Lesson Objectives
• Review the basic definition of absolute value
• Learn how to solve basic absolute value equations/inequalities
• Learn how to find the distance between two points on a number line

## What is Absolute Value?

What is absolute value and how do we find the absolute value of a number? Absolute value is a way to think about a number’s size. Simply put, the absolute value of a number is the distance between the number and zero on the number line. Since absolute value represents a distance, the result will always be non-negative. This means the absolute value of a number can be only 0 or larger. When we want to find the absolute value of a number, this is shown by wrapping the number inside of "| |". Let's think about absolute value using an example. What is the absolute value of 5? We would write this problem as: |5| As we can see from the number line above, 5 is 5 units away from zero. Therefore, the absolute value of 5 is simply 5.
|5| = 5
What would happen if we asked, what is the absolute value of -5?
|-5| = ? As we can see from the number line above, -5 is also 5 units away from zero. Therefore, the absolute value of -5 is also 5.
|-5| = 5
We know from our lecture on integers that 5 and -5 are known as opposites. These are numbers that have the same distance away from zero on the number line but lie on opposite sides. Since the distance from zero is always the same for opposites, their absolute values will always be the same. Let's think about another example of opposites: 3, and -3. We can begin by looking at the absolute value of each.
|-3| = 3
|3| = 3
Each is 3 units away from zero on the number line.
When we want to find the absolute value of a number, we don't need to pull out a number line and count units each time. In the case of 0 or a positive number, the absolute value is just the number itself. In the case of a negative number, we simply find the opposite of the number or make the number positive. Let's look at a few examples.
Example 1: Find the absolute value for 7, -21, 14, and -35.
|7| = 7
7 is a positive number, so the absolute value of 7 is simply 7
|-21| = 21
-21 is a negative number, in this case, we make the number positive, the absolute value will be 21
|14| = 14
14 is a positive number, so the absolute value of 14 is simply 14
|-35| = 35
-35 is a negative number, in this case, we make the number positive, the absolute value will be 35

### Absolute Value and Inequalities

In some cases, we may encounter problems that involve the absolute value operation and finding the correct inequality relationship between two numbers. When these problems occur, we first simplify each side and then determine the proper relationship. Let's take a look at a few examples:
Example 2: Replace the ? with the correct symbol: <, >, or =.
|4| ? |-23|
Start by simplifying each side:
|4| = 4
|-23| = 23
4 ? 23
4 < 23
We can show this in terms of our original problem as:
|4| < |-23|
Let's try one more:
|-55| ? |-18|
|-55| = 55
|-18| = 18
55 ? 18
55 > 18
We can show this in terms of our original problem as:
|-55| > |-18|

### Simplifying with the Absolute Value Operation

We will often have to simplify when the absolute value operation is involved. This operation will lead to some unexpected results. Suppose we saw the following scenario:
-(-1)
At this point, we should know that the result here is +1.
-(-1) = 1
We can obtain this result by starting inside of the parentheses. This would yield: the opposite of -1, which is 1. We could also achieve this result by counting the number of "-" symbols. In the case of an even number, the result is positive. In the case of an odd number, the result is negative. In this scenario, we have an even number (2) of "-" symbols and therefore a positive result. Let's look at one more of these:
-(-(-13))
Here we can see 3 total "-" signs. 3 is an odd number, so the result is negative.
-(-(-13)) = -13
Now, let's look at the result of throwing an absolute value operation in the mix. We will see the results do not follow the same logic.
-(-|-13|)
We can no longer use the same rule as before, the absolute value operation will throw off the result. Let's work through this problem starting from the innermost grouping symbol, which in this case, is the absolute value operation:
-(-|-13|)
First, we will find the absolute value of -13, this will give us 13: Let's now replace |-13| with 13 in our problem:
-(-13)
Now we have the opposite of -13, which is 13.
-(-|-13|) = 13
If we counted the total number of "-" symbols here, we would arrive at 3, an odd number. This suggests from our rule that we would have -13 as our answer. This is not the case because of the absolute value operation. The absolute value operation results in a non-negative result, so we have to only consider "-" symbols outside of the operation to use the rule. In this problem, we would count only 2 "-" symbols outside of the absolute value operation. 2 is an even number and would give us a result that is positive, which is correct. Let's try a few examples:
Example 3: Simplify -(-(-|-4|)).
-(-(-|-4|)) = -4
The result here is -4 since there are 3 "-" symbols outside of the absolute value operation. 3 is an odd number, so we end up with a negative result.
Example 4: Simplify -(-(-(-|-9|))).
-(-(-(-|-9|))) = 9
The result here is 9 since there are 4 "-" symbols outside of the absolute value operation. 4 is an even number, so we end up with a positive result.

