Math worksheet focusing on adding and subtracting rational expressions, including problems with like and unlike denominators.
Worksheet for adding and subtracting rational expressions with like and unlike denominators, featuring eight algebraic problems.
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Show Answer Key & Explanations
Step-by-step solution for: Adding and Subtracting Rational Expressions Worksheets - Math Monks
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Show Answer Key & Explanations
Step-by-step solution for: Adding and Subtracting Rational Expressions Worksheets - Math Monks
Here are the step-by-step solutions for each problem on the worksheet.
1) $\frac{4x + 4}{x^2 - 2x - 8} - \frac{3x + 2}{x^2 - 2x - 8}$
* Step 1: Since the denominators are the same, subtract the numerators. Be careful to distribute the negative sign to the second numerator.
$$ \frac{(4x + 4) - (3x + 2)}{x^2 - 2x - 8} $$
$$ \frac{4x + 4 - 3x - 2}{x^2 - 2x - 8} $$
* Step 2: Combine like terms in the numerator ($4x - 3x = x$ and $4 - 2 = 2$).
$$ \frac{x + 2}{x^2 - 2x - 8} $$
* Step 3: Factor the denominator to see if it simplifies. The factors of $-8$ that add to $-2$ are $-4$ and $+2$.
$$ \frac{x + 2}{(x - 4)(x + 2)} $$
* Step 4: Cancel the common factor $(x + 2)$.
$$ \frac{1}{x - 4} $$
2) $\frac{2x^2 + 3}{x^2 - 6x + 5} - \frac{x^2 - 5x + 9}{x^2 - 6x + 5}$
* Step 1: Subtract the numerators. Distribute the negative sign carefully.
$$ \frac{(2x^2 + 3) - (x^2 - 5x + 9)}{x^2 - 6x + 5} $$
$$ \frac{2x^2 + 3 - x^2 + 5x - 9}{x^2 - 6x + 5} $$
* Step 2: Combine like terms ($2x^2 - x^2 = x^2$, $+5x$, and $3 - 9 = -6$).
$$ \frac{x^2 + 5x - 6}{x^2 - 6x + 5} $$
* Step 3: Factor both the numerator and the denominator.
* Numerator ($x^2 + 5x - 6$): Factors are $(x + 6)(x - 1)$.
* Denominator ($x^2 - 6x + 5$): Factors are $(x - 5)(x - 1)$.
$$ \frac{(x + 6)(x - 1)}{(x - 5)(x - 1)} $$
* Step 4: Cancel the common factor $(x - 1)$.
$$ \frac{x + 6}{x - 5} $$
3) $\frac{3v - 4}{2v^2 + 8v + 6} - \frac{2v - 7}{2v^2 + 8v + 6}$
* Step 1: Subtract the numerators.
$$ \frac{(3v - 4) - (2v - 7)}{2v^2 + 8v + 6} $$
$$ \frac{3v - 4 - 2v + 7}{2v^2 + 8v + 6} $$
* Step 2: Combine like terms ($3v - 2v = v$ and $-4 + 7 = 3$).
$$ \frac{v + 3}{2v^2 + 8v + 6} $$
* Step 3: Factor the denominator. First, factor out the 2: $2(v^2 + 4v + 3)$. Then factor the quadratic: $2(v + 3)(v + 1)$.
$$ \frac{v + 3}{2(v + 3)(v + 1)} $$
* Step 4: Cancel the common factor $(v + 3)$.
$$ \frac{1}{2(v + 1)} $$
4) $\frac{3k - 2}{3k^2 - 19k + 6} + \frac{1}{3k^2 - 19k + 6}$
* Step 1: Add the numerators.
$$ \frac{3k - 2 + 1}{3k^2 - 19k + 6} $$
$$ \frac{3k - 1}{3k^2 - 19k + 6} $$
* Step 2: Factor the denominator. We need numbers that multiply to $18$ ($3 \times 6$) and add to $-19$. Those are $-18$ and $-1$.
$$ 3k^2 - 18k - k + 6 = 3k(k - 6) - 1(k - 6) = (3k - 1)(k - 6) $$
So the expression is:
$$ \frac{3k - 1}{(3k - 1)(k - 6)} $$
* Step 3: Cancel the common factor $(3k - 1)$.
$$ \frac{1}{k - 6} $$
---
5) $\frac{4xy}{x^2 - y^2} + \frac{x - y}{x + y}$
* Step 1: Find the Least Common Denominator (LCD). Factor the first denominator: $x^2 - y^2 = (x - y)(x + y)$. The second denominator is just $(x + y)$. The LCD is $(x - y)(x + y)$.
