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The absolute value of an integer or number refers to the distance of a number from zero or origin in a number line. It is expressed as |a|, which states the magnitude of any integer or number ‘a’. The absolute value of a positive or negative integer or number will be the real numbers, regardless of the sign. It is denoted by two vertical lines |a|, which is known as the modulus of ‘a’.
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Keywords: Absolute value; absolute value formula; mod; modulus; integers; number line; real numbers
Also Read: Determinants
Absolute Value
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The absolute value of an integer or number is the actual distance of that integer from zero, in a number line. Thus, the absolute value is always positive and not a negative number.
The symbol of absolute value is expressed by the modulus symbol, ‘| |’, with the numbers between it. For example, the absolute value of 8 is denoted as |8|.
|x| is read as 'mod x' or 'modulus of x.'
The distance of any integer or number from the origin on the number line is the absolute value of that number. It also displays the polarity of the number whether it is positive or negative. It cannot be negative as it shows the distance and the distance can’t be negative. So, it is always positive.
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| Related Articles | ||
|---|---|---|
| Relations and functions | Binary operations | Onto functions |
| Real valued functions | Polynomials | Linear programme |
Absolute Value Formula
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The absolute values are defined like the following:
- |x| = +x for x > 0
- |x| = - x for x < 0
There is no absolute value for zero as the absolute value changes the sign of the numbers into positive and zero has no sign.
If the integer or number is positive then it will result in a positive number only. And if the integer or number is negative, then the modulus of that integer or number will also be a positive number. It is symbolized as |n|, where n is an integer.
Absolute Value Properties
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| Sl.No. | Absolute value properties | Expression |
|---|---|---|
| 1 | Non negativity | | x | ≥ 0 |
| 2 | Positive definiteness | | x | = 0 ↔ a = 0 |
| 3 | Multiplicativity | | x × y| = |x| × |y| |
| 4 | Subadditivity | | x + y| ≤ | x | + | y | |
| 5 | Symmetry | |-x| = |x| |
| 6 | Identity of indiscernible (comparable to positive definiteness) | | x – y | = 0 ↔ a = b |
| 7 | Triangle inequality (comparable to subadditivity) | | x – y | ≤ | x – z | + | z – x | |
| 8 | Preservation of division (comparable to multiplicativity) | | x / y| = | x | / | y | |
| 9 | Equivalent to subadditivity | | x – y | ≥ | | x | – | y | | |
Absolute Value
Things to Remember
- The absolute value of any integer or number signifies the distance of the number from zero in a number line.
- The symbol to characterize absolute value is |x|, where x is an integer.
- Absolute value signifies an integer’s distance from zero on the number line without taking direction into account. An integer’s absolute value is never negative.
- Absolute values are necessary because sometimes we just need to use positive values, and the absolute value is a handy tool for this. If we see an absolute value in a problem or equation, it signifies that everything inside it is always positive. Absolute values are often useful in distance questions and are sometimes used in inequalities.
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Sample Questions
Ques: What is the absolute value of a number -12/5? (2 marks)
Ans: |-12/5|
|-12/5| = 12/5
Thus, the absolute value of a number -12/5 is 12/5
Ques: Find the absolute value for the following numbers: (3 marks)
(a) |-1/2|
(b) |72|
(c) |3/4|
Ans: The absolute value of the numbers is:
(a) |-1/2| = ½
(b) |72| = 72
(c) |3/4| = ¾
Ques: What is the value of 2 | 5x – 1 | if x = – 2?(3 marks)
=> 2| 5x – 1| = 2| -10 – 1|
=>2| 5x – 1| = 2| -11|
=>2| 5x – 1| = 2 × 11
=>2| 5x – 1| = 22
Ques: Solve 4|x – 2| = 16 (3 marks)
=>|x – 2| = 16/4
=> x – 2| = 4
=>x – 2 = 4 or x – 2 = – 4
=>x = 4 + 2 or x = – 4 + 2
=>x = 6 or – 2
Ques: Solve |5 – 3x| = 12(3 marks)
Ans: |5 – 3x| = 12
5 – 3x = 12 or 5 – 3x = –12
–3x = 7 or –3x = –7
x = –7/3 or x = 17/3
Ques: Solve |4x – 3|= |x + 6| (3 marks)
Ans: |4x – 3|= |x + 6|
⇒4x – 3 = x + 6 or ⇒ 4x – 3 = – (x + 6)
⇒ 3x = 9 or ⇒ 4x – 3 = – x – 6
⇒ x = 3 or ⇒ 5x = –3
⇒ x = 3 or ⇒ x = –3/5
Ques: Solve |2x+3|<6 (3 marks)
Ans: |2x+3|<6
⇒–6 < 2x+3 < 6
⇒–6 –3 <2x + 3 – 3 < 6 –3
⇒–9 < 2x < 3
⇒−9/2 < x < 3/2
Thus, the solution to the given absolute value inequality is the interval (−9/2 < x < 3/2).
Ques: Find all the possible values of xx which satisfy |2x|=16 (2 marks)
Ans: Given, |2x|=16
⇒2x=±16
⇒2x=16or 2x=–16
⇒x=8or x=–8
Ques: Simplify –|–8+4–3| (2 marks)
Ans: Given, –|–8+4–3|
⇒–|–11+4|
⇒–|–7|
⇒–7
Ques. What is the absolute value of 20–36(2) +2(40–37)? (2 marks)
Ans: Given, |20–36(2) +2(40–37) |
⇒|20–72+6|
⇒|–46|=46
Ques. Solve the equation given below |x+4|=5 (2 marks)
Ans: |x+4|=5
⇒x+4=±5
⇒x+4=5⇒x+4=5 or x+4=–5x+4=–5
⇒x=1⇒x=1 or x=–9x=–9
Ques. Solve the equation |x|=x+2. (2 marks)
Ans: |x|=x+2
⇒x=±(x+2)
⇒x=x+2⇒x=x+2 or ⇒x=–x–2⇒x=–x–2
⇒x=–1
Ques: Solve |2x – 5| = x + 4 (3 marks)
Ans: 2x – 5 = x + 4 or 2x – 5 = -(x + 4)
2x – x = 5 + 4 or 2x – 5 = -x – 4
x = 9 or 2x + x = 5 – 4
3x = 1
x = 1/3
Thus, the possible values of x after solving the equation |2x – 5| = x + 4 are 9, 1/3.
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