Question

Resolva, em R as seguintes inequações:

A) 2(3-x) /5 + x/2 > 1/4 + 2(x-1)/3

B) 3x-1/4 - x-3/2 > x+7/4

C) (x-3)^2 - ( 4-x)^2 < x/2

D) 4x-3/5 - 2+x/3 < 3x/5 + 1 - 2x/15


● BOTEI A FOTO DA QUESTÃO ●

Answer 1

$\Large\boxed{\begin{array}{l}\tt b)~\sf\dfrac{2\cdot(3-x)}{5}+\dfrac{x}{2}\geqslant\dfrac{1}{4}+\dfrac{2\cdot(x-1)}{3}\\\\\underline{\rm multiplique\,por\,60\,os\,2\,lados}\\\sf 60\cdot\dfrac{2(3-x)}{5}+60\cdot\dfrac{x}{2}\geqslant60\cdot\dfrac{1}{4}+60\cdot\dfrac{2\cdot(x-1)}{3}\\\\\sf 24(3-x)+30x\geqslant15+40(x-1)\\\sf 72-24x+30x\geqslant15+40x-40\\\sf -24x+30x-40x\geqslant15-40-72\\\sf-34x\geqslant-97\cdot(-1)\\\sf 34x\geqslant97\\\sf x\leqslant\dfrac{97}{34}\end{array}}$

$\Large\boxed{\begin{array}{l}\tt c)~\sf\dfrac{3x-1}{4}-\dfrac{x-3}{2}\geqslant\dfrac{x+7}{4}\\\\\underline{\rm multiplicando\,por\,4\,nos\,2\,lados\,temos:}\\\\\sf4\cdot\dfrac{3x-1}{4}-4\cdot\dfrac{x-3}{2}\geqslant4\cdot\dfrac{x+7}{4}\\\\\sf 3x-1-2\cdot(x-3)\geqslant x+7\\\sf 3x-1-2x+6\geqslant x+7\\\sf 3x-2x-x\geqslant7+1-6\\\sf 0x\geqslant2\\\sf S=\bigg\{\bigg\}\end{array}}$

$\Large\boxed{\begin{array}{l}\tt c)~\sf (x-3)^2-(4-x)^2\leqslant\dfrac{x}{2}\\\\\sf x^2-6x+9-(16-8x+x^2)\leqslant\dfrac{x}{2}\\\\\sf\diagup\!\!\!\!\!x^2-6x+9-16+8x-\diagup\!\!\!\!\!x^2-\dfrac{x}{2}\leqslant0\\\underline{\rm multiplique\,por\,2\,os\,dois\,lados}\\\sf 2\cdot\bigg(-6x+9-16+8x-\dfrac{x}{2}\bigg)\leqslant 2\cdot0\\\sf -12x+18-32+16x-x\leqslant0\\\sf -12x+16x-x\leqslant32-18\\\sf 3x\leqslant14\\\sf x\leqslant\dfrac{14}{3}\\\sf S=\bigg\{x\in\mathbb{R}/ x\leqslant\dfrac{14}{3}\bigg\}\end{array}}$$\Large\boxed{\begin{array}{l}\tt d)~\sf\dfrac{4x-3}{5}-\dfrac{2+x}{3}<\dfrac{3x}{5}+1-\dfrac{2x}{15}\\\\\sf\dfrac{4x-3}{5}+\dfrac{-2-x}{3}<\dfrac{3x}{5}+1-\dfrac{2x}{15}\\\\\sf\dfrac{3(4x-3)+5(-2-x)<9x+15-2x}{15}\\\\\sf 12x-9-10-5x<9x+15-2x\\\sf 12x-5x-9x+2x<15+9+10\\\sf 0x<34\\\sf S=\bigg\{\bigg\}\end{array}}$