Materiál pripravila: hodinovaucitelka.sk
Predpokladané vedomosti: Pred štúdiom tohto materiálu by si mal/a ovládať základné rovnice s absolútnou hodnotou typu \(|x| = a\) a \(|x - b| = a\). Pozri Absolútna hodnota - rovnice a nerovnice.
Lineárnou rovnicou s absolútnou hodnotou nazývame každú rovnicu s neznámou \(x \in \mathbb{R}\), ktorá obsahuje jeden alebo viac výrazov v absolútnej hodnote, pričom neznáma \(x\) sa vyskytuje v prvej mocnine.
Príklady: - \(|2x - 3| = 5\) - \(|x + 2| + |x - 2| = 8\) - \(|3x - 1| = |x + 5|\)
Táto metóda je univerzálna a funguje pre všetky lineárne rovnice s absolútnou hodnotou.
\[ |A| = \begin{cases} A & \text{ak } A \geq 0 \\ -A & \text{ak } A < 0 \end{cases} \]
Úloha: Riešte v \(\mathbb{R}\) rovnicu \(|x + 2| + |x - 2| = 8\)
Položíme výrazy v absolútnych hodnotách rovné nule:
absolútna hodnota \(|x + 2|\): \[x + 2 = 0 \quad \Rightarrow \quad x_1 = -2\]
absolútna hodnota \(|x - 2|\): \[x - 2 = 0 \quad \Rightarrow \quad x_2 = 2\]
Nulové body \(-2\) a \(2\) rozdelia číselnú os na tri intervaly: - \(I_1 = (-\infty, -2)\) - \(I_2 = \langle -2, 2 \rangle\) - \(I_3 = (2, +\infty)\)
| \(x\) | \((-\infty, -2)\) | \(\langle -2, 2 \rangle\) | \((2, +\infty)\) |
|---|---|---|---|
| \(x + 2\) | \(-\) | \(+\) | \(+\) |
| \(x - 2\) | \(-\) | \(-\) | \(+\) |
Pre \(x \in I_1 = (-\infty, -2)\):
\[|x + 2| = -(x + 2) = -x - 2\] \[|x - 2| = -(x - 2) = -x + 2\]
\[\begin{align*} -x - 2 + (-x + 2) &= 8 \\ -2x &= 8 \\ x &= -4 \end{align*}\]
Overenie: \(-4 \in (-\infty, -2)\) ✓
\[K_1 = \{-4\}\]
Pre \(x \in I_2 = \langle -2, 2 \rangle\):
\[|x + 2| = x + 2\] \[|x - 2| = -(x - 2) = -x + 2\]
\[\begin{align*} x + 2 + (-x + 2) &= 8 \\ 4 &= 8 \end{align*}\]
Spor! Rovnica nemá riešenie na tomto intervale.
\[K_2 = \emptyset\]
Pre \(x \in I_3 = (2, +\infty)\):
\[|x + 2| = x + 2\] \[|x - 2| = x - 2\]
\[\begin{align*} x + 2 + x - 2 &= 8 \\ 2x &= 8 \\ x &= 4 \end{align*}\]
Overenie: \(4 \in (2, +\infty)\) ✓
\[K_3 = \{4\}\]
\[K = K_1 \cup K_2 \cup K_3 = \{-4\} \cup \emptyset \cup \{4\} = \boxed{\{-4; 4\}}\]
Úloha: Riešte v \(\mathbb{R}\) rovnicu \(|2x - 1| = |x + 3|\)
\[2x - 1 = 0 \quad \Rightarrow \quad x_1 = \frac{1}{2}\] \[x + 3 = 0 \quad \Rightarrow \quad x_2 = -3\]
Zoradíme: \(-3 < \frac{1}{2}\)
| \(x\) | \((-\infty, -3)\) | \(\langle -3, \frac{1}{2} \rangle\) | \((\frac{1}{2}, +\infty)\) |
|---|---|---|---|
| \(2x - 1\) | \(-\) | \(-\) | \(+\) |
| \(x + 3\) | \(-\) | \(+\) | \(+\) |
Pre \(x \in I_1 = (-\infty, -3)\):
\[|2x - 1| = -(2x - 1) = -2x + 1\] \[|x + 3| = -(x + 3) = -x - 3\]
\[\begin{align*} -2x + 1 &= -x - 3 \\ -2x + x &= -3 - 1 \\ -x &= -4 \\ x &= 4 \end{align*}\]
Overenie: \(4 \notin (-\infty, -3)\) ✗
\[K_1 = \emptyset\]
Pre \(x \in I_2 = \langle -3, \frac{1}{2} \rangle\):
\[|2x - 1| = -2x + 1\] \[|x + 3| = x + 3\]
\[\begin{align*} -2x + 1 &= x + 3 \\ -2x - x &= 3 - 1 \\ -3x &= 2 \\ x &= -\frac{2}{3} \end{align*}\]
Overenie: \(-\frac{2}{3} \in \langle -3, \frac{1}{2} \rangle\) ✓
\[K_2 = \left\{-\frac{2}{3}\right\}\]
Pre \(x \in I_3 = (\frac{1}{2}, +\infty)\):
\[|2x - 1| = 2x - 1\] \[|x + 3| = x + 3\]
\[\begin{align*} 2x - 1 &= x + 3 \\ 2x - x &= 3 + 1 \\ x &= 4 \end{align*}\]
Overenie: \(4 \in (\frac{1}{2}, +\infty)\) ✓
\[K_3 = \{4\}\]
\[K = K_1 \cup K_2 \cup K_3 = \emptyset \cup \left\{-\frac{2}{3}\right\} \cup \{4\} = \boxed{\left\{-\frac{2}{3}; 4\right\}}\]
Riešte metódou nulových bodov.
