Adding and Subtracting Rational Expressions Worksheets - Math Monks - Free Printable
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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. $\frac{2z}{z-1} - \frac{3z}{z+1}$
* Find the Common Denominator: The denominators are $(z-1)$ and $(z+1)$. The least common denominator (LCD) is their product: $(z-1)(z+1)$.
* Adjust Fractions: Multiply the top and bottom of the first fraction by $(z+1)$ and the second by $(z-1)$.
$$ \frac{2z(z+1)}{(z-1)(z+1)} - \frac{3z(z-1)}{(z+1)(z-1)} $$
* Combine Numerators:
$$ \frac{2z^2 + 2z - (3z^2 - 3z)}{(z-1)(z+1)} $$
$$ \frac{2z^2 + 2z - 3z^2 + 3z}{(z-1)(z+1)} $$
* Simplify: Combine like terms ($2z^2 - 3z^2 = -z^2$ and $2z + 3z = 5z$).
$$ \frac{-z^2 + 5z}{(z-1)(z+1)} $$
2. $\frac{4x}{x^2-1} - \frac{2}{x} - \frac{2}{x+1}$
* Factor Denominators: $x^2 - 1$ becomes $(x-1)(x+1)$. The other denominators are $x$ and $(x+1)$.
* Find LCD: The LCD is $x(x-1)(x+1)$.
* Adjust Fractions:
* First term needs $x$: $\frac{4x(x)}{x(x-1)(x+1)} = \frac{4x^2}{LCD}$
* Second term needs $(x-1)(x+1) = x^2-1$: $\frac{2(x^2-1)}{LCD} = \frac{2x^2-2}{LCD}$
* Third term needs $x(x-1) = x^2-x$: $\frac{2(x^2-x)}{LCD} = \frac{2x^2-2x}{LCD}$
* Combine Numerators: Be careful with subtraction signs.
$$ 4x^2 - (2x^2 - 2) - (2x^2 - 2x) $$
$$ 4x^2 - 2x^2 + 2 - 2x^2 + 2x $$
* Simplify: $4x^2 - 2x^2 - 2x^2 = 0$. Remaining terms are $2x + 2$.
$$ \frac{2x + 2}{x(x-1)(x+1)} $$
* Final Simplification: Factor out 2 from the top: $2(x+1)$. Cancel $(x+1)$ with the bottom.
$$ \frac{2}{x(x-1)} $$
3. $3 + \frac{t}{t+2} - \frac{2}{t^2-4}$
* Factor Denominators: $t^2 - 4$ becomes $(t-2)(t+2)$. The number 3 can be written as $\frac{3}{1}$.
* Find LCD: The LCD is $(t-2)(t+2)$.
* Adjust Fractions:
* $3$ needs $(t-2)(t+2) = t^2-4$: $\frac{3(t^2-4)}{LCD} = \frac{3t^2-12}{LCD}$
* $\frac{t}{t+2}$ needs $(t-2)$: $\frac{t(t-2)}{LCD} = \frac{t^2-2t}{LCD}$
* Last term stays same: $\frac{2}{LCD}$
* Combine Numerators:
$$ (3t^2 - 12) + (t^2 - 2t) - 2 $$
$$ 4t^2 - 2t - 14 $$
* Result:
$$ \frac{4t^2 - 2t - 14}{(t-2)(t+2)} $$
4. $\frac{4}{y+1} + \frac{y+2}{y^2-1} + \frac{3}{y-1}$
* Factor Denominators: $y^2 - 1$ becomes $(y-1)(y+1)$.
* Find LCD: The LCD is $(y-1)(y+1)$.
* Adjust Fractions:
* First term needs $(y-1)$: $\frac{4(y-1)}{LCD} = \frac{4y-4}{LCD}$
* Middle term is already good: $\frac{y+2}{LCD}$
* Last term needs $(y+1)$: $\frac{3(y+1)}{LCD} = \frac{3y+3}{LCD}$
* Combine Numerators:
$$ (4y - 4) + (y + 2) + (3y + 3) $$
$$ 8y + 1 $$
* Result:
$$ \frac{8y + 1}{(y-1)(y+1)} \quad \text{or} \quad \frac{8y+1}{y^2-1} $$
5. $\frac{p+6}{5p+10} - \frac{p-2}{4p+8}$
* Factor Denominators:
* $5p + 10 = 5(p+2)$
* $4p + 8 = 4(p+2)$
* Find LCD: The LCD is $20(p+2)$.
* Adjust Fractions:
* First term needs 4: $\frac{4(p+6)}{20(p+2)} = \frac{4p+24}{20(p+2)}$
* Second term needs 5: $\frac{5(p-2)}{20(p+2)} = \frac{5p-10}{20(p+2)}$
* Combine Numerators:
$$ (4p + 24) - (5p - 10) $$
$$ 4p + 24 - 5p + 10 $$
$$ -p + 34 $$
* Result:
$$ \frac{34 - p}{20(p+2)} $$
6. $2 + \frac{x-3}{x+1}$
* Find LCD: Treat 2 as $\frac{2}{1}$. The LCD is $(x+1)$.
* Adjust Fractions:
$$ \frac{2(x+1)}{x+1} + \frac{x-3}{x+1} $$
$$ \frac{2x + 2 + x - 3}{x+1} $$
* Simplify:
$$ \frac{3x - 1}{x+1} $$
7. $\frac{5ab}{a^2-b^2} - \frac{a-b}{a+b}$
* Factor Denominators: $a^2 - b^2 = (a-b)(a+b)$.
