Equation with absolute value

Task number: 2763

Solve this equation over the real domain: \(||x-2|-3|=5\).

  • Resolution

    • \(x\le 2: |-x+2-3|=5 \longrightarrow |-x-1|=5 \longrightarrow |-x-1|=5\)
      1. \(x\le -1: -x-1=5 \longrightarrow x=-6\)
      2. \(x\ge -1: x+1=5 \longrightarrow x=4\) has no solution
    • \(x\ge 2: |x-2-3|=5 \longrightarrow |x-5|=5\)
      1. \(x\le 5: -x+5=5 \longrightarrow x=0\) has no solution
      2. \(x\ge -1: x-5=5 \longrightarrow x=10\)

  • Result

    The equation has two solutions: \(x=-6\) and \(x=10\).

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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