1. \(|2x - 4| = 6\)
Nulový bod: \(2x - 4 = 0 \Rightarrow x = 2\)
Pre \(x < 2\): \[\begin{align*} -2x + 4 &= 6 \\ x &= -1 \in (-\infty, 2) \checkmark \end{align*}\]
Pre \(x \geq 2\): \[\begin{align*} 2x - 4 &= 6 \\ x &= 5 \in \langle 2, +\infty) \checkmark \end{align*}\]
\[K = \{-1; 5\}\]
2. \(|3x + 6| = 9\)
Nulový bod: \(x = -2\)
Pre \(x < -2\): \[\begin{align*} -3x - 6 &= 9 \\ x &= -5 \checkmark \end{align*}\]
Pre \(x \geq -2\): \[\begin{align*} 3x + 6 &= 9 \\ x &= 1 \checkmark \end{align*}\]
\[K = \{-5; 1\}\]
3. \(|5 - x| = 3\)
Nulový bod: \(x = 5\)
Pre \(x < 5\): \[\begin{align*} 5 - x &= 3 \\ x &= 2 \checkmark \end{align*}\]
Pre \(x \geq 5\): \[\begin{align*} -5 + x &= 3 \\ x &= 8 \checkmark \end{align*}\]
\[K = \{2; 8\}\]
4. \(|4x + 2| = 10\)
Nulový bod: \(x = -\frac{1}{2}\)
Pre \(x < -\frac{1}{2}\): \[\begin{align*} -4x - 2 &= 10 \\ x &= -3 \checkmark \end{align*}\]
Pre \(x \geq -\frac{1}{2}\): \[\begin{align*} 4x + 2 &= 10 \\ x &= 2 \checkmark \end{align*}\]
\[K = \{-3; 2\}\]
5. \(|2 - 3x| = 7\)
Nulový bod: \(x = \frac{2}{3}\)
Pre \(x < \frac{2}{3}\): \[\begin{align*} 2 - 3x &= 7 \\ x &= -\frac{5}{3} \checkmark \end{align*}\]
Pre \(x \geq \frac{2}{3}\): \[\begin{align*} -2 + 3x &= 7 \\ x &= 3 \checkmark \end{align*}\]
\[K = \left\{-\frac{5}{3}; 3\right\}\]
6. \(|x - 5| = 2x + 1\)
Nulový bod: \(x = 5\)
Podmienka: \(2x + 1 \geq 0 \Rightarrow x \geq -\frac{1}{2}\)
Pre \(-\frac{1}{2} \leq x < 5\): \[\begin{align*} -x + 5 &= 2x + 1 \\ x &= \frac{4}{3} \checkmark \end{align*}\]
Pre \(x \geq 5\): \[\begin{align*} x - 5 &= 2x + 1 \\ x &= -6 \notin \langle 5, +\infty) \end{align*}\]
\[K = \left\{\frac{4}{3}\right\}\]
7. \(|x - 1| + |x + 3| = 6\)
Nulové body: \(x = 1\), \(x = -3\)
Pre \(x < -3\): \[\begin{align*} -x + 1 - x - 3 &= 6 \\ -2x &= 8 \\ x &= -4 \checkmark \end{align*}\]
Pre \(-3 \leq x < 1\): \[\begin{align*} -x + 1 + x + 3 &= 6 \\ 4 &= 6 \quad \text{spor} \end{align*}\]
Pre \(x \geq 1\): \[\begin{align*} x - 1 + x + 3 &= 6 \\ 2x &= 4 \\ x &= 2 \checkmark \end{align*}\]
\[K = \{-4; 2\}\]
8. \(|x + 1| + |x - 4| = 7\)
Nulové body: \(x = -1\), \(x = 4\)
Pre \(x < -1\): \[\begin{align*} -x - 1 - x + 4 &= 7 \\ x &= -2 \checkmark \end{align*}\]
Pre \(-1 \leq x < 4\): \[\begin{align*} x + 1 - x + 4 &= 7 \\ 5 &= 7 \quad \text{spor} \end{align*}\]
Pre \(x \geq 4\): \[\begin{align*} x + 1 + x - 4 &= 7 \\ x &= 5 \checkmark \end{align*}\]
\[K = \{-2; 5\}\]
9. \(|2x - 1| + |x + 2| = 6\)
