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

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

1.1.1 $4x - 4 > 0$ Step 1: Add 4 to both sides. $4x > 4$ Step 2: Divide by 4. $x > 1$ Number line representation: An open circle at 1, with a line extending to the right. Solution: $\boxed{x > 1}$

1.1.2 $2x + 4 \ge 8$ Step 1: Subtract 4 from both sides. $2x \ge 4$ Step 2: Divide by 2. $x \ge 2$ Number line representation: A closed circle at 2, with a line extending to the right. Solution: $\boxed{x \ge 2}$

1.1.3 $2x + 6 < 0$ Step 1: Subtract 6 from both sides. $2x < -6$ Step 2: Divide by 2. $x < -3$ Number line representation: An open circle at -3, with a line extending to the left. Solution: $\boxed{x < -3}$

1.1.4 $-3x - 8 \le -5$ Step 1: Add 8 to both sides. $-3x \le 3$ Step 2: Divide by -3 and reverse the inequality sign. $x \ge -1$ 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 $-3 \le 3x \le 6$ Step 1: Divide all parts by 3. $\frac{-3}{3} \le \frac{3x}{3} \le \frac{6}{3}$ $-1 \le x \le 2$ 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 $-1 \le x - 3 < 5$ Step 1: Add 3 to all parts. $-1 + 3 \le x - 3 + 3 < 5 + 3$ $2 \le x < 8$ 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 $1 \le 1 - 2x < 7$ Step 1: Subtract 1 from all parts. $1 - 1 \le 1 - 2x - 1 < 7 - 1$ $0 \le -2x < 6$ Step 2: Divide all parts by -2 and reverse the inequality signs. $\frac{0}{-2} \ge \frac{-2x}{-2} > \frac{6}{-2}$ $0 \ge x > -3$ Step 3: Rewrite in standard order (smallest to largest). $-3 < x \le 0$ 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)