Absolute Value Equations

Definition

An absolute value equation is a mathematical equation that involves the absolute value of a variable or expression. The absolute value of a number represents its distance from zero on the number line, regardless of its sign. Solving absolute value equations is a key skill in the context of 2.7 Solve Absolute Value Inequalities.

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 isolate the absolute value expression and then consider the two cases: when the expression is positive and when it is negative.
3. Graphically, the solutions to an absolute value equation represent the points where the graph of the absolute value function intersects the $x$-axis.
4. Absolute value equations can be linear or quadratic, and the solution methods vary depending on the type of equation.
5. Solving absolute value equations is a crucial skill for understanding and solving absolute value inequalities, which involve comparing the absolute value of an expression to a constant.

Review Questions

• Explain the process of solving a linear absolute value equation.
  • To solve a linear absolute value equation of the form $|ax + b| = c$, where $a$, $b$, and $c$ are constants, you can follow these steps: 1) Isolate the absolute value expression by subtracting $b$ from both sides: $|ax| = c - b$. 2) Consider the two cases: when $ax$ is positive and when $ax$ is negative. For the positive case, solve for $x$ using $ax = c - b$. For the negative case, solve for $x$ using $ax = -(c - b)$. 3) The final solution is the union of the two sets of solutions.
• Describe the graphical interpretation of the solutions to an absolute value equation.
  • The solutions to an absolute value equation $|f(x)| = g(x)$ represent the points where the graph of the absolute value function $|f(x)|$ intersects the graph of the function $g(x)$. Graphically, this means finding the $x$-coordinates where the absolute value graph and the other graph cross the $x$-axis. The number of solutions depends on the structure of the equation, as there can be one, two, or no points of intersection.
• Analyze the similarities and differences between solving linear and quadratic absolute value equations.
  • Both linear and quadratic absolute value equations involve isolating the absolute value expression and considering the positive and negative cases. However, the solution methods differ. For a linear absolute value equation $|ax + b| = c$, you can solve for $x$ directly in each case. For a quadratic absolute value equation $|ax^2 + bx + c| = d$, you may need to use techniques like factoring, completing the square, or the quadratic formula to find the solutions. Additionally, quadratic absolute value equations can have up to four solutions, while linear absolute value equations have at most two solutions.

Related terms

Absolute Value:

The absolute value of a number is its distance from zero on the number line, regardless of its sign. It is denoted by vertical bars, such as |x|.

Linear Equation:

A linear equation is an equation that can be written in the form $ax + b = 0$, where $a$ and $b$ are constants and $x$ is the variable.

Quadratic Equation:

A quadratic equation is an equation that can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $x$ is the variable.