Nulové body: \(x = \frac{1}{2}\), \(x = -2\)
Pre \(x < -2\): \[\begin{align*} -2x + 1 - x - 2 &= 6 \\ -3x &= 7 \\ x &= -\frac{7}{3} \checkmark \end{align*}\]
Pre \(-2 \leq x < \frac{1}{2}\): \[\begin{align*} -2x + 1 + x + 2 &= 6 \\ x &= -3 \notin \langle -2, \frac{1}{2}) \end{align*}\]
Pre \(x \geq \frac{1}{2}\): \[\begin{align*} 2x - 1 + x + 2 &= 6 \\ 3x &= 5 \\ x &= \frac{5}{3} \checkmark \end{align*}\]
\[K = \left\{-\frac{7}{3}; \frac{5}{3}\right\}\]
10. \(|x - 3| + |x + 1| = 8\)
Nulové body: \(x = 3\), \(x = -1\)
Pre \(x < -1\): \[\begin{align*} -x + 3 - x - 1 &= 8 \\ x &= -3 \checkmark \end{align*}\]
Pre \(-1 \leq x < 3\): \[\begin{align*} -x + 3 + x + 1 &= 8 \\ 4 &= 8 \quad \text{spor} \end{align*}\]
Pre \(x \geq 3\): \[\begin{align*} x - 3 + x + 1 &= 8 \\ x &= 5 \checkmark \end{align*}\]
\[K = \{-3; 5\}\]
11. \(|x + 2| + |x - 5| = 9\)
Nulové body: \(x = -2\), \(x = 5\)
Pre \(x < -2\): \[\begin{align*} -x - 2 - x + 5 &= 9 \\ x &= -3 \checkmark \end{align*}\]
Pre \(-2 \leq x < 5\): \[\begin{align*} x + 2 - x + 5 &= 9 \\ 7 &= 9 \quad \text{spor} \end{align*}\]
Pre \(x \geq 5\): \[\begin{align*} x + 2 + x - 5 &= 9 \\ x &= 6 \checkmark \end{align*}\]
\[K = \{-3; 6\}\]
12. \(|3x - 2| + |x - 1| = 5\)
Nulové body: \(x = \frac{2}{3}\), \(x = 1\)
Pre \(x < \frac{2}{3}\): \[\begin{align*} -3x + 2 - x + 1 &= 5 \\ -4x &= 2 \\ x &= -\frac{1}{2} \checkmark \end{align*}\]
Pre \(\frac{2}{3} \leq x < 1\): \[\begin{align*} 3x - 2 - x + 1 &= 5 \\ x &= 3 \notin \langle \frac{2}{3}, 1) \end{align*}\]
Pre \(x \geq 1\): \[\begin{align*} 3x - 2 + x - 1 &= 5 \\ x &= 2 \checkmark \end{align*}\]
\[K = \left\{-\frac{1}{2}; 2\right\}\]
13. \(|x - 2| = |x + 4|\)
Nulové body: \(x = 2\), \(x = -4\)
Pre \(x < -4\): \[\begin{align*} -x + 2 &= -x - 4 \\ 2 &= -4 \quad \text{spor} \end{align*}\]
Pre \(-4 \leq x < 2\): \[\begin{align*} -x + 2 &= x + 4 \\ x &= -1 \checkmark \end{align*}\]
Pre \(x \geq 2\): \[\begin{align*} x - 2 &= x + 4 \\ -2 &= 4 \quad \text{spor} \end{align*}\]
\[K = \{-1\}\]
14. \(|2x + 1| = |x - 3|\)
Nulové body: \(x = -\frac{1}{2}\), \(x = 3\)
Pre \(x < -\frac{1}{2}\): \[\begin{align*} -2x - 1 &= -x + 3 \\ x &= -4 \checkmark \end{align*}\]
Pre \(-\frac{1}{2} \leq x < 3\): \[\begin{align*} 2x + 1 &= -x + 3 \\ x &= \frac{2}{3} \checkmark \end{align*}\]
Pre \(x \geq 3\): \[\begin{align*} 2x + 1 &= x - 3 \\ x &= -4 \notin \langle 3, +\infty) \end{align*}\]
\[K = \left\{-4; \frac{2}{3}\right\}\]
