key term - Absolute Value Equation

Definition

An absolute value equation is a mathematical equation that involves the absolute value function. The absolute value of a number represents the distance of that number from zero on the number line, regardless of whether the number is positive or negative.

5 Must Know Facts For Your Next Test

1. Absolute value equations can have one, two, or no solutions, depending on the structure of the equation.
2. To solve an absolute value equation, you can use the property that $|x| = a$ if and only if $x = a$ or $x = -a$.
3. Absolute value equations can be linear or quadratic, and the solution methods vary depending on the equation type.
4. Graphically, the solution to an absolute value equation represents the point(s) where the graph of the absolute value function intersects a horizontal line.
5. Absolute value equations are often used to model real-world situations involving distances, magnitudes, or deviations from a reference point.

Review Questions

• Explain how the absolute value function relates to the concept of distance on the number line.
  • The absolute value function, $|x|$, represents the distance of the input $x$ from the origin (0) on the number line. Regardless of whether $x$ is positive or negative, the absolute value function always returns a non-negative value that corresponds to the distance of $x$ from 0. This property of the absolute value function is crucial in understanding and solving absolute value equations, as the solutions to these equations represent the point(s) where the distance from 0 satisfies the given equation.
• Describe the different types of solutions that an absolute value equation can have and the factors that determine the number of solutions.
  • Absolute value equations can have one, two, or no solutions, depending on the structure of the equation. If the equation is of the form $|x| = a$, where $a$ is a non-negative constant, then the equation will have two solutions: $x = a$ and $x = -a$. If the equation is of the form $|x| \geq a$ or $|x| \leq a$, then the equation will have one solution. If the equation is of the form $|x| > a$ or $|x| < a$, then the equation will have no solutions. The number of solutions is determined by the relationship between the absolute value expression and the constant on the other side of the equation.
• Analyze how the graph of an absolute value equation can be used to visualize and understand the solutions to the equation.
  • The graph of an absolute value equation can provide valuable insights into the solutions of the equation. Graphically, the solution to an absolute value equation represents the point(s) where the graph of the absolute value function intersects a horizontal line. For example, the graph of the equation $|x| = 5$ would intersect the horizontal line $y = 5$ at the points $x = 5$ and $x = -5$, corresponding to the two solutions of the equation. Similarly, the graph of $|x| \leq 5$ would show the range of values for $x$ where the absolute value is less than or equal to 5, providing a visual representation of the solution set. Understanding the relationship between the graph of an absolute value equation and its solutions can help students develop a deeper understanding of these types of equations.

Related terms

Absolute Value Function:

The absolute value function, denoted as $|x|$, returns the distance of the input $x$ from zero on the number line.

Linear Equation:

A linear equation is an equation in which the variables are raised to the first power and the variables are combined using only addition and subtraction.

Quadratic Equation:

A quadratic equation is a polynomial equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.