Absolute Value

Definition

The absolute value of a number is the distance between that number and zero on the number line, regardless of whether the number is positive or negative. It represents the magnitude or size of a quantity without regard to its sign or direction.

5 Must Know Facts For Your Next Test

1. The absolute value of a number is always a non-negative value, as it represents the distance from zero.
2. Absolute value is denoted using vertical bars, such as |x|, which means the absolute value of x.
3. Absolute value is a key concept in linear equations, as it can be used to represent constraints or inequalities.
4. The absolute value function is a piecewise function, with different formulas for positive and negative inputs.
5. Absolute value can be used to find the distance between two points on a number line or in a coordinate plane.

Review Questions

• How can the absolute value of a number be used to represent constraints or inequalities in linear equations?
  • The absolute value of a number can be used to represent constraints or inequalities in linear equations. For example, the equation |x - 3| < 5 represents the set of all values of x that are within 5 units of 3 on the number line. This can be used to model real-world situations where a value must be within a certain range or distance from a target value.
• Explain how the absolute value function is defined and how it differs for positive and negative inputs.
  • The absolute value function is a piecewise function, with different formulas for positive and negative inputs. For a positive input x, the absolute value is simply x. For a negative input x, the absolute value is -x. This is because the absolute value represents the distance of a number from zero on the number line, regardless of its sign. The absolute value function is always non-negative, as it represents the magnitude or size of a quantity without regard to its direction.
• Describe how absolute value can be used to calculate the distance between two points on a number line or in a coordinate plane.
  • Absolute value can be used to calculate the distance between two points on a number line or in a coordinate plane. On a number line, the distance between two points a and b is simply |b - a|, as this represents the absolute value of the difference between the two points. In a coordinate plane, the distance between two points (x1, y1) and (x2, y2) is calculated using the formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, which can be rewritten as $\sqrt{|x_2 - x_1|^2 + |y_2 - y_1|^2}$, highlighting the use of absolute value to represent the magnitude of the differences between the coordinates.

Related terms

Signed Number: A number that has a sign (positive or negative) indicating its direction relative to zero on the number line.

Number Line: A visual representation of the set of real numbers, where each number is assigned a unique position.

Magnitude: The size or quantity of something, without regard to its direction or sign.