|x|, also known as the absolute value of x, is a mathematical operation that represents the distance between the real number x and zero on the number line. It is a fundamental concept in algebra that is essential for understanding and working with absolute value functions.
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The absolute value of a number is always non-negative, as it represents the distance from zero on the number line.
The absolute value function, $f(x) = |x|$, has a V-shaped graph that is symmetric about the y-axis.
Absolute value can be used to define the distance between two points on the number line, as well as to solve certain types of equations and inequalities.
The absolute value of a positive number is equal to the number itself, while the absolute value of a negative number is the opposite of the number.
Absolute value is a fundamental concept in algebra and is essential for understanding and working with various mathematical topics, including functions, equations, and inequalities.
Review Questions
Explain how the absolute value of a number, |x|, represents the distance of that number from zero on the number line.
The absolute value of a number, |x|, represents the distance of that number from zero on the number line, regardless of whether the number is positive or negative. For example, the absolute value of 5 is 5, as it is 5 units away from zero on the number line. Similarly, the absolute value of -3 is also 3, as it is 3 units away from zero on the number line. This property of absolute value is essential for understanding and working with various mathematical concepts, such as absolute value functions and inequalities.
Describe the key features of the absolute value function, $f(x) = |x|$, and how it differs from other types of functions.
The absolute value function, $f(x) = |x|$, is a unique type of function that has a V-shaped graph that is symmetric about the y-axis. Unlike many other functions, the absolute value function is always non-negative, meaning that its range is restricted to values greater than or equal to zero. This property of the absolute value function is crucial for understanding its behavior and applications in various mathematical contexts, such as solving equations and inequalities involving absolute values.
Analyze how the absolute value of a number can be used to solve certain types of equations and inequalities, and explain the significance of this in the context of 3.6 Absolute Value Functions.
The absolute value of a number is essential for solving equations and inequalities that involve absolute values, which is a key focus of the 3.6 Absolute Value Functions topic. For example, an equation like $|x - 2| = 5$ can be solved by considering two cases: $x - 2 = 5$ and $x - 2 = -5$, which leads to the solutions $x = 7$ and $x = -3$. Similarly, an inequality like $|x + 4|
geq 3$ can be solved by considering the two cases where $x + 4
geq 3$ and $x + 4
leq -3$, which gives the solution set $x
leq -7$ or $x
geq -1$. Understanding how to use the properties of absolute value to solve these types of equations and inequalities is crucial for success in the 3.6 Absolute Value Functions topic.
An absolute value function is a function that takes the absolute value of its input, typically denoted as f(x) = |x|. These functions have a V-shaped graph and are always non-negative.
Signed Number: A signed number is a real number that has a positive or negative sign, indicating its direction on the number line. The absolute value of a signed number represents its distance from zero, regardless of its sign.
An inequality is a mathematical statement that compares two expressions, such as x < 5 or |x| ≥ 3. Absolute value is often used in inequalities to define the range of values for a variable.