* Step 2: Rewrite the second fraction with the LCD. Multiply top and bottom by $(x - y)$.
$$ \frac{4xy}{(x - y)(x + y)} + \frac{(x - y)(x - y)}{(x + y)(x - y)} $$
* Step 3: Expand the numerator of the second fraction: $(x - y)^2 = x^2 - 2xy + y^2$.
$$ \frac{4xy + (x^2 - 2xy + y^2)}{(x - y)(x + y)} $$
* Step 4: Combine like terms in the numerator ($4xy - 2xy = 2xy$).
$$ \frac{x^2 + 2xy + y^2}{(x - y)(x + y)} $$
* Step 5: Factor the new numerator. It is a perfect square: $(x + y)^2$.
$$ \frac{(x + y)(x + y)}{(x - y)(x + y)} $$
* Step 6: Cancel the common factor $(x + y)$.
$$ \frac{x + y}{x - y} $$
6) $\frac{2}{a^2 - 5a + 4} + \frac{-2}{a^2 - 4}$
* Step 1: Factor the denominators to find the LCD.
* $a^2 - 5a + 4 = (a - 4)(a - 1)$
* $a^2 - 4 = (a - 2)(a + 2)$
* LCD = $(a - 4)(a - 1)(a - 2)(a + 2)$
* Step 2: Rewrite fractions with the LCD.
$$ \frac{2(a - 2)(a + 2)}{\text{LCD}} + \frac{-2(a - 4)(a - 1)}{\text{LCD}} $$
* Step 3: Expand the numerators.
* First part: $2(a^2 - 4) = 2a^2 - 8$
* Second part: $-2(a^2 - 5a + 4) = -2a^2 + 10a - 8$
* Step 4: Add the numerators.
$$ (2a^2 - 8) + (-2a^2 + 10a - 8) = 10a - 16 $$
* Step 5: Write over the LCD and simplify. Factor out 2 from the numerator.
$$ \frac{2(5a - 8)}{(a - 4)(a - 1)(a - 2)(a + 2)} $$
*(Note: Nothing cancels out further).*
7) $2 + \frac{x - 3}{x + 1}$
* Step 1: Turn the whole number 2 into a fraction with the denominator $(x + 1)$.
$$ \frac{2(x + 1)}{x + 1} + \frac{x - 3}{x + 1} $$
* Step 2: Distribute the 2 in the numerator.
$$ \frac{2x + 2}{x + 1} + \frac{x - 3}{x + 1} $$
* Step 3: Add the numerators.
$$ \frac{2x + 2 + x - 3}{x + 1} $$
$$ \frac{3x - 1}{x + 1} $$
8) $\frac{4x}{x^2 - 2x - 3} - \frac{3}{x^2 - 5x + 6}$
* Step 1: Factor the denominators.
* $x^2 - 2x - 3 = (x - 3)(x + 1)$
* $x^2 - 5x + 6 = (x - 3)(x - 2)$
* LCD = $(x - 3)(x + 1)(x - 2)$
* Step 2: Adjust the fractions to have the LCD.
* First fraction needs $(x - 2)$: $\frac{4x(x - 2)}{\text{LCD}}$
* Second fraction needs $(x + 1)$: $\frac{3(x + 1)}{\text{LCD}}$
* Step 3: Expand the numerators.
* Top 1: $4x^2 - 8x$
* Top 2: $3x + 3$
* Step 4: Subtract the second numerator from the first.
$$ (4x^2 - 8x) - (3x + 3) = 4x^2 - 8x - 3x - 3 = 4x^2 - 11x - 3 $$
* Step 5: Check if the numerator factors to cancel anything.
* Factors of $4 \times -3 = -12$ that add to $-11$ are $-12$ and $1$.
* $4x^2 - 12x + x - 3 = 4x(x - 3) + 1(x - 3) = (4x + 1)(x - 3)$.
