Basic Addition with Variable Numerators

Add: $\frac{3x}{x+2} + \frac{5}{x+2}$

Verify denominators are the same: Both fractions have denominator $(x+2)$ = Like denominators confirmed

Add the numerators: $3x + 5$ = $3x + 5$

Write over the common denominator: $\frac{3x + 5}{x+2}$ = $\frac{3x + 5}{x+2}$

Check if simplification is possible: $3x + 5$ and $x + 2$ share no common factors = Cannot simplify

Answer: $\frac{3x + 5}{x+2}$

Adding with Binomial Numerators

Add: $\frac{x+3}{x-1} + \frac{2x-5}{x-1}$

Confirm like denominators: Both have $(x-1)$ = Like denominators

Add the numerators: $(x+3) + (2x-5)$ = $x + 3 + 2x - 5$

Combine like terms: $x + 2x + 3 - 5$ = $3x - 2$

Write the result: $\frac{3x-2}{x-1}$ = $\frac{3x-2}{x-1}$

Answer: $\frac{3x-2}{x-1}$

Addition with Polynomial Denominators

Add: $\frac{x^2+1}{x^2-4} + \frac{3x-5}{x^2-4}$

Verify same denominators: Both have $(x^2-4)$ = Like denominators

Add numerators: $(x^2+1) + (3x-5)$ = $x^2 + 3x + 1 - 5$

Combine like terms: $x^2 + 3x - 4$ = $x^2 + 3x - 4$

Factor numerator if possible: $x^2 + 3x - 4 = (x+4)(x-1)$ = $(x+4)(x-1)$

Factor denominator: $x^2 - 4 = (x+2)(x-2)$ = $(x+2)(x-2)$

Check for common factors: No common factors between numerator and denominator = Cannot simplify

Answer: $\frac{(x+4)(x-1)}{(x+2)(x-2)}$ or $\frac{x^2+3x-4}{x^2-4}$

Addition That Simplifies

Add: $\frac{x+1}{x+3} + \frac{2}{x+3}$

Confirm same denominators: Both have $(x+3)$ = Like denominators

Add the numerators: $(x+1) + 2$ = $x + 3$

Write over common denominator: $\frac{x+3}{x+3}$ = $\frac{x+3}{x+3}$

Simplify: $\frac{x+3}{x+3} = 1$ (when $x \neq -3$) = $1$

Answer: $1$ (where $x \neq -3$)

Adding Three Rational Expressions

Add: $\frac{2x}{x-5} + \frac{3}{x-5} + \frac{x+1}{x-5}$

Confirm all denominators match: All three have $(x-5)$ = Like denominators

Add all numerators: $2x + 3 + (x+1)$ = $2x + 3 + x + 1$

Combine like terms: $2x + x + 3 + 1$ = $3x + 4$

Write the result: $\frac{3x+4}{x-5}$ = $\frac{3x+4}{x-5}$

Check for simplification: No common factors = Final answer

Answer: $\frac{3x+4}{x-5}$

Complex Simplification After Adding

Add: $\frac{x^2-9}{x^2+5x+6} + \frac{6}{x^2+5x+6}$

Verify same denominators: Both have $x^2+5x+6$ = Like denominators

Add the numerators: $(x^2-9) + 6$ = $x^2 - 3$

Write over common denominator: $\frac{x^2-3}{x^2+5x+6}$ = Initial result

Factor denominator: $x^2+5x+6 = (x+2)(x+3)$ = $(x+2)(x+3)$

Check if numerator factors: $x^2-3$ does not factor over integers = Cannot factor

Write final answer: $\frac{x^2-3}{(x+2)(x+3)}$ = Simplified form

Answer: $\frac{x^2-3}{(x+2)(x+3)}$

Mistake: Adding the denominators along with the numerators

Why: Students sometimes apply addition to both parts: $\frac{a}{c} + \frac{b}{c} \neq \frac{a+b}{c+c}$

Correct: Keep the denominator the same: $\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$

Mistake: Forgetting to distribute the negative sign

Why: When subtracting, the minus must apply to all terms in the numerator being subtracted.

Correct: $\frac{x+3}{d} - \frac{x-1}{d} = \frac{(x+3)-(x-1)}{d} = \frac{x+3-x+1}{d} = \frac{4}{d}$

Mistake: Not checking for simplification after adding

Why: The sum may contain common factors that should be cancelled.

Correct: Always factor the numerator and denominator after adding to check for common factors.

Mistake: Forgetting domain restrictions

Why: Values that make the denominator zero are excluded from the domain.

Correct: State restrictions: for $\frac{x}{x-2} + \frac{3}{x-2}$, note that $x \neq 2$.

Electrical Circuits

When resistors are in parallel, the total resistance involves adding fractions with the same form.

Two circuit paths contribute $\frac{I_1}{V}$ and $\frac{I_2}{V}$ amperes per volt. Total conductance: $\frac{I_1 + I_2}{V}$.

Work Rate Problems

When two machines work at the same cycle time, their combined output rates add directly.

Machine A produces $\frac{x}{t}$ units and Machine B produces $\frac{y}{t}$ units per hour. Together: $\frac{x+y}{t}$ units per hour.

Q: Why don't we add the denominators?

A: Adding fractions is about combining parts of the same whole. If both fractions are split into the same number of parts (same denominator), you just count how many parts you have total. The size of each part (denominator) stays the same.

Q: What if the result equals 1?

A: If after adding, the numerator equals the denominator (like $\frac{x+3}{x+3}$), the expression simplifies to 1, but only for values where the denominator is not zero.

Q: Can I cancel before adding?

A: No, you can only cancel common factors after combining the numerators. Each fraction must be treated as a whole unit until they are combined.