Lineární rovnice s absolutní hodnotou

Lineární rovnice s absolutní hodnotou

1. V množině reálných čísel řešte rovnice

$$(a)\ |x|=7$$$$(b)\ |x-1|=3$$
$$(c)\ |x+4|=1$$$$(d)\ |x+2|=7$$
$$(e)\ |x+\pi|=6$$$$(f)\ |6-x|=2$$
$$(g)\ |x-\sqrt{3}+1|=2-\sqrt{3}$$$$(h)\ |4x-7|=-1$$

$$(a)\ x\in\{\pm7\}$$$$(b)\ x\in\{-2;4\}$$
$$(c)\ x\in\{-5;-3\}$$$$(d)\ x\in\{-9;5\}$$
$$(e)\ x\in\{-6-\pi;6-\pi\}$$$$(f)\ x\in\{4;8\}$$
$$(g)\ x\in\{1;2\sqrt{3}-3\}$$$$(h)\ x\in\varnothing$$

2. V množině reálných čísel řešte rovnice

$$(a)\ |3x-1|+5=3$$$$(b)\ |5-3x|=7$$
$$(c)\ |2x-4|=0$$$$(d)\ |7x+2|=10$$
$$(e)\ 2|x-1|-6=0$$$$(f)\ 2|-2x-2|=16$$

$$(a)\ x\in\varnothing$$$$(b)\ x\in\left\{-\frac{2}{3};4\right\}$$
$$(c)\ x\in\left\{2\right\}$$$$(d)\ x\in\left\{-\frac{12}{7};\frac{8}{7}\right\}$$
$$(e)\ x\in\left\{-2;4\right\}$$$$(f)\ x\in\left\{-5;3\right\}$$

3. V množině reálných čísel řešte rovnice

$$(a)\ |2x-3|=x$$$$(b)\ |2x+3|-3x=5$$
$$(c)\ |x+1|=2x$$$$(d)\ |x-2|=2x-3$$
$$(e)\ |3x+2|=4x+5$$$$(f)\ |2x+1|=x+5$$
$$(g)\ |x-2|=3x+1$$$$(h)\ |4x+3|=3-x$$
$$(i)\ |x-7|=x-7$$$$(j)\ |8-5x|=5x-8$$
$$(k)\ |x-1|=2x+1$$$$(l)\ |3x+1|=2x+3$$
$$(m)\ x-7=|2x-1|$$$$(n)\ |x+1|=1-x$$
$$(o)\ |7+3x|=11-x$$$$(p)\ |3x-10|=x+6$$

$$(a)\ x\in\{1;3\}$$$$(b)\ x\in\left\{-\frac{8}{5}\right\}$$
$$(c)\ x\in\{1\}$$$$(d)\ x\in\left\{\frac53\right\}$$
$$(e)\ x\in\{-1\}$$$$(f)\ x\in\{-2;4\}$$
$$(g)\ x\in\left\{\frac14\right\}$$$$(h)\ x\in\{-2;0\}$$
$$(i)\ x\in\langle 7;\infty)$$$$(j)\ x\in\left\langle\frac85;\infty\right)$$
$$(k)\ x\in\{0\}$$$$(l)\ x\in\left\{-\frac{4}{5};2\right\}$$
$$(m)\ x\in\O$$$$(n)\ x\in\{0\}$$
$$(o)\ x\in\left\{-9;1\right\}$$$$(p)\ x\in\{1;8\}$$

4. V množině reálných čísel řešte rovnice

$$(a)\ |x|+ |x+2|=4$$$$(b)\ |5-x|=|x-1|$$$$(c)\ |x+2|=4|x-3|$$
$$(d)\ x+|x|=0$$$$(e)\ |x+3|=2|x-3|$$$$(f)\ |x-1|+|x+2|=3$$
$$(g)\ |3x-2|+|2x+5|=7$$$$(h)\ |8-5x|=5x-8$$$$(i)\ |4-x|-|2x+3|=7$$
$$(j)\ |2x+1|-|2x|+1=2x$$$$(k)\ |4x-7|+|6x+15|=18$$$$(l)\ |x-5|-|2x+11|=6$$
$$(m)\ |x-3|=|3x+2|-1$$$$(n)\ |3x+4|=|2x-3|$$$$(o)\ 4|x+\sqrt{2}|-2|x-\sqrt{2}|=x$$

$$(a)\ x\in\left \{ -3;1 \right \}$$$$(b)\ x\in\left \{ 3 \right \}$$$$(c)\ x\in\left \{ 2;\frac{14}{3} \right \}$$
$$(d)\ x\in(-\infty ;0\rangle$$$$(e)\ x\in\left \{ 1;9 \right \}$$$$(f)\ x\in\left \langle -2;1 \right \rangle$$
$$(g)\ x\in\left \{ 0;\frac{4}{5} \right \}$$$$(h)\ x\in\left \langle \frac{8}{5};\infty \right )$$$$(i)\ x\in\varnothing$$
$$(j)\ x\in\left \{ 1 \right \}$$$$(k)\ x\in\left \{ -\frac{13}{5};-2 \right \}$$$$(l)\ x\in\left \{ -10;-4 \right \}$$
$$(m)\ x\in\left \{ -3;\frac{1}{2} \right \}$$$$(n)\ x\in\left \{ -7;-\frac{1}{5} \right \}$$$$(o)\ x\in\left \{ -2\sqrt{2};-\frac{2\sqrt{2}}{5} \right \}$$

5. V množině reálných čísel řešte rovnice

$$(a)\ |2x-4|-|x+3|=2-|x-5|$$$$(b)\ |x-1|+3|2-x|=x-|1-x|$$
$$(c)\ |x-4|+|2x-1|=|x|+3$$$$(d)\ 2|x|-|x+1|=3|x-1|-x$$
$$(e)\ \big||x+1|-3\big|=1$$$$(f)\ \big||x-3|+1\big|=5$$
$$(g)\ \big||x+1|+|x-1|\big|=2$$$$(h)\ \Big|\big||x|-1\big|-1\Big|=\frac{1}{2}$$
$$(i)\ \Big|\big||x-1|-2\big|-3\Big|=1$$$$(j)\ \Big|\big||x-3|+2\big|-1\Big|=2$$
$$(k)\ \frac{2+|x|}{3-|x|}=3$$$$(l)\ \left | \frac{x^2+4x-77}{x-7} \right |=2$$

$$(a)\ x\in\left \{ 1;7 \right \}$$$$(b)\ x\in\left \{ 2 \right \}$$
$$(c)\ x\in\left \langle \frac{1}{2};4 \right \rangle$$ $$(d)\ x\in\left \{ \frac{4}{5};2 \right \}$$
$$(e)\ x\in\left \{ -5;-3;1;3 \right \}$$$$(f)\ x\in\left \{ -1;7 \right \}$$
$$(g)\ x\in\left \langle -1;1 \right \rangle$$$$(h)\ x\in\left \{ -\frac{5}{2};-\frac{3}{2};-\frac{1}{2};\frac{1}{2} \right \}$$
$$(i)\ x\in\left \{ -5;-3;1;5;7 \right \}$$$$(j)\ x\in\left \{ 2;4 \right \}$$
$$(k)\ x\in\left \{ \pm\frac{7}{4}\right \}$$$$(l)\ x\in\left \{ -13;-9\right \}$$

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