* Step 6: Write the final fraction and cancel.
$$ \frac{(4x + 1)(x - 3)}{(x - 3)(x + 1)(x - 2)} $$
Cancel $(x - 3)$.
$$ \frac{4x + 1}{(x + 1)(x - 2)} $$
Final Answer:
1) $\frac{1}{x - 4}$
2) $\frac{x + 6}{x - 5}$
3) $\frac{1}{2(v + 1)}$
4) $\frac{1}{k - 6}$
5) $\frac{x + y}{x - y}$
6) $\frac{2(5a - 8)}{(a - 4)(a - 1)(a - 2)(a + 2)}$
7) $\frac{3x - 1}{x + 1}$
8) $\frac{4x + 1}{(x + 1)(x - 2)}$
With Like Denominators
1) $\frac{4x + 4}{x^2 - 2x - 8} - \frac{3x + 2}{x^2 - 2x - 8}$
* Step 1: Since the denominators are the same, subtract the numerators. Be careful to distribute the negative sign to the second numerator.
$$ \frac{(4x + 4) - (3x + 2)}{x^2 - 2x - 8} $$
$$ \frac{4x + 4 - 3x - 2}{x^2 - 2x - 8} $$
* Step 2: Combine like terms in the numerator ($4x - 3x = x$ and $4 - 2 = 2$).
$$ \frac{x + 2}{x^2 - 2x - 8} $$
* Step 3: Factor the denominator to see if it simplifies. The factors of $-8$ that add to $-2$ are $-4$ and $+2$.
$$ \frac{x + 2}{(x - 4)(x + 2)} $$
* Step 4: Cancel the common factor $(x + 2)$.
$$ \frac{1}{x - 4} $$
2) $\frac{2x^2 + 3}{x^2 - 6x + 5} - \frac{x^2 - 5x + 9}{x^2 - 6x + 5}$
* Step 1: Subtract the numerators. Distribute the negative sign carefully.
$$ \frac{(2x^2 + 3) - (x^2 - 5x + 9)}{x^2 - 6x + 5} $$
$$ \frac{2x^2 + 3 - x^2 + 5x - 9}{x^2 - 6x + 5} $$
* Step 2: Combine like terms ($2x^2 - x^2 = x^2$, $+5x$, and $3 - 9 = -6$).
$$ \frac{x^2 + 5x - 6}{x^2 - 6x + 5} $$
* Step 3: Factor both the numerator and the denominator.
* Numerator ($x^2 + 5x - 6$): Factors are $(x + 6)(x - 1)$.
* Denominator ($x^2 - 6x + 5$): Factors are $(x - 5)(x - 1)$.
$$ \frac{(x + 6)(x - 1)}{(x - 5)(x - 1)} $$
* Step 4: Cancel the common factor $(x - 1)$.
$$ \frac{x + 6}{x - 5} $$
3) $\frac{3v - 4}{2v^2 + 8v + 6} - \frac{2v - 7}{2v^2 + 8v + 6}$
* Step 1: Subtract the numerators.
$$ \frac{(3v - 4) - (2v - 7)}{2v^2 + 8v + 6} $$
$$ \frac{3v - 4 - 2v + 7}{2v^2 + 8v + 6} $$
* Step 2: Combine like terms ($3v - 2v = v$ and $-4 + 7 = 3$).
$$ \frac{v + 3}{2v^2 + 8v + 6} $$
* Step 3: Factor the denominator. First, factor out the 2: $2(v^2 + 4v + 3)$. Then factor the quadratic: $2(v + 3)(v + 1)$.
$$ \frac{v + 3}{2(v + 3)(v + 1)} $$
* Step 4: Cancel the common factor $(v + 3)$.
$$ \frac{1}{2(v + 1)} $$
4) $\frac{3k - 2}{3k^2 - 19k + 6} + \frac{1}{3k^2 - 19k + 6}$
* Step 1: Add the numerators.
$$ \frac{3k - 2 + 1}{3k^2 - 19k + 6} $$
$$ \frac{3k - 1}{3k^2 - 19k + 6} $$
* Step 2: Factor the denominator. We need numbers that multiply to $18$ ($3 \times 6$) and add to $-19$. Those are $-18$ and $-1$.
$$ 3k^2 - 18k - k + 6 = 3k(k - 6) - 1(k - 6) = (3k - 1)(k - 6) $$
So the expression is:
$$ \frac{3k - 1}{(3k - 1)(k - 6)} $$
* Step 3: Cancel the common factor $(3k - 1)$.
$$ \frac{1}{k - 6} $$
---
With Unlike Denominators
5) $\frac{4xy}{x^2 - y^2} + \frac{x - y}{x + y}$
* Step 1: Find the Least Common Denominator (LCD). Factor the first denominator: $x^2 - y^2 = (x - y)(x + y)$. The second denominator is just $(x + y)$. The LCD is $(x - y)(x + y)$.
