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

How to solve integers and absolute value problems, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

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})

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

What are integers and absolute value?

Integers are the whole numbers, their opposites and zero: …, −3, −2, −1, 0, 1, 2, 3, …. The absolute value of a number is its distance from zero on a number line, written with bars. So |−6| = 6 and |6| = 6, because both numbers sit 6 units from zero. Distance is never negative, so an absolute value is always zero or positive.

What is the difference between an integer and its absolute value?

An integer tells you both direction and size, while its absolute value tells you only the size. The integer −12 means 12 units to the left of zero, but |−12| = 12 just measures the distance. In real life, a temperature of −12 degrees and 12 degrees are on opposite sides of zero, yet both are 12 degrees away from it.

How do you add integers using absolute value?

If the signs match, add the absolute values and keep the shared sign: −4 + (−9) = −13. If the signs differ, subtract the smaller absolute value from the larger and use the sign of the number with the larger absolute value. For −8 + 5, find 8 − 5 = 3; since 8 is larger and negative, the answer is −3.

Is an integer and its absolute value always opposites?

No. For positive integers and zero, the absolute value is the same number: |7| = 7 and |0| = 0. Only for negative integers is the absolute value the opposite, as in |−7| = 7. A quick way to remember it: absolute value never changes a positive number, and it removes the negative sign from a negative one.

How do you solve integers and absolute value problems?

Treat absolute value bars like parentheses: simplify inside them first, take the absolute value, then finish the rest of the expression. For |3 − 10| + 2, work inside to get |−7|, which is 7, and then 7 + 2 = 9. Watch for a negative sign outside the bars, since −|−4| equals −4, not 4.

Related Topics

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