Resolução livro Cálculo A de Diva Flemming, Seção 1.6 página 10

1.Determinar todos os intervalos de números que satisfazem as desigualdades abaixo. Fazer a representação gráfica.

b)  $\mathrm{2x-5<\frac{1}{3}+\frac{3}{4}x+\frac{1-x}{3}}$

 $\mathrm{2x-5<\frac{1}{3}+\frac{3}{4}x+\frac{1-x}{3}}$
$\mathrm{2x{\color{Red} -}\frac{3}{4}{\color{Red} -}\frac{1-x}{3}}x<\frac{1}{3}{\color{Red} +}5$
$\mathrm{2x{\color{Red} -}\frac{3}{4}{\color{Red} -}\frac{1-x}{3}x<\frac{1}{3}{\color{Red} +}5 \Rightarrow \frac{1+15}{3}}$
$\mathrm{2x-\frac{9x-4(1-x)}{12}< \frac{16}{3}}$
$\mathrm{\frac{24x-9x-4+4x)}{12}< \frac{16}{3}}$
$\mathrm{\frac{19x-4}{12}< \frac{16}{3}}$
$\mathrm{\frac{19x}{12}-\frac{4}{12}< \frac{16}{3}}$
$\mathrm{\frac{19x}{12}< \frac{16}{3}}{\color{Red} +}\frac{4}{12}$
$\mathrm{\frac{19x}{12}< \frac{16}{3}{\color{Red} +}\frac{4}{12}\Rightarrow \frac{12+192}{36}}$
$\mathrm{\frac{19x}{12}< \frac{204}{36}\Rightarrow {\color{Red} \div 6}\Rightarrow \frac{34}{6} {\color{Red} \div 2}\Rightarrow \frac{17}{3}}$
$\mathrm{\frac{19x}{12}< \frac{17}{3}}$
$\mathrm{3{\color{Red} \cdot} 19x< 17{\color{Red} \cdot} 12}$
$\mathrm{57x< 204}$
$\mathrm{x< \frac{204}{57}}$
$\mathrm{x< \frac{204}{57}\Rightarrow {\color{Red} \div 3}\Rightarrow \frac{68}{19} }$
$\mathrm{x<\frac{68}{19} }$






Resposta: {x ∈ R| -∞ ≤ x ≤ 68/19}

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