GED Math: Absolute Value

In real situations, absolute value gives the size of a change or gap without a direction. The distance between two numbers is the absolute value of their difference: |a − b|. Use it for temperature changes, score gaps, elevations, and total distance walked (every leg counts as positive, even when you walk backward). Absolute error is the absolute value of the difference between a measurement and the true value. An overdraft works the same way: a balance of −$86 is thought of simply as “$86 short.”

1. What Absolute Value Means

The absolute value of a number is its distance from zero on the number line. It is written with two straight bars around the number, like |−5|. Because distance is never negative, the absolute value of any number is positive (or zero for 0 itself). That is why |5| = 5 and |−5| = 5: both numbers are five spaces from zero. In other words, absolute value keeps the size (magnitude) of a number and drops its sign. Test tip: a signed number shows direction from zero, but a number inside the bars is only about distance.

2. Absolute Value Bars as Grouping Symbols

In the order of operations, absolute value bars work like parentheses. First simplify whatever is inside the bars (add, subtract, multiply, or divide), then take the absolute value of that result, and only then do the operations outside. Be careful with a negative sign outside the bars: it is not part of the absolute value, so it stays. For example, −|7| = −7, because the bars give 7 and the negative sign in front of them is applied afterward.

3. Combining Absolute Value with Other Operations

When a problem has several operations, use the usual order: simplify inside bars (and parentheses) first, take absolute values, then handle exponents, multiplication and division, and finally addition and subtraction. Watch two traps. First, |a + b| is not the same as |a| + |b|: |−3 + 8| = |5| = 5, but |−3| + |8| = 11. Second, bars can sit inside parentheses, as in 63 ÷ (−|−9|). Work from the inside out: |−9| = 9, then −9, then 63 ÷ (−9) = −7. Decimals work the same way.

4. Absolute Value in Real Life: Distance, Difference, and Error

In real situations, absolute value gives the size of a change or gap without a direction. The distance between two numbers is the absolute value of their difference: |a − b|. Use it for temperature changes, score gaps, elevations, and total distance walked (every leg counts as positive, even when you walk backward). Absolute error is the absolute value of the difference between a measurement and the true value. An overdraft works the same way: a balance of −$86 is thought of simply as “$86 short.”

Common Mistakes to Avoid

• Leaving a negative answer: the absolute value of a number is never negative, so |−9| = 9.
• Dropping a negative sign outside the bars: −|7| = −7, not 7.
• Treating |a + b| as |a| + |b|. Add or subtract inside the bars first.
• Taking absolute value of each number before doing the operation inside the bars.
• Forgetting that the bars act like parentheses in the order of operations.
• Subtracting in the wrong order for distance and forgetting the absolute value, which gives a negative distance.
• In walking problems, counting backward legs as negative. Total distance walked adds every leg as a positive.

Frequently Asked Questions