Simplify the following expression: 2/(x-3) + 5/(x+4) - 7/(x+2)

Steps to Solve

1. Find a Common Denominator

The common denominator for the three fractions is the product of their individual denominators: $(x-3)(x+4)(x+2)$. We will rewrite each fraction with this common denominator.

2. Rewrite Each Fraction

• For the first fraction, $\frac{2}{x-3}$, multiply the numerator and denominator by $(x+4)(x+2)$: $$ \frac{2(x+4)(x+2)}{(x-3)(x+4)(x+2)} = \frac{2(x^2 + 6x + 8)}{(x-3)(x+4)(x+2)} = \frac{2x^2 + 12x + 16}{(x-3)(x+4)(x+2)} $$
• For the second fraction, $\frac{5}{x+4}$, multiply the numerator and denominator by $(x-3)(x+2)$: $$ \frac{5(x-3)(x+2)}{(x-3)(x+4)(x+2)} = \frac{5(x^2 -x - 6)}{(x-3)(x+4)(x+2)} = \frac{5x^2 - 5x - 30}{(x-3)(x+4)(x+2)} $$
• For the third fraction, $\frac{7}{x+2}$, multiply the numerator and denominator by $(x-3)(x+4)$: $$ \frac{7(x-3)(x+4)}{(x-3)(x+4)(x+2)} = \frac{7(x^2 + x - 12)}{(x-3)(x+4)(x+2)} = \frac{7x^2 + 7x - 84}{(x-3)(x+4)(x+2)} $$

3. Combine the Fractions

Now, combine the fractions using the common denominator: $$ \frac{(2x^2 + 12x + 16) + (5x^2 - 5x - 30) - (7x^2 + 7x - 84)}{(x-3)(x+4)(x+2)} $$ Simplify the numerator: $$ \frac{2x^2 + 12x + 16 + 5x^2 - 5x - 30 - 7x^2 - 7x + 84}{(x-3)(x+4)(x+2)} $$ $$ \frac{(2x^2 + 5x^2 - 7x^2) + (12x - 5x - 7x) + (16 - 30 + 84)}{(x-3)(x+4)(x+2)} $$ $$ \frac{0x^2 + 0x + 70}{(x-3)(x+4)(x+2)} $$ $$ \frac{70}{(x-3)(x+4)(x+2)} $$

4. Final Simplified Expression

The simplified expression is: $$ \frac{70}{(x-3)(x+4)(x+2)} $$