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InequalitiesAlgebraHigh schoolabout 1:05

Solve |x - 2| < 3: animated lesson

A narrated, step-by-step inequalities video lesson with captions. Final answer: -1 < x < 5; interval notation: (-1, 5).

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.4 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    ∣x−2∣<3|x - 2| < 3
  2. 2. Step by step
    1. ∣x−2∣<3|x - 2| < 3
      Write the problem with the absolute value isolated
    2. −3<x−2<3-3 < x - 2 < 3
      Rewrite as a compound ("and") inequality
    3. x−2=−3  ⇒  x=−3+2=−1x−2=3  ⇒  x=3+2=5x - 2 = -3 \;\Rightarrow\; x = -3 + 2 = -1 \\ x - 2 = 3 \;\Rightarrow\; x = 3 + 2 = 5
      Find the endpoints by solving each boundary equation
    4. (−1, 5)(-1,\ 5)
      Write the solution set in interval notation
  3. 3. The answer
    −1<x<5(−1, 5)-1 < x < 5 \qquad (-1,\ 5)

    |x - 2| < 3 → -1 < x < 5

  4. 4. Check
    • Test point x = 2 (midpoint): |0| = 0 < 3, so the inside of the interval satisfies the inequality.

Transcript

The problem. Here is the problem: |x minus 2| less than 3. Solve an absolute value inequality by turning it into a compound inequality and finding the boundary points.

Step by step. Write the problem with the absolute value isolated. The absolute value is already alone on one side, so it can be split into cases. Rewrite as a compound ("and") inequality. The inside must lie within c units of zero. Find the endpoints by solving each boundary equation. The solution lies between the endpoints. Write the solution set in interval notation.

The answer. So the answer is negative 1 less than x less than 5; interval notation: (negative 1, 5). An absolute value inequality compares a distance with c: "less than" keeps the inside within c of zero (one interval), "greater than" pushes it beyond c (two rays).

Check. Test point x equals 2 (midpoint): |0| equals 0 less than 3, so the inside of the interval satisfies the inequality.

About inequalities

Solve linear and quadratic inequalities and write the answer as an interval.

More lessons like this

Solve |x - 2| < 3: Inequalities Step-by-Step Lesson