Example: Simplify the fraction $\frac{2x + 6}{x + 3} .$

Solution:

• Factor the numerator: $\ 2x + 6 = 2(x + 3) .$
• The fraction becomes $\ \frac{2(x + 3)}{x + 3} .$
• Since $\ x + 3$ is common to both the numerator and denominator, cancel it out: $\frac{2(x + 3)}{x + 3} = 2$ So, the simplified fraction is $\ 2$.

Example: Add $\frac{1}{x + 2} \ and \ \frac{3}{x - 1} .$

Solution:

1. Find the common denominator: $\ (x + 2)(x - 1) .$
2. Rewrite each fraction with the common denominator: $\frac{1}{x + 2} = \frac{x - 1}{(x + 2)(x - 1)}, \quad \frac{3}{x - 1} = \frac{3(x + 2)}{(x + 2)(x - 1)}$
3. Add the fractions: $\frac{x - 1}{(x + 2)(x - 1)} + \frac{3(x + 2)}{(x + 2)(x - 1)} = \frac{x - 1 + 3(x + 2)}{(x + 2)(x - 1)}$
4. Simplify the numerator: $x - 1 + 3(x + 2) = x - 1 + 3x + 6 = 4x + 5$ $\text{So, } \frac{1}{x + 2} + \frac{3}{x - 1} = \frac{4x + 5}{(x + 2)(x - 1)}.$

Example: Solve $\ \frac{2}{x + 1} + \frac{3}{x - 2} = 1 .$

Solution:

5. Find the common denominator: $\ (x + 1)(x - 2) .$
6. Multiply each term by the common denominator to eliminate the fractions: $\frac{2(x - 2)}{x + 1} + \frac{3(x + 1)}{x - 2} = (x + 1)(x - 2)$ This simplifies to: $2(x - 2) + 3(x + 1) = (x + 1)(x - 2)$
7. Expand and simplify: $2x - 4 + 3x + 3 = x^2 - x - 2$ Combine like terms: $5x - 1 = x^2 - x - 2$
8. Rearrange to form a quadratic equation: $0 = x^2 - 6x - 1$
9. Solve using the quadratic formula: $x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-1)}}{2(1)}$ Simplify: $x = \frac{6 \pm \sqrt{36 + 4}}{2} = \frac{6 \pm \sqrt{40}}{2} = \frac{6 \pm 2\sqrt{10}}{2} = 3 \pm \sqrt{10}$ So, the solutions are $\ x = 3 + \sqrt{10} \ and \ x = 3 - \sqrt{10}$.

Solve the equation $\frac{1}{x + 2} - \frac{2}{x - 3} = 0 .$