Resolve as seguintes inequações
Os números reais: Matematicamente Falando 9 - Parte 1 Pág. 36 Ex. 16
Resolve as seguintes inequações, representando o conjunto-solução sob a forma de intervalo de números reais.
h) \(2x – \frac{3}{2} < 4x + \frac{{2x – 1}}{2}\)
i) \(\frac{{5x + 1}}{2} – \frac{{x – 7}}{3} \ge x\)
j) \(\frac{3}{4}\left( {x + 1} \right) \le 7 – \frac{2}{3}\left( {1 – x} \right)\)
k) \(\frac{{3 – x}}{5} + 0,1\left( {2x + 1} \right) \ge 0\)
l) \(\frac{{0,2x – 1}}{3} \ge \frac{{0,8 – x}}{2}\)
m) \( – x – \frac{{2\left( {8x – 3} \right)}}{{21}} > \frac{{3x}}{7} – \frac{{5 – 3x}}{7}\)
h)
\[\begin{array}{*{20}{l}}{\mathop {2x}\limits_{\left( 2 \right)} – \frac{3}{{\mathop 2\limits_{\left( 1 \right)} }} < \mathop {4x}\limits_{\left( 2 \right)} + \frac{{2x – 1}}{{\mathop 2\limits_{\left( 1 \right)} }}}& \Leftrightarrow &{4x – 3 < 8x + 2x – 1}\\{}& \Leftrightarrow &{ – 6x < 2}\\{}& \Leftrightarrow &{x > – \frac{1}{3}}\\{}&{}&{}\\{}&{}&{S = \left] { – \frac{1}{3},\; + \infty } \right[}\end{array}\]
i)
\[\begin{array}{*{20}{l}}{\frac{{5x + 1}}{{\mathop 2\limits_{\left( 3 \right)} }} – \frac{{x – 7}}{{\mathop 3\limits_{\left( 2 \right)} }} \ge \mathop x\limits_{\left( 6 \right)} }& \Leftrightarrow &{15x + 3 – 2x + 14 \ge 6x}\\{}& \Leftrightarrow &{7x \ge – 17}\\{}& \Leftrightarrow &{x \ge – \frac{{17}}{7}}\\{}&{}&{}\\{}&{}&{S = \left[ { – \frac{{17}}{7},\; + \infty } \right[}\end{array}\]
j)
\[\begin{array}{*{20}{l}}{\frac{3}{{\mathop 4\limits_{\left( 3 \right)} }}\left( {x + 1} \right) \le \mathop 7\limits_{\left( {12} \right)} – \frac{2}{{\mathop 3\limits_{\left( 4 \right)} }}\left( {1 – x} \right)}& \Leftrightarrow &{9\left( {x + 1} \right) \le 84 – 8\left( {1 – x} \right)}\\{}& \Leftrightarrow &{9x + 9 \le 84 – 8 + 8x}\\{}& \Leftrightarrow &{x \le 67}\\{}&{}&{}\\{}&{}&{S = \left] { – \infty ,\;67} \right]}\end{array}\]
k)
\[\begin{array}{*{20}{l}}{\frac{{3 – x}}{{\mathop 5\limits_{\left( 2 \right)} }} + \mathop {0,1}\limits_{\left( {10} \right)} \left( {2x + 1} \right) \ge \mathop 0\limits_{\left( {10} \right)} }& \Leftrightarrow &{6 – 2x + 1 \times \left( {2x + 1} \right) \ge 0}\\{}& \Leftrightarrow &{0x \ge – 7}\\{}& \Leftrightarrow &{x \in \mathbb{R}}\\{}&{}&{}\\{}&{}&{S = \left] { – \infty ,\; + \infty } \right]}\end{array}\]
l)
\[\begin{array}{*{20}{l}}{\frac{{0,2x – 1}}{{\mathop 3\limits_{\left( 2 \right)} }} \ge \frac{{0,8 – x}}{{\mathop 2\limits_{\left( 3 \right)} }}}& \Leftrightarrow &{0,4x – 2 \ge 2,4 – 3x}\\{}& \Leftrightarrow &{4x – 20 \ge 24 – 30x}\\{}& \Leftrightarrow &{34x \ge 44}\\{}& \Leftrightarrow &{x \ge \frac{{22}}{{17}}}\\{}&{}&{}\\{}&{}&{S = \left[ {\frac{{22}}{{17}},\; + \infty } \right[}\end{array}\]
m)
\[\begin{array}{*{20}{l}}{\mathop { – x}\limits_{\left( {21} \right)} – \frac{{2\left( {8x – 3} \right)}}{{\mathop {21}\limits_{\left( 1 \right)} }} \ge \frac{{3x}}{{\mathop 7\limits_{\left( 3 \right)} }} – \frac{{5 – 3x}}{{\mathop 7\limits_{\left( 3 \right)} }}}& \Leftrightarrow &{ – 21x – 16x + 6 \ge 9x – 15 + 9x}\\{}& \Leftrightarrow &{ – 55x \ge – 21}\\{}& \Leftrightarrow &{x \le \frac{{21}}{{55}}}\\{}&{}&{}\\{}&{}&{S = \left] { – \infty ,\;\frac{{21}}{{55}}} \right]}\end{array}\]
