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Inequalities

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

An inequality (die Ungleichung) compares two terms with \(<\), \(>\), \(\leq\) or \(\geq\). Its solution is usually a whole range of numbers.
📄 Open worksheet (PDF, German)

The video

In the video we buy cinema tickets for at most \(20\) €, solve \(3x + 4 < 19\) and see when the sign turns round.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:30 The signs
  3. 0:51 Solving an inequality
  4. 1:18 Careful, mistake!
  5. 1:47 Try it yourself

The most important points

\(20\) €, a ticket costs \(6\) €: \(6x \leq 20\). Three tickets (\(18\) €) are fine, four (\(24\) €) are not.

From the cinema to the inequality.
From the cinema to the inequality.

Solving

Transform as with equations: \(3x + 4 < 19 \;|-4\), \(3x < 15 \;|:3\), \(x < 5\). Solution: all numbers less than \(5\).

On the number line: left of 5, 5 itself does not belong.
On the number line: left of \(5\), \(5\) itself does not belong.

Turning the sign

If you multiply or divide by a negative number, the sign turns round: \(-2x < 6 \;|:(-2)\) gives \(x > -3\).

Check with 0: -2 · 0 = 0 < 6.
Check with \(0\): \(-2 \cdot 0 = 0 < 6\).

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
Page 1/4
The worksheet – Page 2
Page 2/4
The worksheet – Page 3
Page 3/4
Show the worksheet solutions
The worksheet – Page 4
Page 4/4

Try it yourself

Task

Solve: a) \(2x - 3 \geq 7\) b) \(-3x > 12\).

Show solution

a) \(2x \geq 10\), \(x \geq 5\) b) turn the sign: \(x < -4\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Ungleichungen aufstellen und lösen (Zeichen <, >, ≤, ≥, Zahlenstrahl, Zeichen umdrehen). Stelle mir nacheinander fünf kurze Aufgaben. Warte immer auf meine Antwort. Sag mir dann, ob sie richtig ist, und erkläre kurz, warum. Benutze einfache deutsche Sätze.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.