* Step 2: Rewrite the second fraction with the LCD. Multiply top and bottom by $(x - y)$.
$$ \frac{4xy}{(x - y)(x + y)} + \frac{(x - y)(x - y)}{(x + y)(x - y)} $$
* Step 3: Expand the numerator of the second fraction: $(x - y)^2 = x^2 - 2xy + y^2$.
$$ \frac{4xy + (x^2 - 2xy + y^2)}{(x - y)(x + y)} $$
* Step 4: Combine like terms in the numerator ($4xy - 2xy = 2xy$).
$$ \frac{x^2 + 2xy + y^2}{(x - y)(x + y)} $$
* Step 5: Factor the new numerator. It is a perfect square: $(x + y)^2$.
$$ \frac{(x + y)(x + y)}{(x - y)(x + y)} $$
* Step 6: Cancel the common factor $(x + y)$.
$$ \frac{x + y}{x - y} $$
6) $\frac{2}{a^2 - 5a + 4} + \frac{-2}{a^2 - 4}$
* Step 1: Factor the denominators to find the LCD.
* $a^2 - 5a + 4 = (a - 4)(a - 1)$
* $a^2 - 4 = (a - 2)(a + 2)$
* LCD = $(a - 4)(a - 1)(a - 2)(a + 2)$
* Step 2: Rewrite fractions with the LCD.
$$ \frac{2(a - 2)(a + 2)}{\text{LCD}} + \frac{-2(a - 4)(a - 1)}{\text{LCD}} $$
* Step 3: Expand the numerators.
* First part: $2(a^2 - 4) = 2a^2 - 8$
* Second part: $-2(a^2 - 5a + 4) = -2a^2 + 10a - 8$
* Step 4: Add the numerators.
$$ (2a^2 - 8) + (-2a^2 + 10a - 8) = 10a - 16 $$
* Step 5: Write over the LCD and simplify. Factor out 2 from the numerator.
$$ \frac{2(5a - 8)}{(a - 4)(a - 1)(a - 2)(a + 2)} $$
*(Note: Nothing cancels out further).*
7) $2 + \frac{x - 3}{x + 1}$
* Step 1: Turn the whole number 2 into a fraction with the denominator $(x + 1)$.
$$ \frac{2(x + 1)}{x + 1} + \frac{x - 3}{x + 1} $$
* Step 2: Distribute the 2 in the numerator.
$$ \frac{2x + 2}{x + 1} + \frac{x - 3}{x + 1} $$
* Step 3: Add the numerators.
$$ \frac{2x + 2 + x - 3}{x + 1} $$
$$ \frac{3x - 1}{x + 1} $$
8) $\frac{4x}{x^2 - 2x - 3} - \frac{3}{x^2 - 5x + 6}$
* Step 1: Factor the denominators.
* $x^2 - 2x - 3 = (x - 3)(x + 1)$
* $x^2 - 5x + 6 = (x - 3)(x - 2)$
* LCD = $(x - 3)(x + 1)(x - 2)$
* Step 2: Adjust the fractions to have the LCD.
* First fraction needs $(x - 2)$: $\frac{4x(x - 2)}{\text{LCD}}$
* Second fraction needs $(x + 1)$: $\frac{3(x + 1)}{\text{LCD}}$
* Step 3: Expand the numerators.
* Top 1: $4x^2 - 8x$
* Top 2: $3x + 3$
* Step 4: Subtract the second numerator from the first.
$$ (4x^2 - 8x) - (3x + 3) = 4x^2 - 8x - 3x - 3 = 4x^2 - 11x - 3 $$
* Step 5: Check if the numerator factors to cancel anything.
* Factors of $4 \times -3 = -12$ that add to $-11$ are $-12$ and $1$.
* $4x^2 - 12x + x - 3 = 4x(x - 3) + 1(x - 3) = (4x + 1)(x - 3)$.
* Step 6: Write the final fraction and cancel.
$$ \frac{(4x + 1)(x - 3)}{(x - 3)(x + 1)(x - 2)} $$
Cancel $(x - 3)$.
$$ \frac{4x + 1}{(x + 1)(x - 2)} $$
Final Answer:
1) $\frac{1}{x - 4}$
2) $\frac{x + 6}{x - 5}$
3) $\frac{1}{2(v + 1)}$
4) $\frac{1}{k - 6}$
5) $\frac{x + y}{x - y}$
6) $\frac{2(5a - 8)}{(a - 4)(a - 1)(a - 2)(a + 2)}$
7) $\frac{3x - 1}{x + 1}$
8) $\frac{4x + 1}{(x + 1)(x - 2)}$
Parent Tip: Review the logic above to help your child master the concept of adding and subtracting algebraic expressions worksheet.
