Resolve as inequações seguintes
Os números reais: Matematicamente Falando 9 - Parte 1 Pág. 31 Ex. 2
Resolve as inequações seguintes:
i) \(4x – 1 < 3x + \frac{1}{2}\)
k) \(5\left( {1 + 3x} \right) + \frac{1}{2} \ge 5x\)
l) \(\frac{1}{3} + \frac{1}{2}\left( {x – 1} \right) < 2x + 1\)
m) \(\frac{{y + 3}}{6} \le 2 – \frac{{4 – 3y}}{2}\)
n) \(\frac{{7x – 3}}{4} – \frac{{9x – 4}}{8} > 0\)
i) \[\begin{array}{*{20}{l}}{\mathop {4x}\limits_{\left( 2 \right)} – \mathop 1\limits_{\left( 2 \right)} < \mathop {3x}\limits_{\left( 2 \right)} + \frac{1}{{\mathop 2\limits_{\left( 1 \right)} }}}& \Leftrightarrow &{8x – 2 < 6x + 1}\\{}& \Leftrightarrow &{2x < 3}\\{}& \Leftrightarrow &{x < \frac{3}{2}}\\{}&{}&{}\\{}&{}&{S = \left] { – \infty ,\;\frac{3}{2}} \right[}\end{array}\]
k) \[\begin{array}{*{20}{l}}{5\left( {1 + 3x} \right) + \frac{1}{2} \ge 5x}& \Leftrightarrow &{5 + 15x + \frac{1}{2} \ge 5x}\\{}& \Leftrightarrow &{10x \ge – \frac{{11}}{2}}\\{}& \Leftrightarrow &{x \ge – \frac{{11}}{{20}}}\\{}&{}&{}\\{}&{}&{S = \left[ { – \frac{{11}}{{20}},\; + \infty } \right[}\end{array}\]
l) \[\begin{array}{*{20}{l}}{\frac{1}{3} + \frac{1}{2}\left( {x – 1} \right) < 2x + 1}& \Leftrightarrow &{\frac{1}{{\mathop 3\limits_{\left( 2 \right)} }} + \frac{x}{{\mathop 2\limits_{\left( 3 \right)} }} – \frac{1}{{\mathop 2\limits_{\left( 3 \right)} }} < \mathop {2x}\limits_{\left( 6 \right)} + \mathop 1\limits_{\left( 6 \right)} }\\{}& \Leftrightarrow &{2 + 3x – 3 < 12x + 6}\\{}& \Leftrightarrow &{ – 9x < 7}\\{}& \Leftrightarrow &{x > – \frac{7}{9}}\\{}&{}&{}\\{}&{}&{S = \left] { – \frac{7}{9},\; + \infty } \right[}\end{array}\]
m) \[\begin{array}{*{20}{l}}{\frac{{y + 3}}{{\mathop 6\limits_{\left( 1 \right)} }} \le \mathop 2\limits_{\left( 6 \right)} – \frac{{4 – 3y}}{{\mathop 2\limits_{\left( 3 \right)} }}}& \Leftrightarrow &{y + 3 \le 12 – 12 + 9y}\\{}& \Leftrightarrow &{ – 8y \le – 3}\\{}& \Leftrightarrow &{y \ge \frac{3}{8}}\\{}&{}&{}\\{}&{}&{S = \left[ {\frac{3}{8},\; + \infty } \right[}\end{array}\]
n) \[\begin{array}{*{20}{l}}{\frac{{7x – 3}}{{\mathop 4\limits_{\left( 2 \right)} }} – \frac{{9x – 4}}{{\mathop 8\limits_{\left( 1 \right)} }} > \mathop 0\limits_{\left( 8 \right)} }& \Leftrightarrow &{14x – 6 – 9x + 4 > 0}\\{}& \Leftrightarrow &{5x > 2}\\{}& \Leftrightarrow &{x > \frac{2}{5}}\\{}&{}&{}\\{}&{}&{S = \left] {\frac{2}{5},\; + \infty } \right[}\end{array}\]
