Adding and Subtracting Rational Expressions

Adding fractions: the same principle as arithmetic

Just like adding $\frac{2}{3} + \frac{1}{4}$ requires a common denominator, adding rational expressions requires a common denominator made from both denominators.

Step-by-step: Add $\dfrac{2}{x+1} + \dfrac{3}{x-2}$

Step 1 — Find the LCD. The denominators are $(x+1)$ and $(x-2)$. Since they share no common factors, the LCD is $(x+1)(x-2)$.

Step 2 — Rewrite each fraction with the LCD:

$\frac{2}{x+1} \cdot \frac{x-2}{x-2} + \frac{3}{x-2} \cdot \frac{x+1}{x+1}$

$= \frac{2(x-2)}{(x+1)(x-2)} + \frac{3(x+1)}{(x+1)(x-2)}$

Step 3 — Combine the numerators (denominators are now the same):

$= \frac{2(x-2) + 3(x+1)}{(x+1)(x-2)}$

Step 4 — Expand the numerator:

$= \frac{2x - 4 + 3x + 3}{(x+1)(x-2)} = \frac{5x - 1}{(x+1)(x-2)}$

Step 5 — Check: can the result be simplified? The numerator $5x - 1$ doesn't factor to include $(x+1)$ or $(x-2)$, so the answer is already in simplest form.

Final answer: $\dfrac{5x - 1}{(x+1)(x-2)}$, where $x \neq -1$ and $x \neq 2$.

Common mistakes

• Forgetting to multiply the numerator. When you multiply the denominator by $(x-2)$, you must also multiply the numerator by $(x-2)$.
• Sign errors when distributing. $3(x+1) = 3x + 3$, not $3x + 1$.
• Not checking for simplification. After combining, always check if the numerator shares a factor with the denominator.

Try it in the visualization

Adjust the numerators and denominators. The LCD is computed, both fractions are rewritten, and the combined result is shown step by step. The graph shows the original two functions and their sum.