Questions: Absolute Value

Question 1

What is |3 − 8|?

• A. −5
• B. 5
• C. 11
• D. −11

Answer: (B)

Evaluate the expression inside first: 3 − 8 = −5. Then take the absolute value (distance from zero): |−5| = 5. The most common error is to 'drop the negative sign' on the 3 or 8 before subtracting, which gives the wrong answer. The bars act like parentheses — everything inside is computed first.

Question 2

If x = −6, what is |x + 2|?

• A. 8
• B. −4
• C. 4
• D. −8

Answer: (C)

Substitute x = −6 and evaluate inside first: −6 + 2 = −4. Then take the absolute value: |−4| = 4. Option A (8) comes from incorrectly treating |x| as 6 before adding 2. Option B keeps the negative sign, which is wrong because absolute value is always non-negative. The key rule: evaluate the expression inside the bars completely, then report the distance from zero.

Question 3

For any two real numbers a and b, |a − b| = |b − a|.

• T. True
• F. False

Answer: True

Absolute value measures distance on the number line, and distance is symmetric — the distance from a to b is the same as the distance from b to a. Algebraically, b − a = −(a − b), and |−(a − b)| = |a − b| since absolute value ignores the sign. This is why order doesn't matter when computing distance between two numbers.

Question 4

If x is a positive number, then |x − 15| = x − 15.

• T. True
• F. False

Answer: False

This is only true when x ≥ 15. When x is positive but less than 15 — say x = 4 — then x − 15 = −11, and |x − 15| = |−11| = 11, not −11. Absolute value depends on whether the expression inside is non-negative, not on whether x itself is positive. A positive x does not guarantee a non-negative expression inside the bars.

Question 5

Why is 'just drop the negative sign' a dangerous shortcut for evaluating absolute value?

Model answer: The shortcut only works when the input is a bare negative number like |−5|. It fails on expressions like |3 − 8|, where you must evaluate the expression first (getting −5) before taking the distance. 'Dropping the negative sign' on 3 or 8 individually gives the wrong answer. Absolute value means distance from zero of the entire expression inside the bars — not a piecewise operation on individual terms.

The shortcut encourages students to act on the negative sign before computing, but absolute value applies to the final result inside the bars. |3 − 8| ≠ 3 + 8. The correct process is: evaluate inside completely, identify the result, then ask how far that result is from zero. This matters especially for expressions with variables, like |a − b|, where you cannot determine the sign without knowing the values.