### Solving Simple Absolute Value Equations/Inequalities

As we move through our course, we will encounter some very challenging equations that involve the absolute value operation. For now, we will think about a few very simple examples. To start things off, let’s revisit our first problem, where we considered finding the absolute value of 5 and -5.
|5| = 5
The absolute value of 5 is 5 since 5 is 5 units away from zero on the number line. Additionally, we should remember that opposites (additive inverses) have the same absolute value. This is because opposites have the same distance from zero on the number line, they just lie on opposite sides. One will be positive and the other negative. We can see this by thinking about the absolute value of -5.
|-5| = 5
The absolute value of -5 is 5 since -5 is 5 units away from zero on the number line. Now suppose we saw an equation such as:
|x| = 5
What number or numbers can replace x and give us a true statement?
Since we are taking the absolute value of what is plugged in for x, x can be 5 or x can be -5.
|5| = 5
|-5| = 5
We would have two solutions for this equation:
x = 5 or x = -5
Example 5: Solve each equation.
|x| = 13
To solve this equation, let's again think about the fact that two numbers will have an absolute value of 13: 13 and (-13).
x = 13 or -13
Additionally, we can use similar logic to work through absolute value inequalities. To see this, let’s think about the absolute value of 8 and -8. From our number line, we can see that 8 is 8 units away from zero on the number line. This means the absolute value of 8 is 8.
|8| = 8
Additionally, the opposite of 8, (-8) will also have an absolute value of 8. Let's look at the absolute value of (-8) using a number line: What could we say is the solution for an inequality such as:
|x| > 8
If we think about this carefully, we are saying that we can replace x with any number whose absolute value or distance from zero is greater than 8. This means any number larger than 8 or any number smaller than -8. This means we can show our solution as:
x > 8 or x < -8
Similarly, if we flipped the sign, we would want to find all numbers whose absolute value is less than 8.
|x| < 8
Here x can be any value between -8 and 8.
-8 < x < 8
Example 6: Solve each inequality.
|x| > 4
To solve this inequality, let's think about what numbers have an absolute value that is larger than 4.
x > 4 or x < -4

### Distance Between Two Points on a Number Line

If P and Q are points on a number line with coordinates a and b, respectively, then we can find the distance between the two points P and Q as:
d(P,Q) = |b - a| or d(P,Q) = |a - b|
The thing to understand here is that the order of the subtraction isn't going to matter since it is done inside of absolute value bars.
Example 7: Find the distance between -3 and 6 on the number line.
|-3 - 6| = |-9| = 9
|6 - (-3)| = |9| = 9
Either way, the distance between -3 and 6 is 9 units. We can prove this by counting the units on the number line below.

#### Skills Check:

Example #1

Simplify: $$-|{-}5^2 + 3|$$

A
$$22$$
B
$$-22$$
C
$$28$$
D
$$-28$$
E
$$0$$

Example #2

Solve for x: $$|x| < 3$$

A
$$-3 < x < 3$$
B
$$x > 3\hspace{.35em}or \hspace{.35em}x < -3$$
C
$$\text{No Solution}$$
D
$$-3 ≤ x ≤ 3$$
E
$$x ≥ 0$$

Example #3

Find the distance between point Q and Point R $$Q=-1, R=-5$$

A
$$5$$
B
$$8$$
C
$$7$$
D
$$4$$
E
$$6$$

Example #4

Find the distance between point Q and Point R $$Q=-11, R=15$$

A
$$26$$
B
$$4$$
C
$$165$$
D
$$19$$
E
$$3$$