Absolute value

Japanese school year: Math I

What you learn

Learn how to solve equations and inequalities containing absolute values by considering different cases based on distance from the origin. This technique is widely used when evaluating error tolerances and intervals on a number line. Prior familiarity with linear equations, inequalities, and number lines will support your understanding.

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Key points

This formula defines the absolute value and shows how to remove the bars. If the real number $a$ is non-negative, keep it as $a$ ; if negative, reverse its sign to $-a$ .

|a| = \begin{cases} a & (a \ge 0) \\ -a & (a < 0) \end{cases}

This formula gives the range of $x$ whose distance from the origin is less than a positive constant $c$ . On the number line, the solution is the open interval $-c < x < c$ .

|x| < c \iff -c < x < c \quad (c > 0)

This formula gives the range of $x$ whose distance from the origin is greater than a positive constant $c$ . On the number line, the solution splits into the outer regions $x < -c$ and $c < x$ .

|x| > c \iff x < -c, \; c < x \quad (c > 0)

This equation finds the values of $x$ whose distance from the origin equals $c$ ( $c \ge 0$ ). On the number line, it corresponds to points at distance $c$ in either direction, yielding $x = \pm c$ .

|x| = c \iff x = \pm c \quad (c \ge 0)

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Practice 1Coming soon