Lineární nerovnice s absolutní hodnotou

Lineární nerovnice s absolutní hodnotou

1. V množině reálných čísel řešte nerovnice

$$(a)\ \left | x \right |\geq 6$$$$(b)\ \left | x+1 \right |>0$$$$(c)\ | x-3 |<2$$
$$(d)\ | x+3 |>4$$$$(e)\ | x+2 |<-1$$$$(f)\ | x-5 |\leq3$$
$$(g)\ | x+5 |\leq7$$$$(h)\ | x-\sqrt{3} |>2+5\sqrt{3}$$$$(i)\ | 2x+1 |<5$$
$$(j)\ | 5x-3 |<0$$$$(k)\ | 3x-1 |\leq5$$$$(l)\ | 2x-3 |\geq6$$

$$(a)\ x\in(-\infty ;-6\rangle\cup\langle6;\infty)$$$$(b)\ x\in\mathbb{R}\setminus\left \{ -1 \right \}$$$$(c)\ x\in\left ( 1;5 \right )$$
$$(d)\ x\in\left (-\infty; -7;\right )\cup(1;\infty)$$$$(e)\ x\in\varnothing$$$$(f)\ x\in\left \langle 2;8 \right \rangle$$
$$(g)\ x\in\left \langle -12;2 \right \rangle$$$$(h)\ x\in\left ( -\infty;-2-4\sqrt{3} \right )\cup \left ( 2+6\sqrt{3};\infty \right )$$$$(i)\ x\in\left ( -3;2 \right )$$
$$(j)\ x\in\varnothing$$$$(k)\ x\in\left \langle -\frac{4}{3};2 \right \rangle$$$$(l)\ x\in\left(\-\infty;-\frac{3}{2}\right\rangle\cup\left\langle\frac{9}{2};\infty\right)$$

2. V množině reálných čísel řešte nerovnice

$$(a)\ 2x-3 \geq|x-2|$$$$(b)\ |2x-3|-5 \leq x+1$$
$$(c)\ |3x-1|< x$$$$(d)\ |x-3|+x< 5$$

$$(a)\ x\in\left\langle \frac53;-\infty\right)$$$$b)\ x\in\left \langle -1;9 \right \rangle$$
$$c)\ x\in\left ( \frac{1}{4};\frac{1}{2} \right )$$$$d)\ x\in\left ( -\infty;4\right )$$

3. V množině reálných čísel řešte nerovnice

$$(a)\ |x+3|>|x-2|$$$$(b)\ |2x-1|<|x+1|$$
$$(c)\ |3x-2|<5+|x+1|$$$$(d)\ |x+1|\leq 2|x|$$
$$(e)\ |x|+|x-1|\geq 2$$$$(f)\ |1-x|>3|x+3|$$
$$(g)\ |x|+|x-3|>10$$$$(h)\ 6-|x+2|\leq 3|x-1|$$

$$a)\ x\in\left (-\frac12;\infty \right )$$$$b)\ x\in\left (0;2 \right )$$
$$c)\ x\in\left (-1;4 \right )$$$$d)\ x\in \left ( -\infty;-\frac13\right\rangle\cup\left\langle1 ;\infty\right )$$
$$e)\ x\in \left ( -\infty;-\frac12\right\rangle\cup\left\langle \frac32 ;\infty\right )$$$$f)\ x\in \left (-5;-2\right )$$
$$g)\ x\in \left (-\infty;-\frac72\right )\cup\left ( \frac{13}{2};\infty \right )$$$$h)\ x\in \left (-\infty;-\frac12\right\rangle \cup\left \langle \frac74;\infty \right )$$

4. V množině reálných čísel řešte nerovnice

$$(a)\ |x-1|+|2-x|>3+x$$$$(b)\ |x|-|x-5|\geq4(x-3)$$
$$(c)\ |2x+3|\leq4+|x|$$$$(d)\ 3|x-1|-2|x+4|<6x-2$$
$$(e)\ |2x-1|-|x+3|<$$$$(f)\ |3-|2-x||\leq2$$
$$(g)\ |x-1|+|4-x|>x+2$$$$(h)\ ||x+4|-|x||>2|x|$$
$$(i)\ |x-3|+3|x-1|<2x+1$$$$(j)\ |x+1|+|x|+|x-3|>2$$

$$a)\ x\in\left (-\infty;0 \right )\cup\left (6;\infty \right )$$$$b)\ x\in\left (-\infty;\frac72 \right \rangle$$
$$c)\ x\in\left \left \langle -7;1 \right \rangle$$$$d)\ x\in\left (-\frac3{11};\infty \right )$$
$$e)\ x\in\left (-\frac23;4 \right )$$ $$f)\ x\in \left \langle -3;1 \right \rangle\cup\left \langle 3;7 \right \rangle$$
$$g)\ x\in \left (-\infty;1 \right )\cup\left (7;\infty \right )$$$$h)\ x\in \left (-1;2 \right )$$
$$i)\ x\in \left (\frac56;\frac72 \right )$$$$j)\ x\in \mathbb{R}$$

5. V množině reálných čísel řešte nerovnice

$$(a)\ |2x-4|+|3x+6|-|5x-2|\leq8-4x$$$$(b)\ 2|x-1|-3|x+2|<3x-|x|$$
$$(c)\ 5|x-1|-3|x-2|+|x-4|+x-5>0$$$$(d)\ 0<|x-5|\leq3$$
$$(e)\ 2\leq|x-4|<5$$$$(f)\ 3<|2x+4|<10$$
$$(g)\ \left\{\begin{array}{l}|x-3|+2|x+1|>4\\|1+2x|-5\leq x\end{array}\right.$$

$$a)\ x\in \left \langle \frac43;\infty \right)$$$$b)\ x\in \left (-\frac49;\infty \right)$$
$$c)\ x\in (-\infty;1)\cup\left(\frac32;\infty \right)$$$$d)\ x\in \left \langle2;5)\cup(5;8 \right \rangle$$
$$e)\ x\in \left (-1;2\rangle \cup\langle6;9 \right )$$$$f)\ x\in \left (-7;-\frac72 \right ) \cup\left (-\frac12;3 \right )$$
$$g)\ x\in \left \langle -2;-1)\cup(-1;4 \right \rangle$$

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