Please don't be rude I'm just a student who wants to know more about inequality I was asking for examples of problems to solve and teach me

This uses the Addition Property of inequalities.

$$2x - 5 + 5 < 11 + 5$$ $$2x < 16$$

Since 2 is a positive number, the direction of the inequality remains the same. This uses the Multiplication/Division Property.

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

The solution is all real numbers less than 8.

$$\boxed{x < 8}$$

Example 2: Inequality with Multiplication/Division by a Negative Number

Solve for $x$:

$$-3x + 7 \ge 19$$

$$-3x + 7 - 7 \ge 19 - 7$$ $$-3x \ge 12$$

Since -3 is a negative number, we must reverse the direction of the inequality sign.

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

The solution is all real numbers less than or equal to -4.

$$\boxed{x \le -4}$$

Example 3: Inequality with Variables on Both Sides

Solve for $x$:

$$4x + 3 \le 2x - 9$$

$$4x + 3 - 2x \le 2x - 9 - 2x$$ $$2x + 3 \le -9$$

$$2x + 3 - 3 \le -9 - 3$$ $$2x \le -12$$

Since 2 is a positive number, the inequality direction remains the same.

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

The solution is all real numbers less than or equal to -6.

$$\boxed{x \le -6}$$