|x|, or the absolute value of x, represents the distance of the number x from zero on the real number line, regardless of direction. This means that |x| is always a non-negative number, making it a crucial concept in understanding distances and magnitudes. Absolute value helps in comparing numbers and solving equations that involve distance or magnitude without concern for their sign.
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|x| = x when x is greater than or equal to 0, and |x| = -x when x is less than 0.
The graph of |x| is a V-shaped figure on the coordinate plane, reflecting the property that negative and positive inputs yield the same output.
Absolute value can be used to define distance in more complex spaces, not just on the real line.
|x| can be involved in solving equations and inequalities, where it expresses conditions that account for both positive and negative solutions.
The absolute value function is continuous everywhere and has a sharp corner at x=0, indicating a change in slope.
Review Questions
How does the concept of absolute value relate to measuring distance on the real number line?
|x| measures how far a number x is from zero on the real number line. It represents this distance without considering the direction, which is essential for understanding how far apart numbers are from each other. For instance, both |3| and |-3| equal 3, indicating they are the same distance from zero, even though one is positive and the other is negative.
In what ways can absolute value be applied to solve equations and inequalities?
Absolute value can simplify solving equations by providing a way to handle both positive and negative solutions. For example, in the equation |x - 2| = 5, we split it into two cases: x - 2 = 5 and x - 2 = -5. Similarly, when working with inequalities involving absolute values, we can create intervals that represent all possible solutions based on distance from a certain point.
Evaluate how absolute value affects graphing functions on a coordinate plane and its implications for function continuity.
When graphing the function y = |x|, it results in a V-shape that reflects both positive and negative inputs equally. The vertex of this graph is at (0,0), which shows that there’s no output less than zero. This V-shape indicates a continuous function everywhere except at x=0 where it has a sharp corner, demonstrating how absolute values influence both visual representation and properties like continuity in mathematical functions.
Related terms
Distance: A numerical measurement of how far apart two points are, which can be represented using absolute values.
Real Numbers: All the numbers that can be found on the number line, including both rational and irrational numbers.
Inequality: A mathematical statement that compares two values, often involving absolute values to express distance or range.