28 lines
990 B
Markdown
28 lines
990 B
Markdown
# Valeurs Absolue
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## Definition
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\\(\lvert x\rvert = \begin{cases} x & \text{si } x \geq 0 \\\ -x & \text{si } x < 0 \end{cases}\\)
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## Exemple
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1) \\(\lvert 3x+5\rvert \leq 2\\)
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- Si \\( 3x + 5 \geq 0 \iff x \geq \frac{-5}{3}\\)
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- Alors \\(\lvert 3x+5\rvert \leq 2\\) devient \\(3x + 5 \leq 2\\)
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- \\(\iff x \leq -1\\)
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- > Ensemble des solutions trouvées pour ce cas :
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\\[ ]-\infty, -1] \cap \left[\frac{-5}{3}, +\infty\right[ = \left[\frac{-5}{3}, -1\right]\\]
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- Si \\( 3x + 5 < 0 \iff x \leq \frac{-5}{3}\\)
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- Alors \\(\lvert 3x+5\rvert \leq 2\\) devient \\-(3x + 5 \leq 2\\)
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- \\(\iff x \leq \frac{-7}{3}\\)
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- > Ensemble des solutions trouvées pour ce cas :
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\\[ \left[\frac{-7}{3}, +\infty\right[\cap\left]-\infty, \frac{-5}{3}\right[ = \left[\frac{-7}{3}, \frac{-5}{3}\right[\\]
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- > Conclusion:
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\\[\\{x \vert\lvert 3x+5\rvert \leq 2 \\} = \left[\frac{-5}{3}, -1\right] \cup \left[\frac{-7}{3}, \frac{-5}{3}\right[ = \left[\frac{-7}{3}, -1\right] \\]
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