Absolute Value Functions, Equations, and Inequalities

Definition The absolute value of a real number $a$, written $\lvert a \rvert$, is the distance from $a$ to $0$ on the number line. For real numbers $a$ and $b$, the distance between them is $\lvert a - b \rvert$, and the order does not matter, so $\lvert a - b \rvert = \lvert b - a \rvert$. A distance is never negative, so $\lvert a \rvert \ge 0$ for every real $a$, and $\lvert a \rvert = 0$ only when $a = 0$.

Definition For every real number $x$,

When $x < 0$ the number $-x$ is positive, so the output is never negative. Applied to an expression, the same rule gives $\lvert x - c \rvert = x - c$ for $x \ge c$ and $\lvert x - c \rvert = c - x$ for $x < c$. Also $\sqrt{x^2} = \lvert x \rvert$ for every real $x$, because the square root sign means the nonnegative root.

Definition Let $a$, $h$ and $k$ be real numbers with $a \ne 0$. By the piecewise definition, the function $f(x) = a\lvert x - h \rvert + k$ equals $a(x - h) + k$ for $x \ge h$ and $-a(x - h) + k$ for $x < h$. So its graph consists of two half-lines that meet at the vertex $(h, k)$, with slope $-a$ on the left and slope $a$ on the right, and it is symmetric about the vertical line $x = h$. If $a > 0$ the graph opens upward and $k$ is the minimum value of $f$. If $a < 0$ it opens downward and $k$ is the maximum value.

Definition Let $u$ be an expression in $x$ and let $c > 0$ be a constant. The equation $\lvert u \rvert = c$ holds exactly when

For two expressions $u$ and $v$, the equation $\lvert u \rvert = \lvert v \rvert$ holds exactly when $u = v$ or $u = -v$.

Definition Let $u$ be an expression in $x$ and let $c > 0$ be a constant. Then

The same holds with $\le$ in place of $<$ and $\ge$ in place of $>$ on both sides. A “less than” inequality becomes a single chain, which is an “and” statement. A “greater than” inequality becomes an “or” statement, which cannot be written as one chain.

Definition For a center $a$ and a radius $r > 0$,

so the solutions form the open interval $(a - r, a + r)$ centered at $a$. On the other hand, $\lvert x - a \rvert > r$ holds exactly on $(-\infty, a - r) \cup (a + r, \infty)$, the outside of the closed interval $[a - r, a + r]$. With $\le$ the endpoints join the inside, and with $\ge$ they join the outside. Conversely, an open interval $(p, q)$ with $p < q$ is the solution set of $\lvert x - m \rvert < r$, where

Definition The rules for equations and inequalities above need $c > 0$. When the comparison value is $0$ or negative, use the fact that $\lvert u \rvert \ge 0$ always holds, with equality only when $u = 0$. This gives the table below, where $c$ is a negative constant and only inputs at which $u$ is defined count.

Definition When the right side contains the variable, as in $\lvert u \rvert = v$, the equation holds exactly when

Solving $u = v$ and $u = -v$ without the condition $v \ge 0$ can produce candidates that fail the original equation. Such a number is called an extraneous candidate and must be thrown out. Squaring both sides causes the same problem, since $u^2 = v^2$ only says that $\lvert u \rvert = \lvert v \rvert$. The safe habit is to substitute every candidate into the original equation.