Equations et inéquations : Exercice 17 première année secondaire

a) $2(x+5)>(x+3)-(x-1)$
$\iff 2x+10>x+3-x+1$
$\iff 2x>4$
$\iff x>\frac{4}{2}=2$
Donc $\boxed{S_\Bbb R=\left]{2,+\infty}\right[}$

b) $4-(2x-1)\leqslant 3(4x+1)$
$\iff 4-2x+1\leqslant 12x+3$
$\iff 5-2x\leqslant 12x+3$
$\iff -14x\leqslant -2$
$\iff x\geqslant \frac{-2}{-14}$
$\iff x\geqslant \frac{1}{7}$
Donc $\boxed{S_\Bbb R=\left[{\dfrac{1}{7},+\infty}\right[}$

c) $5-2(x+3)\geqslant 2(x+1)-3(x-2)$
$\iff 5-2x-6\geqslant 2x+2-3x+6$
$\iff -2x-1\geqslant -x+8$
$\iff -2x+x\geqslant 8+1$
$\iff -x\geqslant 9$ $\iff x\leqslant -9$
Donc $\boxed{S_\Bbb R=\left]{-\infty,-9}\right]}$

d) $\frac{3}{14}x-1<\frac{5}{7}$
$\iff \frac{3}{14}x<1+\frac{5}{7}$
$\iff \frac{3}{14}x<\frac{12}{7}$
$\iff x<\frac{12\times 14}{7\times 3}$
$\iff x<8$
Donc $\boxed{S_\Bbb R=\left]{-\infty,8}\right[}$

e) $\frac{1}{4}-x>-\frac{5}{12}$
$\iff -x>\frac{5}{12}-\frac{1}{4}$
$\iff -x>\frac{5}{12}-\frac{3}{12}$
$\iff -x>\frac{2}{12}$
$\iff x<-\frac{1}{6}$
Donc $\boxed{S_\Bbb R=\left]{-\infty,-\dfrac{1}{6}}\right[}$