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https://github.com/danbulant/notes
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130 lines
2.4 KiB
Markdown
130 lines
2.4 KiB
Markdown
---
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tags: [mat]
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---
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# Absolutní hodnota
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$|x|$
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```ad-sentence
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$|x|$ = $x<0$ ? $-x$ : $x$
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```
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## Nulový bod
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*"bod zlomu"*
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$|x|$ -> $x=0$...nulový bod
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$|x-3|$ -> x=3...nulový bod
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---
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$|x-5|=11$ nulový bod $x=5$
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$x<5$
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$-(x-5)=11$
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$-x+5=11 \space | -5$
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$-x=6 | *(-1)$
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$x=-6$ ✔️ $[-6\in(-\infty;5)]$
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$x \geq 5$
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$x-5=11 | -5$
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$x=11+5$
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$x=16$ ✔️ $[-16\in<5;+\infty)]$
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$K={-6;16}$
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zkouška:
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$|x-5|=|-6-5|=11$ ✔️
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$|x-5|=|16-5|=11$ ✔️
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---
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$|x+3|=63$ nulový bod $x = -3$
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$x<3$
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$-(x+3)=63$
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$-x-3=63$
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$-x=66$
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$x=66$ ❌ $[66\not\in x<3]$
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$x \geq 3$
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---
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$$\textcolor{red}{|x+1|}+\textcolor{green}{|x-1|}=4$$
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$\textcolor{red}{nulový \space bod \dots -1}$ $\textcolor{green}{nulový \space bod \dots 1}$
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![[data/Absolutní hodnota_2022-02-23 09.05.43.excalidraw.md]]
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| - | $x \lt 1$ | $x\ge1$ |
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| ----- | --------------- | ------------- |
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| $x\lt-1$ | $-(x+1)+-(x-1)=4$ | $\emptyset$ |
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| $x\ge-1$ | $(x+1)+-(x-1)=4$ | $(x+1)+(x-1)=4$ |
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$k_1$:
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$$-(x+1)+-(x-1)=4$$
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$$-x-1-x+1=4$$
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$$-2x=4$$
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$$x=-2$$
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$$k_1=-2$$
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$k_2: \emptyset$
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$k_3$:
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$$(x+1)-(x-1)=4$$
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$$x+1-x+1=4$$
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$$0x+2=4 \space | -2$$
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$$0x=2$$
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$$k_4 = \emptyset$$
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$k_4$:
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$$(x+1)+(x-1)=4$$
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$$2x=4$$
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$$x=2$$
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$$k_4 = 2$$
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$2 \ge -1 \vee 2\ge 1$
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---
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$|3x-2|-|2x-3|=3$
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nulový bod=$\frac23$ nulový bod=$\frac32$
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| - | $x\lt\frac32$ | $x\gt\frac32$ |
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| --------------- | --------------------------- | ---------------------- |
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| $x \lt \frac23$ | $k_1$: $-(3x-2)- -(2x-3)=3$ | $k_2: \emptyset$ |
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| $x\ge\frac23$ | $k_3: (3x-2)- -(2x-3)=3$ | $k_4: (3x-2)-(2x-3)=3$ |
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$k_1: -3x+2+2x-3=3$
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$-x-1=3$
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$-x=4$
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$x=-4$
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$k_1=\{-4\}$
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$-4 < \frac32 \vee x\lt\frac23$
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$k_2: \emptyset$
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$k_3: 3x-2+2x-3=3$
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$5x-5=3$
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$5x=8$
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$x=\frac85$
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$\frac85 \not\in <\frac23;\frac32)$
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$k_3 = \emptyset$
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---
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$3|x-5|+2|x+1|=2$
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nulový bod...$5$ nulový bod...$-1$
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| - | $x<5$ | $x \ge 5$ |
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| -------- | ------------------ | ----------------- |
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| $x<-1$ | $-3(x-5)-2(x+1)=2$ | $\emptyset$ |
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| $x\ge-1$ | -3(x-5)+2(x+1)=2$ | $3(x-5)+2(x+1)=2$ |
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$k_1: -3(x-5)-2(x+1)=2$
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$-3x+15-2x-2=2$
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$-5x+13=2$
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$-5x=-11$
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$5x=11$
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$x=\frac{11}5$; $\frac{11}5 \not\in (-\infty; -1)$
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$k_1=\emptyset$
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$k_2: \emptyset$
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$k_3: -3(x-5)+2(x+1)=2$
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$-3x+15+2x+2=2$
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$-x+17=2$
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$-x=-15$
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$x=15$; $15\not\in <-1; 5)$
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$k_3 = \emptyset$
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$k_4: 3(x-5)+2(x+1)=2$
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$3x-15+2x+2=2$
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$5x-13=2$
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$5x=15$
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$x=3$; $3\not\in <5; \infty)$
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$k_4=\emptyset$
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