Absolute Value Equations

Definition

An absolute value equation is a mathematical equation that involves the absolute value function, which represents the distance of a number from zero on the number line. These equations are used to solve for the unknown value that satisfies the given absolute value condition.

5 Must Know Facts For Your Next Test

1. Absolute value equations can be used to model real-world situations, such as finding the distance between two points or the magnitude of a vector.
2. Solving absolute value equations often involves considering two cases: one where the expression inside the absolute value is greater than or equal to zero, and one where it is less than zero.
3. Absolute value equations can be transformed into equivalent equations or inequalities by isolating the absolute value expression and applying the properties of absolute value.
4. The solution set for an absolute value equation may consist of one, two, or no solutions, depending on the specific equation and the values of the variables involved.
5. Graphing absolute value equations can provide a visual representation of the solution set and help in understanding the behavior of the equation.

Review Questions

• Explain how the absolute value function is used in the context of solving absolute value equations.
  • The absolute value function, $|x|$, represents the distance of the number $x$ from zero on the number line. In the context of solving absolute value equations, the absolute value is used to set up an equation where the unknown variable must satisfy a given distance or magnitude condition. For example, the equation $|x - 3| = 5$ can be solved by considering two cases: when $x - 3 extbackslash geq 0$ and when $x - 3 < 0$, which leads to the solutions $x = 8$ and $x = -2$.
• Describe the process of transforming an absolute value equation into an equivalent equation or inequality that can be solved more easily.
  • To transform an absolute value equation into an equivalent form, one can isolate the absolute value expression and then apply the properties of absolute value. For example, the equation $|2x - 1| = 4$ can be rewritten as $(2x - 1) = extbackslash pm 4$, which leads to the solutions $x = rac{5}{2}$ and $x = -rac{3}{2}$. Alternatively, the equation can be rewritten as two separate inequalities: $2x - 1 extbackslash geq 4$ and $2x - 1 extbackslash leq -4$, which also yields the same solutions.
• Analyze the relationship between the graph of an absolute value equation and the solutions to the equation.
  • The graph of an absolute value equation is a V-shaped function that opens upward or downward, depending on the coefficient of the variable. The vertices of the V-shaped graph represent the points where the absolute value expression is equal to zero, which correspond to the solutions of the absolute value equation. The solution set for an absolute value equation can be determined by identifying the x-intercepts of the graph, which represent the values of the variable that satisfy the equation. The number of solutions depends on the specific equation and the values of the variables involved.

Related terms

Absolute Value Function:

The absolute value function, denoted as $|x|$, represents the distance of the number $x$ from zero on the number line, regardless of whether $x$ is positive or negative.

Inequality:

An inequality is a mathematical statement that compares two expressions using symbols such as $<$, $>$, $ extbackslash leq$, or $ extbackslash geq$, and absolute value equations can be rewritten as inequalities.

Graphing:

Absolute value equations can be graphed on the coordinate plane, and the graph of an absolute value equation is a V-shaped function that opens upward or downward.