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Negation and Absolute Value

When you attach a minus sign to any number, you negate that number. Negating a number means changing its sign to the opposite sign, so

  • Attaching a minus sign to a positive number makes it negative.

  • Attaching a minus sign to a negative number makes it positive. The two adjacent (side-by-side) minus signs cancel each other out.

  • Attaching a minus sign to 0 doesn’t change its value, so –0 = 0.

In contrast to negation, placing two bars around a number gives you the absolute value of that number. Absolute value is the number’s distance from 0 on the number line — that is, it’s the positive value of a number, regardless of whether you started out with a negative or positive number:

  • The absolute value of a positive number is the same number.

  • The absolute value of a negative number makes it a positive number.

  • Placing absolute value bars around 0 doesn’t change its value, so |0| = 0.

  • Placing a minus sign outside absolute value bars gives you a negative result — for example, –|6| = –6, and –|–6| = –6.

Sample questions

  1. Negate the number 7.

    –7. Negate 7 by attaching a negative sign to it: –7.

  2. Find the negation of –3.

    3. The negation of –3 is – (–3). The two adjacent minus signs cancel out, which gives you 3.

  3. What’s the negation of 7 – 12?

    5. First do the subtraction, which tells you 7 – 12 = –5. To find the negation of –5, attach a minus sign to the answer: –(–5). The two adjacent minus signs cancel out, which gives you 5.

  4. What number does |9| equal?

    9. The number 9 is already positive, so the absolute value of 9 is also 9.

  5. What number does |–17| equal?

    17. Because –17 is negative, the absolute value of –17 is 17.

  6. Solve this absolute value problem: –|9 – 13| = ?

    –4. Do the subtraction first: 9 – 13 = –4, which is negative, so the absolute value of –4 is 4. But the minus sign on the left (outside the absolute value bars in the original expression) negates this result, so the answer is –4.

Practice questions

  1. Negate each of the following numbers and expressions by attaching a minus sign and then canceling out minus signs when possible:

    a. 6
    b. –29
    c. 0
    d. 10 + 4
    e. 15 – 7
    f. 9 – 10
  2. Solve the following absolute value problems:

    a. |7| = ?
    b. |–11| = ?
    c. |3 + 15| = ?
    d. –|10 – 1| = ?
    e. |1 – 10| = ?
    f. |0| = ?

Following are answers to the practice questions:

  1. Negating numbers

    a. –6. To negate 6, attach a minus sign: –6.
    b. 29. To negate –29, attach a minus sign: –(–29). Now cancel adjacent minus signs: –(–29) = 29.
    c. 0. Zero is its own negation.
    d. –14. Add first: 10 + 4 = 14, and the negation of 14 is –14.
    e. –8. Subtract first: 15 – 7 = 8, and the negation of 8 is –8.
    f. 1. Begin by subtracting: 9 – 10 = –1, and the negation of –1 is 1.
  2. Absolute value problems

    a. |7| = 7. Seven is already positive, so the absolute value of 7 is also 7.
    b.    |–11| = 11. The number –11 is negative, so the absolute value of –11 is 11.
    c.    |3 + 15| = 18. First, do the addition inside the absolute value bars: 3 + 15 = 18, which is positive. The absolute value of 18 is 18.
    d.    –|10 – 1| = –9. First do the subtraction: 10 – 1 = 9, which is positive. The absolute value of 9 is 9. You have a negative sign outside the absolute value bars, so negate your answer to get –9.
    e.    |1 – 10| = 9. Begin by subtracting: 1 – 10 = –9, which is negative. The absolute value of –9 is 9.
    f.    |0| = 0. The absolute value of 0 is 0.
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