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. It is always non-negative, meaning it cannot be less than zero, and is denoted by two vertical bars surrounding the number, like this: $$|x|$$. This concept is crucial for understanding intervals, inequalities, and functions that involve distances or magnitudes.

5 Must Know Facts For Your Next Test

1. The absolute value of a number is defined as $$|x| = x$$ if $$x \geq 0$$ and $$|x| = -x$$ if $$x < 0$$.
2. Absolute values can be used to express distances in real-world situations, such as how far a temperature is from freezing.
3. When solving inequalities involving absolute values, you often need to split the problem into two cases based on the definition of absolute value.
4. The graph of the absolute value function $$f(x) = |x|$$ forms a 'V' shape with its vertex at the origin (0,0).
5. In intervals, the notation $$[a, b]$$ indicates that both endpoints are included, while $$(a, b)$$ indicates that neither endpoint is included; absolute value can help define these boundaries.

Review Questions

• How does the definition of absolute value relate to the concept of distance on the number line?
  • Absolute value represents the distance from zero on the number line regardless of direction. For any real number $$x$$, its absolute value $$|x|$$ quantifies how far it is from zero, making it an essential tool in measuring distances. This relationship helps us understand not only individual numbers but also intervals and their properties.
• In what ways can absolute values influence solving inequalities and forming intervals?
  • When dealing with inequalities that involve absolute values, we must consider different cases based on whether the expression inside the absolute value is positive or negative. For instance, an inequality like $$|x - 3| < 5$$ leads to two separate inequalities: $$-5 < x - 3 < 5$$. This case division is crucial in forming valid intervals that satisfy the original inequality.
• Analyze how understanding absolute value enhances problem-solving in real-world contexts like temperature fluctuations.
  • Understanding absolute value allows us to effectively analyze scenarios like temperature changes where we care about how far a temperature is from a reference point like freezing (0°C). For example, if the temperature drops to -3°C, the absolute value $$|-3| = 3$$ tells us it is 3 degrees below freezing. This perspective aids in interpreting data accurately and making decisions based on how extreme conditions are relative to normal benchmarks.

Related terms

Distance: A numerical measure of how far apart two points are in space, which can also be represented using absolute value when referring to distances on a number line.

Intervals: A range of numbers between two endpoints, which can include or exclude the endpoints, and can be represented using absolute values to express distances from a center point.

Inequality: A mathematical expression that compares two values or expressions, indicating whether one is greater than, less than, or equal to the other, often involving absolute values to express limits.