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Algebraic fractions and fractional equations

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

If the variable is in the denominator, the term is an algebraic fraction (der Bruchterm). The denominator must never be \(0\): this gives the domain (die Definitionsmenge).
📄 Open worksheet (PDF, German)

The video

In the video we share pizza among \(x\) people and solve equations like \(\frac{12}{x} = 3\).

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

Contents of the video

  1. 0:00 Introduction
  2. 0:26 The domain
  3. 0:52 Solving a fractional equation
  4. 1:17 Denominator with a bracket
  5. 1:46 Careful, mistake!
  6. 2:16 Try it yourself

The most important points

Domain

The denominator must never be \(0\). For \(\frac{x+1}{x-3}\), \(D = \mathbb{Q} \setminus \{3\}\).

For x = 3 the denominator becomes 0.
For \(x = 3\) the denominator becomes \(0\).

Solving

Multiply both sides by the denominator: \(\frac{12}{x} = 3 \;\big|\cdot x\) gives \(12 = 3x\), so \(x = 4\).

The equals signs are lined up.
The equals signs are lined up.

For \(\frac{6}{x-2} = 3\) multiply by the whole denominator \((x-2)\): \(6 = 3\cdot(x-2) = 3x - 6\), so \(x = 4\).

Expand and continue as usual.
Expand and continue as usual.

Careful: always check the solution with \(D\). For \(\frac{x}{x-2} = \frac{2}{x-2}\) you get \(x = 2\), but \(2 \notin D\): \(L = \{\,\}\).

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
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The worksheet – Page 2
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The worksheet – Page 3
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Show the worksheet solutions
The worksheet – Page 4
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Try it yourself

Task

Solve: a) \(\frac{10}{x} = 2\) b) \(\frac{8}{x+1} = 2\).

Show solution

a) \(10 = 2x\), \(x = 5\) b) \(8 = 2x + 2\), \(x = 3\). Both are in \(D\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Bruchterme und Bruchgleichungen (Definitionsmenge, mit dem Nenner multiplizieren, Probe). 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.