Basic Subtraction with Monomials

Simplify: $\frac{7x}{x + 3} - \frac{2x}{x + 3}$

Verify denominators are the same: Both have denominator $(x + 3)$ = Like denominators confirmed

Subtract the numerators: $7x - 2x = 5x$ = $\frac{5x}{x + 3}$

Check if result can be simplified: $5x$ and $(x + 3)$ share no common factors = Cannot simplify further

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

Subtraction with Binomial Numerators

Simplify: $\frac{4x + 7}{x - 5} - \frac{x + 2}{x - 5}$

Verify denominators match: Both have denominator $(x - 5)$ = Like denominators confirmed

Set up subtraction with parentheses: $\frac{(4x + 7) - (x + 2)}{x - 5}$ = Parentheses protect the second numerator

Distribute the negative sign: $4x + 7 - x - 2$ = The $+2$ becomes $-2$

Combine like terms: $(4x - x) + (7 - 2) = 3x + 5$ = $\frac{3x + 5}{x - 5}$

Check for simplification: No common factors between $3x + 5$ and $x - 5$ = Final answer

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

Subtraction Requiring Simplification

Simplify: $\frac{x^2 + 3x}{x + 2} - \frac{2x}{x + 2}$

Verify like denominators: Both have denominator $(x + 2)$ = Like denominators confirmed

Subtract the numerators: $\frac{x^2 + 3x - 2x}{x + 2}$ = $\frac{x^2 + x}{x + 2}$

Factor the numerator: $x^2 + x = x(x + 1)$ = $\frac{x(x + 1)}{x + 2}$

Check for common factors: No common factors between $x(x+1)$ and $(x+2)$ = Cannot simplify further

Answer: $\frac{x(x + 1)}{x + 2}$ or equivalently $\frac{x^2 + x}{x + 2}$

Subtraction with Opposite Signs

Simplify: $\frac{2x - 3}{x^2 - 4} - \frac{x - 7}{x^2 - 4}$

Confirm like denominators: Both have denominator $x^2 - 4$ = Like denominators confirmed

Write subtraction with parentheses: $\frac{(2x - 3) - (x - 7)}{x^2 - 4}$ = Ready to distribute negative

Distribute the negative sign carefully: $2x - 3 - x + 7$ = The $-7$ becomes $+7$

Combine like terms: $(2x - x) + (-3 + 7) = x + 4$ = $\frac{x + 4}{x^2 - 4}$

Factor denominator and check: $x^2 - 4 = (x+2)(x-2)$ = No common factors with $(x + 4)$

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

Subtraction That Simplifies Completely

Simplify: $\frac{3x + 6}{x + 2} - \frac{x + 4}{x + 2}$

Verify like denominators: Both have $(x + 2)$ = Like denominators confirmed

Subtract with parentheses: $\frac{(3x + 6) - (x + 4)}{x + 2}$ = Set up for distribution

Distribute negative sign: $3x + 6 - x - 4$ = All signs distributed

Combine like terms: $(3x - x) + (6 - 4) = 2x + 2$ = $\frac{2x + 2}{x + 2}$

Factor numerator: $2x + 2 = 2(x + 1)$ = $\frac{2(x + 1)}{x + 2}$

Check for cancellation: $(x + 1)$ and $(x + 2)$ are different factors = No cancellation possible

Answer: $\frac{2(x + 1)}{x + 2}$ or $\frac{2x + 2}{x + 2}$

Mistake: Forgetting to distribute the negative sign to all terms

Why: When you subtract $(x - 3)$, the negative applies to both terms: $-x$ AND $+3$.

Correct: Always use parentheses: $(5x + 2) - (x - 3) = 5x + 2 - x + 3 = 4x + 5$

Mistake: Subtracting the denominators too

Why: With like denominators, only the numerators change. The denominator stays the same.

Correct: $\frac{A}{C} - \frac{B}{C} = \frac{A - B}{C}$, NOT $\frac{A - B}{C - C}$

Mistake: Not simplifying the final answer

Why: After subtracting, always factor and look for common factors to cancel.

Correct: If you get $\frac{2x + 4}{x + 2}$, factor to $\frac{2(x + 2)}{x + 2} = 2$

Mistake: Dropping parentheses before distributing

Why: Writing $\frac{A - B}{C}$ without parentheses around $B$ can lead to sign errors.

Correct: Always write $\frac{(A) - (B)}{C}$ first, then distribute

Profit Margin Analysis

Businesses calculate net profit margins by subtracting cost ratios from revenue ratios.

If revenue per unit is $\frac{5x + 100}{x}$ and cost per unit is $\frac{3x + 60}{x}$, the profit per unit is $\frac{5x + 100 - 3x - 60}{x} = \frac{2x + 40}{x}$.

Physics: Relative Motion

When calculating relative velocities or differences in rates, we often subtract rational expressions.

If object A moves at $\frac{d}{t-2}$ and object B at $\frac{d-10}{t-2}$, their relative position involves $\frac{d - (d-10)}{t-2} = \frac{10}{t-2}$.

Q: Why do I need parentheses when subtracting?

A: Parentheses ensure you distribute the negative sign to all terms. Without them, you might only negate the first term and get the wrong answer.

Q: What if my answer can be factored but not simplified?

A: That's fine! As long as there are no common factors between the numerator and denominator, your factored form is the final answer.

Q: Can the result be zero?

A: Yes! If the numerators are identical, subtracting gives $\frac{0}{C} = 0$ (where $C \neq 0$).