Solve the following inequalities and represent your solutions on a number line.

$$x > 1$$

Number line representation: An open circle at 1, with a line extending to the right.
Solution: $\boxed{x > 1}$

1.1.2 $2x + 4 \ge 8$

$$x \ge 2$$

Number line representation: A closed circle at 2, with a line extending to the right.
Solution: $\boxed{x \ge 2}$

1.1.3 $2x + 6 < 0$

$$x < -3$$

Number line representation: An open circle at -3, with a line extending to the left.
Solution: $\boxed{x < -3}$

1.1.4 $-3x - 8 \le -5$

$$x \ge -1$$

Number line representation: A closed circle at -1, with a line extending to the right.
Solution: $\boxed{x \ge -1}$

1.2 Solve the following inequalities; represent the solutions of each on a number line and explain the solution of each equation in word.

1.2.1 $-3 \le 3x \le 6$

$$\frac{-3}{3} \le \frac{3x}{3} \le \frac{6}{3}$$ $$-1 \le x \le 2$$

Number line representation: A closed circle at -1 and a closed circle at 2, with a line connecting them.
Explanation in words: x is greater than or equal to -1 and less than or equal to 2.
Solution: $\boxed{-1 \le x \le 2}$

1.2.2 $-1 \le x - 3 < 5$

$$-1 + 3 \le x - 3 + 3 < 5 + 3$$ $$2 \le x < 8$$

Number line representation: A closed circle at 2 and an open circle at 8, with a line connecting them.
Explanation in words: x is greater than or equal to 2 and less than 8.
Solution: $\boxed{2 \le x < 8}$

1.2.3 $1 \le 1 - 2x < 7$

$$1 - 1 \le 1 - 2x - 1 < 7 - 1$$ $$0 \le -2x < 6$$

$$\frac{0}{-2} \ge \frac{-2x}{-2} > \frac{6}{-2}$$ $$0 \ge x > -3$$

$$-3 < x \le 0$$

Number line representation: An open circle at -3 and a closed circle at 0, with a line connecting them.
Explanation in words: x is greater than -3 and less than or equal to 0.
Solution: $\boxed{-3 < x \le 0}$

1.3 Represent the following set of linear inequalities on a number line and shade the area where the two inequalities intersect where applicable. Represent the shaded area in set builder notation as in Question 1.2.

1.3.1 $x > -2$ and $x \le 1$
Number line representation: An open circle at -2 and a closed circle at 1, with the line segment between them shaded.
Set builder notation: $\boxed{\{x \mid -2 < x \le 1\}}$

1.3.2 $x < -1$ and $x > 1$
Number line representation: An open circle at -1 with a line extending left, and an open circle at 1 with a line extending right. There is no overlap between these two regions.
Set builder notation: $\boxed{\emptyset}$ (empty set, as there is no intersection)

1.3.3 $x \le 6$ and $x > -1$
Number line representation: An open circle at -1 and a closed circle at 6, with the line segment between them shaded.
Set builder notation: $\boxed{\{x \mid -1 < x \le 6\}}$

1.3.4 $x \le -2$ and $x \ge 3$
Number line representation: A closed circle at -2 with a line extending left, and a closed circle at 3 with a line extending right. There is no overlap between these two regions.
Set builder notation: $\boxed{\emptyset}$ (empty set, as there is no intersection)