Solve the inequalities and represent on a number line: i) −4x > 12 ii) 5x - 7 vert 3x + 9 iii) 3x - 5 7

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

$$\frac{-4x}{-4} < \frac{12}{-4}$$ $$x < -3$$

An open circle at $-3$ indicates that $-3$ is not included, and the arrow points to the left.

$$\text{<--------------------o}$$ $$\quad \quad \quad \quad \quad \quad \quad \quad -3$$

The solution in interval notation is $(-\infty, -3)$.

$$5x - 3x - 7 \ge 9$$ $$2x - 7 \ge 9$$

$$2x \ge 9 + 7$$ $$2x \ge 16$$

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

A closed circle at 8 indicates that 8 is included, and the arrow points to the right.

$$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \bullet-------------------->$$ $$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 8$$

The solution in interval notation is $[8, \infty)$.

$$3x \le 7 + 5$$ $$3x \le 12$$

$$\frac{3x}{3} \le \frac{12}{3}$$ $$x \le 4$$

A closed circle at 4 indicates that 4 is included, and the arrow points to the left.

$$\text{<--------------------}\bullet$$ $$\quad \quad \quad \quad \quad \quad \quad \quad 4$$

The solution in interval notation is $(-\infty, 4]$.

2. Solve the compound inequality: $-3 \le 2x - 1 < 7$ and express your solution set interval notation

$$-3 + 1 \le 2x - 1 + 1 < 7 + 1$$ $$-2 \le 2x < 8$$

$$\frac{-2}{2} \le \frac{2x}{2} < \frac{8}{2}$$ $$-1 \le x < 4$$

$$\boxed{[-1, 4)}$$

3. A trader buys $x$ bags of rice at N8,000 each and spends an additional N4,000 on transportation. He sells all the bags for N10,000 each.

Cost Price (CP) = (Cost per bag $\times$ number of bags) + Transportation cost

$$CP = 8000x + 4000$$

Selling Price (SP) = Selling price per bag $\times$ number of bags

$$SP = 10000x$$

$$10000x > 8000x + 4000$$

This is the inequality representing the condition for profit.

$$\boxed{10000x > 8000x + 4000}$$

$$10000x - 8000x > 4000$$ $$2000x > 4000$$

$$\frac{2000x}{2000} > \frac{4000}{2000}$$ $$x > 2$$

The trader makes a profit if he sells more than 2 bags of rice. Since the number of bags must be a whole number, he must sell at least 3 bags to make a profit.

$$\boxed{x > 2 \text{ (The trader must sell more than 2 bags of rice to make a profit.)}}$$

4. Solve this inequality: $\frac{2x - 1}{3} \le \frac{x + 5}{2}$

$$4x - 3x - 2 \le 15$$ $$x - 2 \le 15$$

$$x \le 15 + 2$$ $$x \le 17$$

The solution in interval notation is $(-\infty, 17]$.

$$\boxed{x \le 17}$$