How to Solve Integers and Absolute Value Problems? (+FREE Worksheet!)

Understanding integers and absolute value is essential for success in Algebra 1. Integers are the building blocks of signed-number arithmetic, and absolute value tells us the size of an integer without regard to its sign. Together, these two concepts appear in everything from ordering numbers on a number line to solving equations involving distance.

What Are Integers?

An integer is any whole number, its negative, or zero. The set of integers is: …, −4, −3, −2, −1, 0, 1, 2, 3, 4, … Integers do not include fractions or decimals. On a number line, positive integers are to the right of zero and negative integers are to the left.

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What Is Absolute Value?

Definition

The absolute value of a number is its distance from zero on the number line, always expressed as a non-negative number. The notation uses vertical bars: \(\color{blue}{|n|}\).

• \(\color{blue}{|7| = 7}\)  (7 is seven units from zero)
• \(\color{blue}{|-7| = 7}\)  (−7 is also seven units from zero)
• \(\color{blue}{|0| = 0}\)  (zero is zero units from itself)

Key Properties

The absolute value of any non-zero number is always positive. The absolute value of zero is zero. The absolute value of a negative number equals its positive counterpart: \(\color{blue}{|-n| = n}\) for any positive n.

Negative of an Absolute Value

The expression \(\color{blue}{-|n|}\) means “take the absolute value, then negate it.” The result is always non-positive.

• \(\color{blue}{-|-8| = -(8) = -8}\)
• \(\color{blue}{-|12| = -12}\)

Step-by-Step Summary

1. Identify the number inside the absolute-value bars.
2. Find its distance from zero — drop the negative sign if present.
3. If there is a negative sign outside the bars, negate the result.
4. To compare integers, use the number line: numbers farther right are greater.

Watch: What Is Absolute Value?

Math with Mr. J explains absolute value with clear examples and a number-line model:

Integers and Absolute Value – Worked Examples

Example 1: Evaluate \(\color{blue}{|-15|}\).

The distance \(\color{blue}{\text{ of } -15}\) from zero is 15. So \(\color{blue}{|-15| = 15}\).

Example 2: Evaluate \(\color{blue}{-|-8|}\).

First find the absolute value: \(\color{blue}{|-8| = 8}\). Then negate: \(\color{blue}{-8}\). So \(\color{blue}{-|-8| = -8}\).

Example 3: Evaluate \(\color{blue}{|-3| + |5|}\).

\(\color{blue}{|-3| = 3}\) and \(\color{blue}{|5| = 5}\). Adding: \(\color{blue}{3 + 5 = 8}\).

Example 4: Evaluate \(\color{blue}{|-10| – |4|}\).

\(\color{blue}{|-10| = 10}\) and \(\color{blue}{|4| = 4}\). Subtracting: \(\color{blue}{10 – 4 = 6}\).

More Practice: Absolute Value and Number Lines

Khan Academy shows how the number line connects to absolute value:

Exercises for Integers and Absolute Value

Evaluate each expression.

1. \(\color{blue}{|-9|}\)
2. \(\color{blue}{|6|}\)
3. \(\color{blue}{|-22|}\)
4. \(\color{blue}{-|-8|}\)
5. \(\color{blue}{|-3| + |5|}\)
6. \(\color{blue}{|-10| – |4|}\)

Answers

1. \(\color{blue}{9}\)
2. \(\color{blue}{6}\)
3. \(\color{blue}{22}\)
4. \(\color{blue}{-8}\)
5. \(\color{blue}{8}\)
6. \(\color{blue}{6}\)

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Want More Practice?

We haven’t published a worksheet built specifically for Integers and Absolute Value just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work — they’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Absolute Value Equations Worksheet
• Download Absolute Value Inequalities Worksheet
• Download Rational and Irrational Numbers Worksheet

Frequently Asked Questions

Can absolute value ever be negative?

No. Absolute value represents distance, which is never negative. The result of \(\color{blue}{|n|}\) is always greater than or equal to zero.

What is the absolute value of zero?

\(\color{blue}{|0| = 0}\). Zero is zero units from itself, so its absolute value is zero.

How is absolute value used in real life?

Absolute value models situations where direction doesn’t matter, only magnitude — such as calculating the distance between two towns, measuring a temperature change, or finding the size of a financial gain or loss.

Related Topics

• Adding and Subtracting Integers
• Multiplying and Dividing Integers
• Order of Operations
• Approximating Irrational Numbers
• Translate Phrases into an Algebraic Statement