15. \(|3x - 2| = |2x + 1|\)
Nulové body: \(x = \frac{2}{3}\), \(x = -\frac{1}{2}\)
Pre \(x < -\frac{1}{2}\): \[\begin{align*} -3x + 2 &= -2x - 1 \\ x &= 3 \notin (-\infty, -\frac{1}{2}) \end{align*}\]
Pre \(-\frac{1}{2} \leq x < \frac{2}{3}\): \[\begin{align*} -3x + 2 &= 2x + 1 \\ x &= \frac{1}{5} \checkmark \end{align*}\]
Pre \(x \geq \frac{2}{3}\): \[\begin{align*} 3x - 2 &= 2x + 1 \\ x &= 3 \checkmark \end{align*}\]
\[K = \left\{\frac{1}{5}; 3\right\}\]
16. \(|x + 5| = |2x - 1|\)
Nulové body: \(x = -5\), \(x = \frac{1}{2}\)
Pre \(x < -5\): \[\begin{align*} -x - 5 &= -2x + 1 \\ x &= 6 \notin (-\infty, -5) \end{align*}\]
Pre \(-5 \leq x < \frac{1}{2}\): \[\begin{align*} x + 5 &= -2x + 1 \\ x &= -\frac{4}{3} \checkmark \end{align*}\]
Pre \(x \geq \frac{1}{2}\): \[\begin{align*} x + 5 &= 2x - 1 \\ x &= 6 \checkmark \end{align*}\]
\[K = \left\{-\frac{4}{3}; 6\right\}\]
17. \(|4x - 3| = |x + 6|\)
Nulové body: \(x = \frac{3}{4}\), \(x = -6\)
Pre \(x < -6\): \[\begin{align*} -4x + 3 &= -x - 6 \\ x &= 3 \notin (-\infty, -6) \end{align*}\]
Pre \(-6 \leq x < \frac{3}{4}\): \[\begin{align*} -4x + 3 &= x + 6 \\ x &= -\frac{3}{5} \checkmark \end{align*}\]
Pre \(x \geq \frac{3}{4}\): \[\begin{align*} 4x - 3 &= x + 6 \\ x &= 3 \checkmark \end{align*}\]
\[K = \left\{-\frac{3}{5}; 3\right\}\]
18. \(|2x - 5| = |3x + 2|\)
Nulové body: \(x = \frac{5}{2}\), \(x = -\frac{2}{3}\)
Pre \(x < -\frac{2}{3}\): \[\begin{align*} -2x + 5 &= -3x - 2 \\ x &= -7 \checkmark \end{align*}\]
Pre \(-\frac{2}{3} \leq x < \frac{5}{2}\): \[\begin{align*} -2x + 5 &= 3x + 2 \\ x &= \frac{3}{5} \checkmark \end{align*}\]
Pre \(x \geq \frac{5}{2}\): \[\begin{align*} 2x - 5 &= 3x + 2 \\ x &= -7 \notin \langle \frac{5}{2}, +\infty) \end{align*}\]
\[K = \left\{-7; \frac{3}{5}\right\}\]
19. \(|x - 1| + |x - 3| + |x - 5| = 6\)
Nulové body: \(x = 1\), \(x = 3\), \(x = 5\)
Pre \(x < 1\): \[\begin{align*} -x + 1 - x + 3 - x + 5 &= 6 \\ -3x &= -3 \\ x &= 1 \notin (-\infty, 1) \end{align*}\]
Pre \(1 \leq x < 3\): \[\begin{align*} x - 1 - x + 3 - x + 5 &= 6 \\ -x &= -1 \\ x &= 1 \checkmark \end{align*}\]
Pre \(3 \leq x < 5\): \[\begin{align*} x - 1 + x - 3 - x + 5 &= 6 \\ x &= 5 \notin \langle 3, 5) \end{align*}\]
Pre \(x \geq 5\): \[\begin{align*} x - 1 + x - 3 + x - 5 &= 6 \\ 3x &= 15 \\ x &= 5 \checkmark \end{align*}\]
\[K = \{1; 5\}\]
20. \(|2x + 1| - |x - 2| = 3\)
Nulové body: \(x = -\frac{1}{2}\), \(x = 2\)
