Lineární rovnice a nerovnice

1.

$\huge \frac{6+25x}{15}-(x-1)=\frac{2x}{3}+\frac{7}{5}$

$\frac{6+25x}{15}-(x-1)=\frac{2x}{3}+\frac{7}{5}\;/\times15$

$6+25x-15(x-1)=10x+21$

$6+25x-15x+15=10x+21$

$25x-15x-10x=21-15-6$

$0=0$

$x=R$

2.

$\huge x-\frac{1-\frac{3x}{2}}{4}-\frac{2-\frac{x}{4}}{3}=2$

$x-\frac{1-\frac{3x}{2}}{4}-\frac{2-\frac{x}{4}}{3}=2\;/\times12$

$12x-3(1-\frac{3x}{2}-4(2-\frac{x}{4}=24$

$12x-3+\frac{9x}{2}-8+x=24\;/times2$

$26x+9x=70$

$35x=70$

$x=2$

3.

$\huge x-\left(\frac{x}{2}-\frac{3+x}{4}\right)\times\frac{1}{2}=\left[3-(1-\frac{6-x}{3})\times\frac{1}{2}\right]\times\frac{1}{2}+\frac{3}{2}$

$x-\left(\frac{x}{4}-\frac{3+x}{8}\right)=\left[3-(\frac{1}{2}1-\frac{6-x}{6})\right]\times\frac{1}{2}+\frac{3}{2}$

$x-\frac{x}{4}+\frac{3+x}{8}=\frac{3}{2}-\frac{1}{4}+\frac{6-x}{12}+\frac{3}{2}$

$\frac{8x-2x+3+x}{8}=\frac{18-3+6-x+18}{12}$

$\frac{7x+3}{8}=\frac{39-x}{12}\;/\times24$

$21x+9=78-2x$

$23x=69$

$x=3$

4.

$\huge 4(x+1)-\frac{5x+1}{2}-\frac{5x-11}{4}=\frac{x-1}{3}-\frac{2(1-4x)}{9}$

$4(x+1)-\frac{5x+1}{2}-\frac{5x-11}{4}=\frac{x-1}{3}-\frac{2(1-4x)}{9}\;/\times36$

$144(x+1)-18(5x+1)-9(5x-11)=12(x-1)-8(1-4x)$

$144x+144-90x-18-45x+99=12x-12-8+32x$

$9x+225=44x-20$

$-35x=-245$

$x=7$

5.

$\huge 2\left(\frac{3x-1}{4}-\frac{3}{2}\right)-\left(\frac{1+x}{4}+1\right)=\frac{1+5x}{7}-\frac{3}{2}(x+1)$

$\frac{3x-1}{2}-3-\frac{1+x}{4}-1=\frac{1+5x}{7}-\frac{3x+3}{2}$

$\frac{3x-1}{2}-\frac{1+x}{4}-4=\frac{1+5x}{7}-\frac{3x+3}{2}\;/\times28$

$14(3x-1)-7(1+x)-112=4(1+5x)-14(3x+3)$

$42x-14-7-7x-112=4+20x-42x-42$

$35x-133=-22x-38$

$57x=95$

$x=\frac{95}{57}=\frac{5}{3}$

6.

$\huge \frac{2x-1}{2}+\frac{x}{6}<\frac{7x+2}{3}-\frac{x+3}{4}$

$6(2x-1)+2x<4(7x+2)-3(x+3)$

$12x-6+2x<28x+8-3x-9$

$14x-6<25x-1$

$-11x<5$

$x>-\frac{5}{11}$

$x\in\mathbb{R}$

$K=(-\frac{5}{11};\infty)$

7.

$\huge \frac{2x-1}{5}-\frac{3-2x}{4}<3-\frac{x-1}{2}$

$\frac{2x-1}{5}-\frac{3-2x}{4}<3-\frac{x-1}{2}\;/\times 20$

$4(2x-1)-5(3-2x)<60-10(x-1)$

$8x-4-15+10x<60-10x+10$

$18x-19<-10x+70$

$28x<89$

$x<3\frac{5}{28}$

$x\in\mathbb{N}$

$K=\left \{1;2;3 \right \}$

8.

$\huge \frac{4x-3}{5}-\frac{3x-4}{2}+\frac{2x-5}{3}<0$

$\frac{4x-3}{5}-\frac{3x-4}{2}+\frac{2x-5}{3}<0\;/\times 30$

$6(4x-3)-15(3x-4)+10(2x-5)<0$

$24x-18-45x+60+20x-50<0$

$-x-8<0$

$x>-8$

$K=\left \{x\in\mathbb{Z}\wedge (-8;\infty)\right \}$

9.

$\huge \frac{2x-1}{3}-\frac{x+3}{2}<3-\frac{x-2}{3}$

$\frac{2x-1}{3}-\frac{x+3}{2}<3-\frac{x-2}{3}\;/\times 6$

$2(2x-1)-3(x+3)<18-2(x-2)$

$4x-2-3x-9<18-2x+4$

$3x<33$

$x<11$

$x\in\mathbb{N}$

$K=\left \{1;2;3;4;5;6;7;8;9;10 \right \}$

10.

$\huge \frac{x+1}{5}-\frac{x-1}{2}-3<\frac{2x-1}{2}$

$\frac{x+1}{5}-\frac{x-1}{2}-3<\frac{2x-1}{2}\;/\times 10$

$2(x+1)-5(x-1)-30<5(2x-1)$

$2x+2-5x+5-30<10x-5$

$-3x-23<10x-5$

$-13x<18$

$x>-\frac{18}{13}$

$x>-1\frac{5}{13}$

$x\in\mathbb{Z^{-}}$

$K=\left \{-1 \right \}$

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