Equations et inéquations : Exercice 1 première année secondaire

1) $$\begin{align*} &(\sqrt{3}-1)x+2=x+\sqrt{3} \\ &\iff  (\sqrt{3}-1)x-x=\sqrt{3}-2 \\ &\iff (\sqrt{3}-1-1)x=\sqrt{3}-2 \\ &\iff (\sqrt{3}-2)x=\sqrt{3}-2 \\ &\iff x=\frac{\sqrt{3}-2}{\sqrt{3}-2} \\ &\iff x=1 \end{align*}$$ Donc     $\boxed{ S_\Bbb R=\left\{{1}\right\}}$

2) $$\begin{align*} &\frac{x-1}{3}-\frac{x+4}{2}=3x+5 \\ &\iff  \frac{2(x-1)}{6}-\frac{3(x+4)}{6}=\frac{6(3x+5)}{6} \\ &\iff 2(x-1)-3(x+4)=6(3x+5) \\ &\iff 2x-2-3x-12=18x+30 \\ &\iff 2x-3x-18x=30+2+12 \\ &\iff -19x=44 \\ &\iff x=-\frac{44}{19} \end{align*}$$ Donc   $\boxed{S_\Bbb R=\left\{{-\frac{44}{19}}\right\}}$

3) $$\begin{align*} &x^3+8=2x^2+4x \\ &\iff  x^3+2^3=2x(x+2) \\ &\iff (x+2)(x^2-2x+4)=2x(x+2) \\ &\iff (x+2)(x^2-2x+4)-2x(x+2)=0 \\ &\iff (x+2)(x^2-2x+4-2x)=0 \\ &\iff (x+2)(x^2-4x+4)=0 \\ &\iff (x+2)(x^2-2\times x\times 2+2^2)=0 \\ &\iff (x+2)(x-2)^2=0 \\ &\iff x+2=0 \; ou \; x-2=0 \\ &\iff x=-2 \; ou \; x=2 \end{align*}$$ Donc   $\boxed{S_\Bbb R=\left\{{-2,2}\right\}}$

4) $$\begin{align*} &\left|{x^2-2x}\right|-3\left|{2-x}\right|=4\left|{x-2}\right| \\ &\iff |x(x-2)|-3|-(x-2)|=4|x-2| \\ &\iff |x(x-2)|-3|x-2|-4|x-2|=0 \\ &\iff |x(x-2)|-7|x-2|=0 \\ &\iff |x||x-2|-7|x-2|=0 \\ &\iff |x-2|(|x|-7)=0 \\ &\iff |x-2|=0\;  ou \;|x|-7=0 \\ &\iff x=2 \; ou \;x=-7 \; ou \;x=7 \end{align*}$$ Donc  $\boxed{S_\Bbb R=\left\{{-7,2,7}\right\}} $