How to Solve Word Problems of Absolute Value and Integers?

How to solve word problems of absolute value and integers, 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.

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.

Writing an Absolute Value Equation From a Word Problem

Some word problems give a target number and say how far a value can be from it. That is a distance, so absolute value fits. Use this pattern: |value − target| = allowed distance.

Example 1

A thermostat is set to 70°F. The room temperature can end up 3 degrees away from that setting in either direction. What are the highest and lowest temperatures?

• Let t be the temperature. Write |t − 70| = 3.
• Either t − 70 = 3 or t − 70 = −3.
• So t = 73 or t = 67.

The temperature can reach 73°F at the most and 67°F at the least.

Example 2

A bag of flour should weigh 16 ounces, and a bag passes inspection if it is within 0.5 ounce of that. Write |w − 16| ≤ 0.5. This means w is between 15.5 and 16.5 ounces, including both. A 15.7 ounce bag passes, because |15.7 − 16| = 0.3, and 0.3 is less than 0.5.

Quick check

Plug each answer back in. For Example 1, |73 − 70| = 3 and |67 − 70| = 3, so both work. If you get only one answer to an equation like this, look again for the second one.

Frequently Asked Questions

What are absolute value word problems in 6th grade?

In 6th grade, absolute value word problems ask how far a number is from zero or how large an amount is, ignoring direction. A typical question: a bank account is overdrawn by $25, shown as −25. Its absolute value, |−25| = 25, tells you the size of the debt is $25, even though the balance itself is negative.

How do you write absolute value equations from word problems?

Find the target value and the allowed distance from it, then write |x − target| = distance. For example, a bakery wants loaves that weigh 500 grams, give or take 15 grams. The extreme weights satisfy |x − 500| = 15. If any weight within that range is fine, write the inequality |x − 500| ≤ 15 instead.

How do you solve absolute value equation word problems?

Split the equation into two cases, one positive and one negative, and solve each. Say a thermostat keeps a room within 4 degrees of 70°F, and you want the extremes: |x − 70| = 4. Then x − 70 = 4 gives x = 74, and x − 70 = −4 gives x = 66, so the temperature stays between 66°F and 74°F.

What are some examples of word problems for absolute value?

A diver is at −35 feet and a seagull flies at 18 feet; how far apart are they? |18 − (−35)| = 53 feet. A stock price changes by −$9 in a day; how big was the change? |−9| = $9. A city’s temperature goes from −4°C to 11°C; by how much did it rise? |11 − (−4)| = 15 degrees.

What is the absolute value of an integer?

The absolute value of an integer is its distance from zero on the number line, so it is always zero or positive. Both 14 and −14 sit 14 units from zero, so |14| = |−14| = 14. In word problems, this is why a loss of 14 yards and a gain of 14 yards describe a change of the same size.

Related Topics

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