The expression |x| = -x if x < 0 defines the absolute value of a negative number. Absolute value represents the distance of a number from zero on the real number line, and when the number is negative, its absolute value is obtained by negating it. This definition emphasizes how absolute value transforms negative values into positive ones, reflecting their distance from zero.
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The absolute value function is defined piecewise: |x| = x when x ≥ 0 and |x| = -x when x < 0.
For any negative number, its absolute value will always be positive since it reflects distance, which cannot be negative.
When graphed, the absolute value function forms a V shape, with the vertex at the origin (0,0).
Absolute values are essential in solving equations and inequalities as they help identify solutions across different intervals.
The concept of absolute value extends beyond numbers to other mathematical objects like vectors, where it represents magnitude.
Review Questions
How does the definition of absolute value change when applied to negative numbers, specifically using |x| = -x if x < 0?
When applying the definition of absolute value to negative numbers, |x| becomes -x if x is less than zero. This means that instead of maintaining its original sign, the negative number is converted into a positive one by negating it. For example, if x is -5, then |x| = -(-5) = 5. This illustrates that absolute value always provides the non-negative distance from zero, regardless of whether the original number was negative.
Discuss how understanding |x| = -x if x < 0 can assist in solving equations involving absolute values.
Understanding |x| = -x if x < 0 is crucial for solving equations involving absolute values because it allows you to properly analyze different cases based on the sign of x. When you encounter an equation like |x| = 3, recognizing that x could be either positive or negative helps you set up two separate cases: x = 3 and x = -3. This approach ensures you capture all possible solutions by accounting for both scenarios where x is less than or greater than zero.
Evaluate the implications of using absolute values in real-world contexts and how |x| = -x if x < 0 plays a role in those applications.
In real-world contexts such as physics or economics, understanding absolute values is essential when dealing with quantities that can have both positive and negative values, such as temperature changes or financial gains/losses. The rule |x| = -x if x < 0 highlights that while losses may be represented as negative numbers, their impact should be considered in terms of magnitude. For instance, a temperature drop of -10 degrees should be understood as an increase in distance from a baseline (like zero degrees), thus converting it into an actionable figure that reflects the true change rather than just its sign.
The absolute value of a number is its non-negative value without regard to its sign, indicating how far the number is from zero.
Distance on the Real Line: The distance between two points on the real number line is the absolute difference between their values, which always results in a non-negative outcome.
A visual representation of numbers in a straight line, where each point corresponds to a number and helps in understanding concepts like absolute value and distance.