2.1.1 Absolute Value Equations¶

Solving equations containing absolute value expressions using algebraic methods and case analysis.

定义¶

An absolute value equation is an equation that contains one or more absolute value expressions. The absolute value of a real number $x$, denoted as $|x|$, is defined as: $$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$ \nTo solve absolute value equations of the form $|f(x)| = k$ where $k \geq 0$, we use the fundamental principle that the expression inside the absolute value can equal either $k$ or $-k$. This leads to solving two separate cases: $f(x) = k$ and $f(x) = -k$. For equations of the form $|f(x)| = |g(x)|$, we solve $f(x) = g(x)$ and $f(x) = -g(x)$. The solution set is the union of all solutions from each case, and we must verify that each solution satisfies the original equation.

核心公式¶

$|x| = k \Rightarrow x = k \text{ or } x = -k$ (where $k \geq 0$)
$|f(x)| = k \Rightarrow f(x) = k \text{ or } f(x) = -k$ (where $k \geq 0$)
$|f(x)| = |g(x)| \Rightarrow f(x) = g(x) \text{ or } f(x) = -g(x)$
$|f(x)| = g(x)$ requires $g(x) \geq 0$, then $f(x) = g(x) \text{ or } f(x) = -g(x)$
$|a - b| = |c - d| \Rightarrow (a-b)^2 = (c-d)^2$ (squaring method)

易错点¶

⚠️ Forgetting to check that solutions satisfy the original equation, especially when the right side is an expression rather than a constant (e.g., in $|x-2| = x+1$, students may forget to verify that $x+1 \geq 0$)
⚠️ Incorrectly handling negative constants: treating $|f(x)| = -5$ as having solutions when in fact no solutions exist since absolute values are always non-negative
⚠️ Making sign errors when setting up the second case: writing $f(x) = -k$ but then solving it incorrectly, or confusing which expression gets the negative sign in equations like $|f(x)| = |g(x)|$
⚠️ Failing to consider the domain restrictions when the right side is a function: in $|f(x)| = g(x)$, forgetting that we must have $g(x) \geq 0$ for any solutions to exist