Absolute value

Definition

Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of direction. This means that absolute value is always non-negative, and it transforms both positive and negative numbers into their positive equivalents. Understanding absolute value is crucial for working with integers, as it allows for comparisons between numbers without concern for their signs.

5 Must Know Facts For Your Next Test

1. The absolute value of a number 'x' is denoted as |x|, which equals 'x' if x is positive or zero and equals '-x' if x is negative.
2. For example, |3| = 3 and |-3| = 3, demonstrating that absolute value removes the sign of the number.
3. Absolute value can be used to solve equations involving integers, such as |x - 5| = 3, leading to multiple solutions.
4. In real-world scenarios, absolute value can represent quantities like temperature changes or distances traveled, where only the magnitude matters.
5. The concept of absolute value is important in functions and inequalities, particularly when determining intervals and ranges.

Review Questions

• How does the concept of absolute value help in comparing two integers?
  • Absolute value allows us to compare two integers by measuring their distances from zero without worrying about their signs. For instance, when comparing -7 and 5, we can look at their absolute values: |-7| = 7 and |5| = 5. This shows that -7 is further from zero than 5 is. Therefore, absolute value provides a clear method to understand the relative sizes of numbers on the number line.
• What are some practical applications of absolute value in everyday life?
  • Absolute value is used in various practical situations like calculating distances, evaluating temperature differences, and analyzing financial data. For example, if a temperature changes from -10 degrees to 5 degrees, the change can be expressed as an absolute value difference of |5 - (-10)| = |15| = 15 degrees. Such applications show how absolute value simplifies understanding quantities where direction does not matter.
• Evaluate the expression |x - 4| + |x + 2| for various integer values of x and analyze how these results demonstrate properties of absolute values.
  • By evaluating |x - 4| + |x + 2| for different integers like -5, 0, and 5, we can observe distinct outcomes based on the position of x relative to 4 and -2. For instance, if x = -5, then |(-5) - 4| + |(-5) + 2| = | -9 | + | -3 | = 9 + 3 = 12. If x = 0, then |0 - 4| + |0 + 2| = | -4 | + | 2 | = 4 + 2 = 6. These results illustrate how the behavior of absolute values can change depending on whether x is less than or greater than the critical points (4 and -2), highlighting its role in piecewise functions.

Related terms

integer: A whole number that can be positive, negative, or zero, but not a fraction or decimal.

number line: A visual representation of numbers in a straight line, where each point corresponds to a number and the distance from zero indicates the absolute value.

distance: The numerical measurement of how far apart two points are, often calculated as the absolute difference between those values.