Pre \(x < -\frac{1}{2}\): \[\begin{align*} -2x - 1 - (-x + 2) &= 3 \\ -x - 3 &= 3 \\ x &= -6 \checkmark \end{align*}\]
Pre \(-\frac{1}{2} \leq x < 2\): \[\begin{align*} 2x + 1 - (-x + 2) &= 3 \\ 3x - 1 &= 3 \\ x &= \frac{4}{3} \checkmark \end{align*}\]
Pre \(x \geq 2\): \[\begin{align*} 2x + 1 - (x - 2) &= 3 \\ x + 3 &= 3 \\ x &= 0 \notin \langle 2, +\infty) \end{align*}\]
\[K = \left\{-6; \frac{4}{3}\right\}\]
21. \(2|x - 1| = |x + 2|\)
Nulové body: \(x = 1\), \(x = -2\)
Pre \(x < -2\): \[\begin{align*} 2(-x + 1) &= -x - 2 \\ -2x + 2 &= -x - 2 \\ x &= 4 \notin (-\infty, -2) \end{align*}\]
Pre \(-2 \leq x < 1\): \[\begin{align*} 2(-x + 1) &= x + 2 \\ -2x + 2 &= x + 2 \\ x &= 0 \checkmark \end{align*}\]
Pre \(x \geq 1\): \[\begin{align*} 2(x - 1) &= x + 2 \\ 2x - 2 &= x + 2 \\ x &= 4 \checkmark \end{align*}\]
\[K = \{0; 4\}\]
22. \(|x + 3| + |2x - 1| = 7\)
Nulové body: \(x = -3\), \(x = \frac{1}{2}\)
Pre \(x < -3\): \[\begin{align*} -x - 3 - 2x + 1 &= 7 \\ -3x &= 9 \\ x &= -3 \notin (-\infty, -3) \end{align*}\]
Pre \(-3 \leq x < \frac{1}{2}\): \[\begin{align*} x + 3 - 2x + 1 &= 7 \\ -x &= 3 \\ x &= -3 \checkmark \end{align*}\]
Pre \(x \geq \frac{1}{2}\): \[\begin{align*} x + 3 + 2x - 1 &= 7 \\ 3x &= 5 \\ x &= \frac{5}{3} \checkmark \end{align*}\]
\[K = \left\{-3; \frac{5}{3}\right\}\]
23. \(|x - 4| = 2|x + 1|\)
Nulové body: \(x = 4\), \(x = -1\)
Pre \(x < -1\): \[\begin{align*} -x + 4 &= 2(-x - 1) \\ -x + 4 &= -2x - 2 \\ x &= -6 \checkmark \end{align*}\]
Pre \(-1 \leq x < 4\): \[\begin{align*} -x + 4 &= 2(x + 1) \\ -x + 4 &= 2x + 2 \\ x &= \frac{2}{3} \checkmark \end{align*}\]
Pre \(x \geq 4\): \[\begin{align*} x - 4 &= 2(x + 1) \\ x - 4 &= 2x + 2 \\ x &= -6 \notin \langle 4, +\infty) \end{align*}\]
\[K = \left\{-6; \frac{2}{3}\right\}\]
24. \(|x - 2| + |x| + |x + 2| = 10\)
Nulové body: \(x = -2\), \(x = 0\), \(x = 2\)
Pre \(x < -2\): \[\begin{align*} -x + 2 - x - x - 2 &= 10 \\ -3x &= 10 \\ x &= -\frac{10}{3} \checkmark \end{align*}\]
Pre \(-2 \leq x < 0\): \[\begin{align*} -x + 2 - x + x + 2 &= 10 \\ -x &= 6 \\ x &= -6 \notin \langle -2, 0) \end{align*}\]
Pre \(0 \leq x < 2\): \[\begin{align*} -x + 2 + x + x + 2 &= 10 \\ x &= 6 \notin \langle 0, 2) \end{align*}\]
Pre \(x \geq 2\): \[\begin{align*} x - 2 + x + x + 2 &= 10 \\ 3x &= 10 \\ x &= \frac{10}{3} \checkmark \end{align*}\]
\[K = \left\{-\frac{10}{3}; \frac{10}{3}\right\}\]
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