Topics
4. Operations with Integers
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Grade 6

4.03 Absolute value of integers

Lesson
Practice

Absolute value of integers

Exploration

The applet below shows the absolute value for different integers on the number line. Move the point left and right and consider the following questions:

  1. What do you notice about the absolute value of a positive number?

  2. What do you notice about the absolute value of a negative number?

  3. Can the absolute value of a number ever be a negative number? Why or why not?

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The mathematical symbol for absolute value is |\,|. For example, we would read \left|-6\right| as "the absolute value of negative six."

The absolute value of a number is its distance from zero on the number line.

Number line from negative 10 to 10 with jumps to the left of 0 by 3 units to negative 3 and to the right of 0 by 3 units to 3

The numbers -3 and 3 are both 3 units from 0, so they have the same absolute value.

The absolute value of a positive number is the number itself.

The absolute value of a negative number is its opposite.

For example, |3|=3 and |-3|=3.

In addition to finding the absolute value of a number, we can simplify expressions that involve an absolute value. We need to remember our order of operations.

  • First, we simplify Grouping Symbols like parentheses or absolute value bars.
  • Next, we do Exponents.
  • Next, we do Multiplication or Division (from left to right)
  • Finally, we do Addition or Subtraction (from left to right)

Examples

Example 1

What is the value of \left|-155\right|?

Worked Solution
Create a strategy

The absolute value of a negative number is positive.

Apply the idea
\displaystyle \left|-155\right|\displaystyle =\displaystyle 155Evaluate

Example 2

Which values are smaller than \left|-20\right|?

A
-15
B
\ \left|-30\right|
C
\left|-5\right|
D
21
Worked Solution
Create a strategy

Evaluate the given expression and options and compare them.

Apply the idea

Original expression:

|-20|=20

Option B

|-30|=30

Option C

|-5|=5

The answers are Option A -15 and C |-5|.

Example 3

Evaluate each of these numbers, and order the results from smallest to largest:

\left| 19 \right|,\,\left| 0 \right|,\, \left| 41 \right|,\, \left| -31 \right|

Worked Solution
Create a strategy

Find the absoulute values of each given number and compare the results.

Apply the idea
\displaystyle \left|19\right|\displaystyle =\displaystyle 19Evaluate
\displaystyle \left|0\right|\displaystyle =\displaystyle 0Evaluate
\displaystyle \left|41\right|\displaystyle =\displaystyle 41Evaluate
\displaystyle \left|-31\right|\displaystyle =\displaystyle 31Evaluate

Compare the evaluated absolute values 0 \lt 19 \lt 31 \lt 41

Therefore, the order from smallest to largest is: \left| 0 \right|,\,\left| 19 \right|,\, \left| -31 \right|,\, \left| 41 \right|

Example 4

Evaluate |7 - 11| and represent the solution on a number line.

Worked Solution
Create a strategy

We need to follow Order of Operations, so we will need to do the subtraction inside the absolute value bars first.

Apply the idea
\displaystyle |7-11|\displaystyle =\displaystyle |-4|Subtract the inside of absolute value bars
\displaystyle =\displaystyle 4The distance between -4 and 0 is 4 on our number line
A numberline from -8 to 8, point at -4 and a dashed arrow from 0 to -4
Reflect and check

We can also think about this as the distance between 7 and 11 on a number line which is 4.

0123456789101112

Example 5

Evaluate \dfrac{-|12|}{2} and represent the absolute value step on a number line.

Worked Solution
Create a strategy

We need to follow Order of Operations, so we will take the absolute value first.

Apply the idea
-5-4-3-2-10123456789101112131415

We can see that 12 is 12 units away from 0 on our number line. So, |12|=12.

\displaystyle \dfrac{-|12|}{2}\displaystyle =\displaystyle \frac{-12}{2}Find the absolute of 12 using number line above
\displaystyle \dfrac{-12}{2}\displaystyle =\displaystyle -6Divide -12 by 2
Reflect and check

Would the answer be different if we changed the problem to \dfrac{|-12|}{2}?

Example 6

Laura is in a hot air balloon 8.5\text{ m} above sea level and Fred is exploring a cave 6.7\text{ m} below sea level. Which of these statements are true? Select all correct responses that apply.

A
Fred has greater elevation than Laura.
B
Fred is futher from sea level than Laura.
C
Laura has greater elevation than Fred.
D
Laura is futher from sea level than Fred.
Worked Solution
Create a strategy

Sea level is at an elevation of 0 \text{ m}. Find the absolute value of each choice and compare the results.

Apply the idea
\displaystyle \left|8.5\right|\displaystyle =\displaystyle 8.5Evaluate
\displaystyle \left|6.7\right|\displaystyle =\displaystyle 6.7Evaluate
\displaystyle 8.7\displaystyle >\displaystyle 6.5Compare the absolute values

The answers are options:

C Laura has greater elevation than Fred.

D Laura is futher from sea level than Fred.

Reflect and check

A was not a valid choice because if Laura is any distance above sea level she will always be higher than Fred who is below sea level.

B was not a valid choice because Fred's absolute value was smaller than Laura's so Laura is a greater distance from 0.

Idea summary

The absolute value of a number is its distance from zero on the number line.

The absolute value of a positive number is the number itself.

The absolute value of a negative number is its opposite.

Outcomes

6.NS.2

The student will reason and use multiple strategies to represent, compare, and order integers.

6.NS.2d

Identify and describe the absolute value of an integer as the distance from zero on the number line.

6.CE.2

The student will estimate, demonstrate, solve, and justify solutions to problems using operations with integers, including those in context.

6.CE.2c

Simplify an expression that contains absolute value bars | | and an operation with two integers (e.g., –|5 – 8| or |−12|/8) and represent the result on a number line.

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