Absolute Value Equations

Absolute Value Equations

with review

Absolute value represents the distance from 0 on a number line.

Any measurement is always positive, including distance. you can't run a negative mile, even if you run it backwards, you still ran the mile.

The Vertical lines you see here surrounding -3 means Absolute Value in Math.

You can remove those vertical lines by changing what is inside them to a positive value. Easy enough for constants(numbers) right? try it.

Multiple Choice

What is the absolute value of -5?

|-5| = ?

• -5

• 5

• +(-5)

• -5 and 5

We can also take the absolute value of a variable.

Just like solving for A in an equation, we can do the same for |A|. remember that the value inside the absolute value symbols is the distance from 0. So you can see A=5 if we travel 5 places to the right. Notice if we travel 5 places to the left, we arrive at -5. So A can equal both -5 or 5, depending on what direction we travel.

How would this math problem look originally? |a| = 5

So when we solve for a, we see if a=5, then |5| also equals 5. We also see if a= -5 that |-5| = 5. We can see that when taking the absolute value of a variable, we will have 2 answers; the answer for moving right(towards positive numbers) and the answer for moving left(towards negative numbers).

Notice we have the absolute value of Y equaling -5.

If we want to solve this like the one before, would we say Y= 5 or -5??? Let us try and see.

y=5 then |5| = -5. That does not seem right, because |5| = 5 not -5. What about if y= -5 then |-5| = -5, that also is not right, because

|-5| = 5. So y has no solution because an absolute value can never equal a negative number. You try some.

Multiple Choice

Solve for X

|X|=7

• X= 7

• X= -7

• X= 7 and X= -7

• No Solution

Multiple Choice

Solve for X

|X| = -3

• X= 3 and X= -3

• X= 3

• X= -3

• No Solution

Notice when solving for X, we have a positive value for A and a negative value for A.

We can apply this process to more advanced absolute value equations.

The equation |X-5|=2

Notice that we have setup two equations. Both equations have the absolute value symbols removed. But one of them has the value switched to a negative. This is how we find out the distance, one going left and one going right from our starting point on a number line. We solve both equations by isolating the variable, just like we have done before. Both equations are solved by adding 5 to both sides. We have our two answers for X; X=7, X=3.

It is very important to isolate the Absolute value equation before attempting to solve.

Think of the absolute value equation as a variable when performing operations to get it isolated. Notice we add 10 to both sides then divide both sides by 3 to get |x-2| isolated. Now the problem |x-2| = 7 can be solved like we just learned.

Why do we isolate the absolute value part first?

Since absolute value is always a positive number, we need to make sure that the equation makes since. Notice in the picture as we start isolating the absolute value part that we end up with |x+2| = -2/3.

Since an absolute value can not equal a negative number, there would be no solution. try some,

use paper and pencil as needed.

Multiple Choice

What would be the correct setup for solving for X?

|X+3| = 5

• X+3= 5 and X+3= -5

• X+3= 5 ad X-3= 5

• X+3= -5 and X-3 = -5

• X+3= 5 and -X-3= -5

Multiple Choice

What would be the correct way to write the problem before solving?

2|x-1|-6=4

• |x-1|=5

• x-1=5

• |x-1|=-5

• x-1=-5

Multiple Choice

Solve for X

2|X+3|+3=1

• X= -4 and X= -2

• X=2 and X= -2

• X=0

• No Solution