How to Solve Word Problems of Absolute Value and Integers?

Absolute value and integers appear in many real-world contexts: temperature changes, sea-level elevations, financial gains and losses, and distances traveled. On the GED Math test, these word problems ask you to recognize when absolute value applies, set up a numeric expression, and solve it. This guide gives you a clear strategy and plenty of practice.

What Do These Word Problems Look Like?

Word problems involving absolute value and integers typically describe situations where direction or sign matters but only magnitude is asked about, or where you need to compare two signed quantities. Common contexts include:

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• Distance: “How far from zero is the temperature?” or “How many units apart are two locations on a number line?”
• Change: “What is the total change in value if it went up 3 then down 8?”
• Comparison: “Which account had the larger change in balance?”

Strategy for Solving These Problems

Step 1: Identify the integer(s)

Assign positive or negative values based on context. “Above sea level” = positive; “below sea level” = negative. “Profit” = positive; “loss” = negative.

Step 2: Decide if absolute value is needed

If the question asks for distance, magnitude of change, or how far — you need absolute value. If it asks which direction or asks for the actual value — you may not.

Step 3: Set up and evaluate the expression

Write the expression with absolute value bars if needed, evaluate any arithmetic inside the bars, then take the absolute value.

Step 4: Answer with context

Include units and make sure your answer matches the question asked.

Step-by-Step Summary

1. Assign positive/negative signs based on the problem context.
2. Write the integer expression described in the problem.
3. Use \(\color{blue}{| |}\) if the problem asks for magnitude, distance, or total change.
4. Evaluate inside the bars first, then apply absolute value.
5. State the answer with appropriate units.

Watch: Absolute Value Word Problems (Video Lesson)

Khan Academy works through word problems that apply absolute value to real-world integer situations:

Worked Examples

Example 1: A diver is at \(\color{blue}{-15}\) feet (below sea level) and rises to \(\color{blue}{-3}\) feet. What is the total distance traveled?

Distance = \(\color{blue}{|-3 – (-15)| = |-3 + 15| = |12| = 12}\) feet.

Example 2: The temperature at noon was \(\color{blue}{4^{\circ}F}\). By midnight it was \(\color{blue}{-9^{\circ}F}\). What was the total change in temperature?

Change = \(\color{blue}{|-9 – 4| = |-13| = 13^{\circ}F}\) decrease.

Example 3: Account A changed by \(\color{blue}{-$45}\) and Account B changed by \(\color{blue}{+$38}\). Which account had the greater magnitude of change?

\(\color{blue}{|-45| = 45}\) and \(\color{blue}{|38| = 38}\). Since \(\color{blue}{45 > 38}\), Account A had the greater magnitude of change.

Example 4: On a number line, Point P is at \(\color{blue}{-6}\) and Point Q is at \(\color{blue}{4}\). What is the distance between them?

Distance = \(\color{blue}{|4 – (-6)| = |4 + 6| = |10| = 10}\) units.

More Practice: Absolute Value Examples Video

Math with Mr. J provides more worked examples of evaluating and applying absolute value:

Exercises

1. A submarine is at \(\color{blue}{-200}\) feet. It surfaces to \(\color{blue}{0}\) feet. How far did it travel?
2. The high temperature was \(\color{blue}{12^{\circ}F}\) and the low was \(\color{blue}{-5^{\circ}F}\). What is the difference in temperature?
3. Two football teams gained and lost yards. Team A had a net of \(\color{blue}{-7}\) yards; Team B had \(\color{blue}{+5}\) yards. Which team had the greater magnitude of change?
4. On a number line, what is the distance between \(\color{blue}{-8}\) and \(\color{blue}{3}\)?
5. A stock was at \(\color{blue}{$50}\). It fell to \(\color{blue}{$32}\). What was the absolute change in price?
6. Which has greater absolute value: a loss of \(\color{blue}{$200}\) or a gain of \(\color{blue}{$175}\)?

Answers

1. \(\color{blue}{|0 – (-200)| = |200| = 200}\) feet.
2. \(\color{blue}{|12 – (-5)| = |17| = 17^{\circ}F}\).
3. \(\color{blue}{|-7| = 7 > 5 = |5|}\); Team A had the greater magnitude.
4. \(\color{blue}{|3 – (-8)| = |11| = 11}\) units.
5. \(\color{blue}{|32 – 50| = |-18| = $18}\).
6. \(\color{blue}{|-200| = 200 > 175 = |175|}\); the loss of $200 has greater absolute value.

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Frequently Asked Questions

Why use absolute value for distance?

Distance is always non-negative. Absolute value ensures we get the magnitude of the difference between two positions, regardless of which is larger. \(\color{blue}{|a – b| = |b – a|}\), so the order of subtraction doesn’t matter.

What is the difference between a change and the absolute value of a change?

A change can be negative (a decrease) or positive (an increase). The absolute value of a change is just the size of that change — it ignores direction. If a temperature drops 8 degrees, the change \(\color{blue}{\text{ is } -8}\) but the absolute change is 8.

How do I know when to use absolute value in a word problem?

Look for the words distance, how far, magnitude, total change, or any situation where only the size (not the direction) of a quantity matters. If the question cares about direction (rise or fall, profit or loss), you may not need absolute value.

Related Topics

• Absolute Value Definition
• Absolute Value of Rational Numbers
• Adding and Subtracting Integers
• Using Number Lines to Represent Integers
• Opposite Integers