Introduction to Absolute Value

Introduction to Solving Absolute Value Equations

Let's start at the beginning!

Multiple Choice

The absolute value of a number represents the ___________ from that number to zero on a number line.

• distance

• journey

• transformation

• dissolution

Multiple Choice

Evaluate: |-6|

• 6

• -6

• 1

• 0

Multiple Choice

Evaluate: |6|

• 6

• -6

• 0

• 1

Absolute Value

The distance from zero on a number line.

| -6 | = 6 means that the value -6 is six steps away from zero on a number line.

So what if we know the distance? How can we figure out what the value is?

Multiple Select

Which of the following could I substitute in for x and get 3?

 $\left|x\right|=3$  

• 3

• -3

• 0

• 1

There are 2 answers!

"What are two numbers that are 3 units away from 0?"

Two values work!

x = -3 or x = 3

Multiple Select

What about this one? Which of the following values could I substitute in for x and it would make a true sentence?

 $\left|x-2\right|=3$  

• 3

• -3

• -1

• 5

• 2

Use Substitution

Let x = -1

| -1 - 2 | = | -3 | = 3

Let x = 5

| 5 - 2 | = | 3 | = 3

What about this one?

|2x+7| = 5

If we weren't given multiple choice answers, we could guess until we found values that worked.

However, there are steps to solve.

Steps for Solving Absolute Value Equations

• ISOLATE the Absolute Value BARS

• SPLIT into two equations: the positive case and the negative case

• SOLVE each equation

• CHECK each solution using substitution into the original equation

Multiple Choice

What does it mean to "isolate the bars?"

How would you isolate the bars for this equation:

 $\left|x\right|-3\ =\ 10$  

• Add 3 to both sides

• Subtract 3 from both sides

• Add 10 to both sides

• Subtract 10 from both sides

We isolate the bars just as we would isolate a variable in a regular equation.

 $\left|x\right|-3=10$  
 $\left|x\right|-3+3\ =\ 10+3$  (Adding 3 to both sides)
 $\left|x\right|=13$  

For this problem, we see a couple of solutions for x.

-13 and 13

Multiple Choice

How about this one? How would you isolate the bars?

 $4\left|x\right|=16$  

• Add 4 to both sides

• Subtract 4 from both sides

• Multiply 4 on both sides

• Divide both sides by 4

Multiple Choice

How about this one?
Isolate the bars:

 $\frac{\left|x\right|}{4}=16$  

• Add 4 to both sides

• Subtract 4 from both sides

• Multiply 4 on both sides

• Divide both sides by 4

Multiple Choice

How about this one?
Isolate the bars:

 $4\left|x+2\right|=16$  

• Add 2 to both sides

• Subtract 2 from both sides

• Multiply 4 on both sides

• Divide both sides by 4

Whatever is inside the bars...

...stays inside the bars when you are isolating. Don't mess with the inside!

Multiple Choice

Isolate the bars......

 $\left|x+3\right|+2=8$  

•  $\left|x+3\right|=6$  

•  $\left|x\right|=3$  

•  $\left|x+5\right|=8$  

•  $\left|5x\right|=8$  

Multiple Choice

Isolate the bars......

 $5\left|x-8\right|-7=13$  

•  $\left|x-8\right|=4$  

•  $\left|x\right|=12$  

•  $\left|5x-40\right|=20$  

•  $\left|5x-47\right|=13$  

So we know how to isolate the bars...then what?

Split into two equations

Split into two equations: 

 $\left|x+3\right|=6$  

• The positive case:  $x+3=6$  which can be solved --> x = 3

• The negative case:  $x+3=-6$  which can be solved --> x = -9

Multiple Select

Split into two equations:

 $\left|2x+7\right|=5$  

•  $2x+7=5$  

•  $-2x+7=5$  

•  $2x+7=-5$  

•  $-2x-7=-5$  

Multiple Select

Isolate the bars and Split into two equations:

 $\left|x-8\right|+2=5$  

•  $x-8=3$  

•  $x-8+2=5$  

•  $x-8=-3$  

•  $x-8+2=-5$