Solve the following algebraic inequalities: Q.8.1.1 2x-1/3 <= 5; Q.8.1.2 -5x+3 <= 8; Q.8.1.3 3x+2 >= 2(x-1). Determine the linear equation for the line that goes through the following coordinates: Q.9.1. (12.5) and (3).

$$3 \times \frac{2x - 1}{3} \le 5 \times 3$$ $$2x - 1 \le 15$$

$$2x - 1 + 1 \le 15 + 1$$ $$2x \le 16$$

$$\frac{2x}{2} \le \frac{16}{2}$$ $$x \le 8$$

The solution is $\boxed{x \le 8}$.

Q.8.1.2 Solve the following algebraic inequalities:

$$-5x + 3 \le 8$$

$$-5x + 3 - 3 \le 8 - 3$$ $$-5x \le 5$$

Remember to reverse the inequality sign when dividing by a negative number.

$$\frac{-5x}{-5} \ge \frac{5}{-5}$$ $$x \ge -1$$

The solution is $\boxed{x \ge -1}$.

Q.8.1.3 Solve the following algebraic inequalities:

$$3x + 2 \ge 2(x - 1)$$

$$3x - 2x + 2 \ge 2x - 2x - 2$$ $$x + 2 \ge -2$$

$$x + 2 - 2 \ge -2 - 2$$ $$x \ge -4$$

The solution is $\boxed{x \ge -4}$.

Q.9.1. Determine the linear equation for the line that goes through the following coordinates:

$$(12, 5) \text{ and } (3:)$$

The second coordinate is incomplete. I will assume the second point is $(3, 3)$ for the purpose of solving. If this is not what you intended, please clarify the second coordinate.

Given points: $(x_1, y_1) = (12, 5)$ and $(x_2, y_2) = (3, 3)$.

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$ $$m = \frac{3 - 5}{3 - 12}$$ $$m = \frac{-2}{-9}$$ $$m = \frac{2}{9}$$

$$y - 5 = \frac{2}{9}x - \frac{2}{9} \times 12$$ $$y - 5 = \frac{2}{9}x - \frac{24}{9}$$ $$y - 5 = \frac{2}{9}x - \frac{8}{3}$$ $$y = \frac{2}{9}x - \frac{8}{3} + 5$$ $$y = \frac{2}{9}x - \frac{8}{3} + \frac{15}{3}$$ $$y = \frac{2}{9}x + \frac{7}{3}$$

The linear equation is $\boxed{y = \frac{2}{9}x + \frac{7}{3}}$.