Solve the systems of inequalities.

Let's solve each system of inequalities step by step. a) First system: \[\begin{cases} 2(x+3) - (x-8) < 4 \\ 6x > 3(x+1) - 1 \end{cases}\] Let's solve the first inequality: \[2(x+3) - (x-8) < 4 \Rightarrow 2x + 6 - x + 8 < 4 \Rightarrow x + 14 < 4 \Rightarrow x < 4 - 14 \Rightarrow x < -10\] Now, let's solve the second inequality: \[6x > 3(x+1) - 1 \Rightarrow 6x > 3x + 3 - 1 \Rightarrow 6x > 3x + 2 \Rightarrow 6x - 3x > 2 \Rightarrow 3x > 2 \Rightarrow x > \frac{2}{3}\] Combining the inequalities, we have: \[x < -10 \text{ and } x > \frac{2}{3}\] Since there is no overlap between these two conditions, there is no solution for this system. b) Second system: \[\begin{cases} 1.6(2-x) - 0.4x > 3 \\ -3(6x-1) - 2x < x \end{cases}\] Let's solve the first inequality: \[1.6(2-x) - 0.4x > 3 \Rightarrow 3.2 - 1.6x - 0.4x > 3 \Rightarrow 3.2 - 2x > 3 \Rightarrow -2x > 3 - 3.2 \Rightarrow -2x > -0.2 \Rightarrow x < \frac{-0.2}{-2} \Rightarrow x < 0.1\] Now, let's solve the second inequality: \[-3(6x-1) - 2x < x \Rightarrow -18x + 3 - 2x < x \Rightarrow -20x + 3 < x \Rightarrow 3 < x + 20x \Rightarrow 3 < 21x \Rightarrow x > \frac{3}{21} \Rightarrow x > \frac{1}{7}\] Combining the inequalities, we have: \[x < 0.1 \text{ and } x > \frac{1}{7}\] Since \(\frac{1}{7} \approx 0.143\), and (0.1 < 0.143), there is no solution for this system. б) Third system: \[\begin{cases} -(x-2) - 3(x-1) < 2x \\ 5x + 4 > 12 - (x-3) \end{cases}\] Let's solve the first inequality: \[-(x-2) - 3(x-1) < 2x \Rightarrow -x + 2 - 3x + 3 < 2x \Rightarrow -4x + 5 < 2x \Rightarrow 5 < 2x + 4x \Rightarrow 5 < 6x \Rightarrow x > \frac{5}{6}\] Now, let's solve the second inequality: \[5x + 4 > 12 - (x-3) \Rightarrow 5x + 4 > 12 - x + 3 \Rightarrow 5x + 4 > 15 - x \Rightarrow 5x + x > 15 - 4 \Rightarrow 6x > 11 \Rightarrow x > \frac{11}{6}\] Combining the inequalities, we have: \[x > \frac{5}{6} \text{ and } x > \frac{11}{6}\] Since \(\frac{11}{6} > \frac{5}{6}\), the solution is: \[x > \frac{11}{6}\] Therefore, the solution to the third system is \(x > \frac{11}{6}\\). Развёрнутый ответ: a) Первая система неравенств не имеет решений, так как условия (x < -10) и \(x > \frac{2}{3}\) несовместимы. b) Вторая система неравенств также не имеет решений, так как условия (x < 0.1) и \(x > \frac{1}{7}\) несовместимы, учитывая, что \(\frac{1}{7} \approx 0.143 > 0.1\). б) Третья система неравенств имеет решение \(x > \frac{11}{6}\\), так как это наиболее строгое условие из \(x > \frac{5}{6}\) и \(x > \frac{11}{6}\\).