we start graphing, let’s look at a quick example of what the function is:
Let’s say, for fun, we have:
Now, let’s take that function, and break it up into two pieces:
So, now, we weirdly have two equations.
But they’re both missing something…..
y
y
And, since we’re math students and want to do some crazy things,
Let’s graph both of these equations!
As we can see, we have the graph of 2|x| - 2 = y
So, let’s see y = 2:
Where the two graphs cross is the answer!
So, x = -2, and x = 2
So, how do we know we have the right answer?
Well, let’s check the only way we know how
Let’s plug and chug!
Let’s try plugging in x = 2 first:
As we can see, that worked out.
So what about x = -2?
So now we know for sure that we have the right answer!
Now let’s look at a few more examples:
Example 1:
Solve the following graphically:
Let’s start this off by breaking this equation into two parts:
And add their missing part:
y
y
Finally, let’s graph them and see where they intersect!
So, to start, let’s do: -7|x - 4|- 3 = y
Now let’s graph: y = -10
So, where do they seem to intersect?
Then, our answer is:
x = 3 x = 5
Solve by graphing (some calculations)
So to solve by graphing the second way, we are going to take the equation we are given.
And set it equal to zero.
Once we do, we just look at the zeroes of the new function, and that will tell us the answer.
So, let’s say, for fun, we have:
Now, let’s take that function, and set it equal to zero:
Now that we have it equal to zero, we just need to look at the graph.
Once we see the zeros of the graph, we’ll know the answer.
So:
Now let’s look at the zeroes:
So then, we can see that:
x = -2, and x = 2
Again, how do we know we have the right answer?
Well, let’s check the only way we know how
Let’s plug and chug!
Let’s try plugging in x = 2 first:
As we can see, that worked out.
So what about x = -2?
So now we know for sure that we have the right answer!
Now let’s look at a few more examples:
Example 1:
Solve the following graphically:
Let’s start this off by setting this equal to zero:
Now let’s graph our new graph:
And let’s take a look at the zeroes of this graph:
So, looking at our graph we can see that our answer is:
x = 3 x = 5
SO THEN HOW DO WE SOLVE ABSOLUTE VALUE FUNCTIONS ALGEBRAICALLY?
Well, we solve absolute value algebraically by realizing the absolute value function is actually two functions pushed into one.
But that makes sense doesn’t it?
Because so far, when we’ve graphed the absolute value function, we have two lines.
One goes to the right, while the other is a reflection of the first.
So it sort of goes without saying that they are two functions, but how do we determine what the two functions are?
DETERMINING THE TWO FUNCTIONS
So, to start, let’s look at an example of absolute value
Say:
Now, we know that there are two answers here.
But how do we determine the second equations?
Well, we know that the function goes both positive and negative
What if we make our answer both positive and negative?
Like so:
Now let’s get rid of those absolute value parts and solve:
So our answer is:
x = 9 and x = -5
So, how do we know we have the right answer?
Well, let’s check the only way we know how
Let’s plug and chug!
Let’s try plugging in x = 9 first:
As we can see, that worked out.
So what about x = -2?
So now we know for sure that we have the right answer!
So what if the function is stretched/compressed?
Well, that’s a bit different.
When a function is stretched or compressed, we want to undo the procedure so we can see the function in it’s natural state.
Let’s look at an example to show what is meant.
Solving algebraically with a number for s:
Let’s say we have something like:
Now, we want to get x by itself
So first we want to add 3 to both sides:
Now we want to get rid of the -5
So, we’re going to divide both sides by -5
_______________
-5 -5
And we’re going to get that:
Now, we do what we’ve been taught to do
We break this up into two equations.
Let’s get rid of the absolute value symbols
And solve:
x = 4
x = 0
So our answer would be:
X = 4, x = 0
Example 1:
Solve the following algebraically:
Now, we want to get x by itself
So first we want to subtract 9 from both sides:
Now we want to get rid of the -4
So, we’re going to divide both sides by -4
_____________
-4 -4
And we’re going to get that:
Now, we break this up into two equations.
Let’s get rid of the absolute value symbols
And solve:
x = -6
x = -8
So our answer would be:
x = -6, x = -8
Example 2:
Solve the following algebraically:
Now, we want to get x by itself
So first we want to add 9 to both sides:
Now we want to get rid of the 2
So, we’re going to divide both sides by 2
_____________
2 2
And we’re going to get that:
Now, we break this up into two equations.
Let’s get rid of the absolute value symbols
And solve:
x = 18
x = 4
So our answer would be:
x = 18, x = 4
Example 3:
Solve the following algebraically:
Well, there is no s, and there is no u
So, let’s skip to the:
Breaking this up into two equations.
Let’s get rid of the absolute value symbols
And solve:
x = -2
x = -12
So our answer would be:
x = -2, x = -12
Example 4:
Solve the following algebraically:
Now, we want to get x by itself
So first we want to subtract 9 from both sides:
Now we want to get rid of the -4
So, we’re going to divide both sides by -4
_____________
-3 -3
And we’re going to get that:
Now, we break this up into two equations.
Let’s get rid of the absolute value symbols
And solve:
x = 11
x = 7
So our answer would be:
x = 11, x = 7
EXCEPTIONS
Sometimes, there are questions which are exceptions, and have no answer.
These are usually when an absolute value expression tries to be equal to a negative number
Here’s an example of an exception:
EXCEPTION EXAMPLE:
So let’s say we have something like:
Now let’s solve this the way we normally would
_____________
-2 -2
However, let’s look at this.
What this is saying, is we can take the absolute value of an expression, and some how get a negative answer.
Is that possible?
Well, let’s look at the graph:
As we can see from the graph
-5 is never touched on.
So we can safely say that this graph has no answer.
EXCEPTION #2:
So let’s say we have something like:
Now let’s solve this the way we normally would
_____________
-2 -2
However, let’s look at this.
If we were to break this equation into two different equations, how would we do it?
Can 0 be positive or negative?
Then, it would seem we have only one answer.
So let’s get rid of the absolute value notation:
Then what we found is this function only has one answer.