Algebra I

Absolute Value: Equations & Inequalities

Big Picture

Absolute value equations and inequalities are very similar to linear equations and inequalities. In both cases, the goal is to solve for a variable. However, unlike linear equations and linear inequalities, the variable is not a specific number. Instead, the variable represents a specific distance from zero. When solving absolute value equations and inequalities, two options need to be considered: when the expression inside the absolute value is not negative and when the expression inside the absolute value is negative.

Key Terms

Absolute Value: The absolute value of a number is the distance of that number from 0.

Absolute Value Equations

An absolute value equation is an equation that contains an absolute value expression

Example: \(|ax+b| = c, \text{ where } c ≥ 0\)

To solve an absolute value equation, split it into two equations and solve individually.

Example: \(|ax+b| = c\), where \(c ≥ 0\)

  • Split into two equations: \(ax+b = c\) and \(ax+b = -c\)
  • Solve: \(x = \frac{c-b}{a}\) and \(x = \frac{c+b}{a}\) are both solutions

Do not start to solve until the absolute value equation is splitinto two equations.

Example: \(|x-4| = 5\)

    \(x-4 = 5\)
    \(x = 9\)
    and
    \(x-4 = -5\)
    \(x = -1\)

    Plotted on the number line:

    • The two solutions are both 5 units away from 4

    Absolute Value Inequalities

    An absolute value inequality is an inequality that contains an absolute value expression

    To solve an absolute value inequality, split into two inequalities, and solve individually.

    Example: \(|ax + b| < c\), where \(c ≥ 0\)

    • Split into two inequalities: \(ax + b > -c\) and \(ax + b < c\) (can be rewritten as \(-c < ax + b < c\))
    • Absolute value inequalities can be rewritten as compound inequalities with “and”
    • Solve: \(x > - \frac{c + b}{a}\) and \(x < \frac{c - b}{a}\) (can be rewritten as \(- \frac{c + b}{a} < x < \frac{c - b}{a}\))

    Example: \(|ax + b| > c\), where \(c ≥ 0\)

    • Split into two inequalities: \(ax + b < -c \text{ or } ax + b > c\)
    • Absolute value inequalities can be rewritten as compound inequalities with “or”
    • Solve:\(x < - \frac{c + b}{a}\) or \(x > \frac{c - b}{a}\)

    If the sign is less thAN, then it’s a compound inequality with AND. If the sign is greatER than, then it’s a compound equality with OR. Remember: less thAND, greatOR.

    Example: \(|x + 12| > 2\)

    \(x + 12 < -2\)
    \(x < -14\)
    or
    \(x + 12 > 2\)
    \(x > -10\)

    Example: \(|4x + 5| ≤ 13\)

    \(4x + 5 ≥ -13\)
    \(4x ≥ -18\)
    \(x ≥ - \frac{9}{2}\)
    and
    \(4x + 5 ≤ 13\)
    \(4x ≤ 8\)
    \(x ≤ 2\)

    Notes