Jeremiah just moved to Boston with his family. He wants to practice Aikido, but he's not sure which dojo to pick. The distance on the bus is probably the deciding factor, but some of them are Outbound, some are Inbound, and some are both. He lives near the Washington St. stop on the Green Line. He finds one dojo in Porter Square and one near the Davis stop on the Red Line. He finds one on Mass Ave, one out at Forest Hills on the Orange Line, and one at Brookline on the Green Line. Which dojo should he pick?

In this concept, you will learn how to take the absolute value of integers.

### Taking Absolute Value of Integers

An **integer **is any positive whole number or its opposite. Here, opposite means sign. So a positive integer has a negative opposite and vice versa.

A **positive number **is a number greater than zero. It can be written with or without a + symbol in front of it. A gain in something is written with a positive number.

A **negative number **is a number that is less than zero. It is always written with a - symbol in front of it. A loss is written with a negative number.

A **number line** is a line on which numbers are marked at intervals, used to illustrate simple numerical operations. Using a number line allows you to see where a number is in relationship to other numbers and from zero.

If you look at this number line, you will see that 2 is two units away from zero. Each little line on the number line is a unit. But -2 is also two units away from zero. Even though these numbers have different signs, they are both two units away from zero.

The **absolute value** of a number is the distance that the integer is from zero. The symbol for absolute value is two parallel lines surrounding the quantity, like this: \begin{align*}|5|\end{align*}

In this example, \begin{align*}|5|\end{align*}

Here is another example.

\begin{align*}|-9| = \underline{\;\;\;\;\;\;\;\;}\end{align*}

The absolute value of negative nine is the number of units that -9 is from zero. It is nine units from zero. So the absolute value of -9 is 9.

\begin{align*}|-9| = 9\end{align*}

Distance from 0 is always a positive quantity. So in practice, all you need to do to find the absolute value is to take away the negative sign. The absolute value of a positive integer will just be that integer. The absolute value of a negative integer will just be that integer without the negative sign.

### Examples

#### Example 1

Earlier, you were given a problem about Jeremiah and his Aikido dilemma.

He can't decide which dojo is the fewest bus stops from his house near Washington St. He has found dojos at Porter, Davis, Mass Ave., Forest Hills, and Brookline.

First, to solve this problem, he figures out how many stops Inbound and Outbound each dojo would require him to travel.

To get to Porter Square, he has to go Inbound 18 stops on the Green Line. Then he transfers to the Red Line at Park Street and goes Out bound 5 stops. To get to Davis, he does the same but goes Outbound 6 stops. To get to Mass Ave he goes Inbound 18 stops on the Green Line, then he transfers to the Red Line and goes Inbound another stop, then he transfers to the Orange Line at Downtown Crossing and goes Outbound 4 stops. To get to Forest Hills, he does the same but goes Outbound 10 stops, instead. But to get to Brookline Village, he only has to go Inbound 13 stops on the Green Line and then he can get off at Kenmore and go 3 stops Outbound on the D.

Next, he decides that any stops going Inbound from his house are positive values and any stops Outbound are negative. Using this convention, he makes a chart.

Inbound | Outbound | |

Porter | +18 | -5 |

Mass Ave | +18 | -6 |

Davis | +18 +1 | -4 |

Forest Hills | +18 +1 | -10 |

Brookline Village | +13 | -3 |

Then, because he only cares about the total number of stops and not what direction they are in, he sets up a series of addition problems using absolute values and solves them.

Porter:

Mass Ave: \begin{align*}18+1+\left|-4\right|=23\end{align*}

Forest Hills: \begin{align*}18+1+\left|-10\right|=29\end{align*}

Finally, he compares the number of total stops. He concludes that the Brookline Village dojo has the fewest total stops, so he will practice there.

#### Example 2

Find the absolute value.

\begin{align*}|-270| = \underline{\;\;\;\;\;\;\;\;}\end{align*}

First, remember that the absolute value is just the distance a value is from zero.

In this case, that is 270.

Next, remember that the distance is always a positive value.

Then, write the absolute value as a positive distance from zero.

The absolute value of \begin{align*}-270\end{align*}

#### Example 3

Find the absolute value.

\begin{align*}|-10| = \underline{\;\;\;\;\;\;\;\;}\end{align*}

First, determine what the distance is.

In this case, it is 10.

Then, remember that the distance is always a positive number.

The answer is 10.

#### Example 4

Find the absolute value.

\begin{align*}|25| = \underline{\;\;\;\;\;\;\;\;}\end{align*}

First, determine what the distance is.

In this case, it is 25.

The answer is 25.

#### Example 5

Find the absolute value.

\begin{align*}|-2| = \underline{\;\;\;\;\;\;\;\;}\end{align*}

First, determine what the distance is.

In this case, it is 2.

Then, remember that the distance is always a positive number.

Therefore, the answer is 2.

### Review

Identify the absolute value of integers. Use a number line for reference.

- \begin{align*}|6|\end{align*}
|6| - \begin{align*}|-6|\end{align*}
|−6| - \begin{align*}|-21|\end{align*}
|−21| - \begin{align*}|8|\end{align*}
|8| - \begin{align*}|12|\end{align*}
|12| - \begin{align*}|-7|\end{align*}
|−7| - \begin{align*}|-17|\end{align*}
|−17| - \begin{align*}|17|\end{align*}
|17| - \begin{align*}|4|\end{align*}
|4| - \begin{align*}|-4|\end{align*}
|−4| - \begin{align*}|-204|\end{align*}
|−204| - \begin{align*}|44|\end{align*}
|44| - \begin{align*}|144|\end{align*}
- \begin{align*}|-144|\end{align*}
- \begin{align*}|-290|\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 11.6.