How to Solve the Absolute Value of Rational Numbers?

How to solve the absolute value of rational numbers, 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.

You already know that the absolute value of an integer (color{blue}{text{ like } -7}) is 7. The same idea extends perfectly to rational numbers, fractions, decimals, and mixed numbers. Whether the number is (color{blue}{-frac{3}{4}}) or (color{blue}{-2.85}), the absolute value simply “removes the negative sign” and gives the distance from zero. This lesson shows you exactly how to handle every case.

What Is the Absolute Value of a Rational Number?

The absolute value of a rational number is its distance from zero on the number line. Because distance is always non-negative:

GED Math for Beginners 2026 The Ultimate Step by Step Guide to Preparing for the GED Math Test

Download

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.

Rated 4.34 out of 5 based on 106 customer ratings

Satisfied 92 Students

• The absolute value of any positive rational number is the number itself.
• The absolute value of any negative rational number is its positive counterpart.
• The absolute value of zero is zero.

In symbols: (color{blue}{|r| = r}) if (color{blue}{r ge 0}), and (color{blue}{|r| = -r}) if (color{blue}{r < 0}).

How to Find the Absolute Value of Rational Numbers

Fractions

Apply the absolute value to the entire fraction. The result has the same numerator and denominator, but is positive.

• (color{blue}{|frac{3}{4}| = frac{3}{4}})
• (color{blue}{|-frac{5}{8}| = frac{5}{8}})
• (color{blue}{|-frac{7}{3}| = frac{7}{3}})

Decimals

The absolute value of a decimal is simply the decimal with no negative sign.

• (color{blue}{|-0.6| = 0.6})
• (color{blue}{|-3.14| = 3.14})

Mixed Numbers

The absolute value of a mixed number removes the negative sign from the entire number.

• (color{blue}{|-2 frac{1}{4}| = 2 frac{1}{4}})
• (color{blue}{|3 frac{7}{10}| = 3 frac{7}{10}})

Comparing Absolute Values

When comparing (color{blue}{|a|}) and (color{blue}{|b|}), first evaluate each absolute value, then compare the resulting non-negative numbers.

(color{blue}{|-frac{5}{4}|}) vs (color{blue}{|frac{7}{8}|}): (color{blue}{frac{5}{4} = 1.25}) and (color{blue}{frac{7}{8} = 0.875}). Since (color{blue}{1.25 > 0.875}), we have (color{blue}{|-frac{5}{4}| > |frac{7}{8}|}).

Step-by-Step Summary

1. Identify whether the rational number is positive, negative, or zero.
2. If positive or zero: the absolute value equals the number.
3. If negative: the absolute value is the positive version of the number.
4. For comparisons: evaluate each absolute value, then compare the non-negative results.

Watch: Interpreting Absolute Value (Video Lesson)

Khan Academy explains what absolute value means in real-world contexts for 6th-grade level learners:

Worked Examples

Example 1: Find (color{blue}{|-frac{9}{10}|})

The number is negative, so the absolute value is its positive counterpart: (color{blue}{|-frac{9}{10}| = frac{9}{10}}).

Example 2: Compare (color{blue}{|-frac{3}{5}|}) and (color{blue}{|frac{1}{2}|}).

(color{blue}{|-frac{3}{5}| = frac{3}{5} = 0.6}) and (color{blue}{|frac{1}{2}| = 0.5}). Since (color{blue}{0.6 > 0.5}), we have (color{blue}{|-frac{3}{5}| > |frac{1}{2}|}).

Example 3: Find (color{blue}{|-4.75|})

(color{blue}{|-4.75| = 4.75})

Example 4: Order from least to greatest by absolute value: (color{blue}{-1 frac{1}{2}, frac{2}{3}, -frac{5}{4}, 0.1})

Absolute values: (color{blue}{frac{3}{2} = 1.5}), (color{blue}{frac{2}{3} &text{ approx }; 0.667}), (color{blue}{frac{5}{4} = 1.25}), (color{blue}{0.1}). Ordering: (color{blue}{0.1 < frac{2}{3} < frac{5}{4} < frac{3}{2}}).

More Practice: What Is Absolute Value? Video

Math with Mr. J covers absolute value with clear examples that extend to rational numbers:

Exercises

1. Find (color{blue}{|-frac{7}{8}|})
2. Find (color{blue}{|-0.45|})
3. Compare: (color{blue}{|-2 frac{3}{4}|}) ___ (color{blue}{|2 frac{1}{2}|})
4. Order by absolute value from least to greatest: (color{blue}{-0.9, frac{3}{4}, -frac{1}{8}, 1.1})
5. Which has the greater absolute value: (color{blue}{-frac{4}{3}}) or (color{blue}{frac{5}{4}})?
6. Find (color{blue}{|-frac{3}{5}| + |-frac{7}{10}|})

Answers

1. (color{blue}{frac{7}{8}})
2. (color{blue}{0.45})
3. (color{blue}{|-2 frac{3}{4}| = 2.75 > 2.5 = |2 frac{1}{2}|}), so (color{blue}{>})
4. Absolute values: (color{blue}{0.9, 0.75, 0.125, 1.1}). Order: (color{blue}{-frac{1}{8} < frac{3}{4} < -0.9 < 1.1})
5. (color{blue}{|-frac{4}{3}| = frac{4}{3} &text{ approx }; 1.333}) and (color{blue}{|frac{5}{4}| = 1.25}). So (color{blue}{-frac{4}{3}}) has the greater absolute value.
6. (color{blue}{frac{3}{5} + frac{7}{10} = frac{6}{10} + frac{7}{10} = frac{13}{10} = 1.3})

Frequently Asked Questions

Is |−a| always equal to |a|?

Yes. Since absolute value measures distance from zero, and both (color{blue}{a}) and (color{blue}{-a}) are the same distance from zero (just on opposite sides), (color{blue}{|a| = |-a|}) for any number (color{blue}{a}).

How do absolute values of fractions compare to those of whole numbers?

The same rules apply. (color{blue}{|-frac{1}{2}| = frac{1}{2} = 0.5}), which is less than (color{blue}{|3| = 3}). Just convert fractions to decimals when a numeric comparison is needed.

Can the absolute value of a rational number be irrational?

No. The absolute value of any rational number is a rational number (the non-negative version of that rational number).

Related Topics

• Absolute Value Definition
• Solve Word Problems of Absolute Value and Integers
• Using Number Lines to Compare and Order Rational Numbers
• Convert Rational Numbers to a Fraction
• Opposite Integers