Solve the following inequalities and represent their solutions on a number line: -6x + 12 > 0, 3(4x - 2) igh tarrow= 6, and 3 − 4x < 11.

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

$$\frac{-6x}{-6} < \frac{-12}{-6}$$ $$x < 2$$

The solution is $\boxed{x < 2}$.

Graphical Representation:
Draw a number line. Place an open circle at 2 (since $x$ is strictly less than 2, 2 is not included). Draw an arrow extending to the left from the open circle, indicating all numbers less than 2.

2) Solve $3(4x - 2) \ge 6$

$$12x \ge 6 + 6$$ $$12x \ge 12$$

$$\frac{12x}{12} \ge \frac{12}{12}$$ $$x \ge 1$$

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

Graphical Representation:
Draw a number line. Place a closed circle (or shaded circle) at 1 (since $x$ is greater than or equal to 1, 1 is included). Draw an arrow extending to the right from the closed circle, indicating all numbers greater than or equal to 1.

3) Solve $3 - 4x < 11$

$$-4x < 11 - 3$$ $$-4x < 8$$

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

$$\frac{-4x}{-4} > \frac{8}{-4}$$ $$x > -2$$

The solution is $\boxed{x > -2}$.

Graphical Representation:
Draw a number line. Place an open circle at -2 (since $x$ is strictly greater than -2, -2 is not included). Draw an arrow extending to the right from the open circle, indicating all numbers greater than -2.