Qual é o conjunto-solução de cada uma das seguintes condições?
Os números reais: Matematicamente Falando 9 - Parte 1 Pág. 34 Ex. 2
Enunciado
Qual é o conjunto-solução de cada uma das seguintes condições?
- \({\begin{array}{*{20}{c}}{x \ge – 3}& \vee &{x \ge 2}\end{array}}\)
- \({ – 5 < x \le 7}\)
- \({\begin{array}{*{20}{c}}{3y + 1 < 7}& \vee &{y – 8 > 11}\end{array}}\)
- \({\begin{array}{*{20}{c}}{2x – 1 > 5}& \wedge &{4 – x < 7}\end{array}}\)
- \({\begin{array}{*{20}{c}}{a – 4 > \frac{a}{2} – \frac{1}{5}}& \vee &{a + 1 < 3a + 2}\end{array}}\)
- \({\begin{array}{*{20}{c}}{2\left( {2 – x} \right) < x – 14}& \wedge &{\frac{x}{2} \le 7 – \frac{x}{5}}\end{array}}\)
Resolução
- Ora,
\[\begin{array}{*{20}{l}}{\begin{array}{*{20}{c}}{x \ge – 3}& \vee &{x \ge 2}\end{array}}& \Leftrightarrow &{x \ge – 3}\\{}&{}&{}\\{}&{}&{S = \left[ { – 3,\; + \infty } \right[}\end{array}\] - Ora,
\[\begin{array}{*{20}{l}}{ – 5 < x \le 7}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x > – 5}& \wedge &{x \le 7}\end{array}}\\{}&{}&{}\\{}&{}&{S = \left] { – 5,\;7} \right]}\end{array}\] - Ora,
\[\begin{array}{*{20}{l}}{\begin{array}{*{20}{c}}{3y + 1 < 7}& \vee &{y – 8 > 11}\end{array}}& \Leftrightarrow &{\begin{array}{*{20}{c}}{3y < 6}& \vee &{y > 19}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{y < 2}& \vee &{y > 19}\end{array}}\\{}&{}&{}\\{}&{}&{S = \left] { – \infty ,\;2} \right[ \cup \left] {19,\; + \infty } \right[}\end{array}\] - Ora,
\[\begin{array}{*{20}{l}}{\begin{array}{*{20}{c}}{2x – 1 > 5}& \wedge &{4 – x < 7}\end{array}}& \Leftrightarrow &{\begin{array}{*{20}{c}}{2x > 6}& \wedge &{ – x < 3}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x > 3}& \wedge &{x > – 3}\end{array}}\\{}& \Leftrightarrow &{x > 3}\\{}&{}&{}\\{}&{}&{S = \left] {3,\; + \infty } \right[}\end{array}\] - Ora,
\[\begin{array}{*{20}{l}}{\begin{array}{*{20}{c}}{\mathop a\limits_{\left( {10} \right)} – \mathop 4\limits_{\left( {10} \right)} > \frac{a}{{\mathop 2\limits_{\left( 5 \right)} }} – \frac{1}{{\mathop 5\limits_{\left( 2 \right)} }}}& \vee &{a + 1 < 3a + 2}\end{array}}& \Leftrightarrow &{\begin{array}{*{20}{c}}{10a – 40 > 5a – 2}& \vee &{ – 2a < 1}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{5a > 38}& \vee &{a > – \frac{1}{2}}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{a > \frac{{38}}{5}}& \vee &{a > – \frac{1}{2}}\end{array}}\\{}& \Leftrightarrow &{a > – \frac{1}{2}}\\{}&{}&{}\\{}&{}&{S = \left] { – \frac{1}{2},\; + \infty } \right[}\end{array}\] - Ora,
\[\begin{array}{*{20}{l}}{\begin{array}{*{20}{c}}{2\left( {2 – x} \right) < x – 14}& \wedge &{\frac{x}{2} \le 7 – \frac{x}{5}}\end{array}}& \Leftrightarrow &{\begin{array}{*{20}{c}}{4 – 2x < x – 14}& \wedge &{5x \le 70 – 2x}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{ – 3x < – 18}& \wedge &{7x \le 70}\end{array}}\\{}& \Leftrightarrow &{\begin{array}{*{20}{c}}{x > 6}& \wedge &{x \le 10}\end{array}}\\{}&{}&{}\\{}&{}&{S = \left] {6,\;10} \right]}\end{array}\]