* Find LCD: The LCD is $(a-b)(a+b)$.
* Adjust Fractions:
* First term is already good.
* Second term needs $(a-b)$: $\frac{(a-b)(a-b)}{(a+b)(a-b)} = \frac{a^2 - 2ab + b^2}{LCD}$
* Combine Numerators:
$$ 5ab - (a^2 - 2ab + b^2) $$
$$ 5ab - a^2 + 2ab - b^2 $$
$$ -a^2 + 7ab - b^2 $$
* Result:
$$ \frac{-a^2 + 7ab - b^2}{(a-b)(a+b)} $$
8. $\frac{7x+4}{x^2+3x+2} + \frac{3x-2}{x^2+3x+2}$
* Check Denominators: They are already the same!
* Combine Numerators:
$$ (7x + 4) + (3x - 2) = 10x + 2 $$
* Current Fraction:
$$ \frac{10x + 2}{x^2 + 3x + 2} $$
* Simplify: Factor both top and bottom to see if anything cancels.
* Top: $2(5x + 1)$
* Bottom: $(x+2)(x+1)$
* Nothing cancels, so we leave it factored or expanded. Usually, factored form is preferred for the denominator.
$$ \frac{2(5x+1)}{(x+2)(x+1)} $$
Final Answer:
1. $\frac{-z^2 + 5z}{(z-1)(z+1)}$
2. $\frac{2}{x(x-1)}$
3. $\frac{4t^2 - 2t - 14}{(t-2)(t+2)}$
4. $\frac{8y + 1}{(y-1)(y+1)}$
5. $\frac{34 - p}{20(p+2)}$
6. $\frac{3x - 1}{x+1}$
7. $\frac{-a^2 + 7ab - b^2}{(a-b)(a+b)}$
8. $\frac{2(5x+1)}{(x+2)(x+1)}$
1. $\frac{2z}{z-1} - \frac{3z}{z+1}$
* Find the Common Denominator: The denominators are $(z-1)$ and $(z+1)$. The least common denominator (LCD) is their product: $(z-1)(z+1)$.
* Adjust Fractions: Multiply the top and bottom of the first fraction by $(z+1)$ and the second by $(z-1)$.
$$ \frac{2z(z+1)}{(z-1)(z+1)} - \frac{3z(z-1)}{(z+1)(z-1)} $$
* Combine Numerators:
$$ \frac{2z^2 + 2z - (3z^2 - 3z)}{(z-1)(z+1)} $$
$$ \frac{2z^2 + 2z - 3z^2 + 3z}{(z-1)(z+1)} $$
* Simplify: Combine like terms ($2z^2 - 3z^2 = -z^2$ and $2z + 3z = 5z$).
$$ \frac{-z^2 + 5z}{(z-1)(z+1)} $$
2. $\frac{4x}{x^2-1} - \frac{2}{x} - \frac{2}{x+1}$
* Factor Denominators: $x^2 - 1$ becomes $(x-1)(x+1)$. The other denominators are $x$ and $(x+1)$.
* Find LCD: The LCD is $x(x-1)(x+1)$.
* Adjust Fractions:
* First term needs $x$: $\frac{4x(x)}{x(x-1)(x+1)} = \frac{4x^2}{LCD}$
* Second term needs $(x-1)(x+1) = x^2-1$: $\frac{2(x^2-1)}{LCD} = \frac{2x^2-2}{LCD}$
* Third term needs $x(x-1) = x^2-x$: $\frac{2(x^2-x)}{LCD} = \frac{2x^2-2x}{LCD}$
* Combine Numerators: Be careful with subtraction signs.
$$ 4x^2 - (2x^2 - 2) - (2x^2 - 2x) $$
$$ 4x^2 - 2x^2 + 2 - 2x^2 + 2x $$
* Simplify: $4x^2 - 2x^2 - 2x^2 = 0$. Remaining terms are $2x + 2$.
$$ \frac{2x + 2}{x(x-1)(x+1)} $$
* Final Simplification: Factor out 2 from the top: $2(x+1)$. Cancel $(x+1)$ with the bottom.
$$ \frac{2}{x(x-1)} $$
3. $3 + \frac{t}{t+2} - \frac{2}{t^2-4}$
* Factor Denominators: $t^2 - 4$ becomes $(t-2)(t+2)$. The number 3 can be written as $\frac{3}{1}$.
* Find LCD: The LCD is $(t-2)(t+2)$.
* Adjust Fractions:
* $3$ needs $(t-2)(t+2) = t^2-4$: $\frac{3(t^2-4)}{LCD} = \frac{3t^2-12}{LCD}$
* $\frac{t}{t+2}$ needs $(t-2)$: $\frac{t(t-2)}{LCD} = \frac{t^2-2t}{LCD}$
* Last term stays same: $\frac{2}{LCD}$
* Combine Numerators:
$$ (3t^2 - 12) + (t^2 - 2t) - 2 $$
$$ 4t^2 - 2t - 14 $$
* Result:
$$ \frac{4t^2 - 2t - 14}{(t-2)(t+2)} $$
4. $\frac{4}{y+1} + \frac{y+2}{y^2-1} + \frac{3}{y-1}$
* Factor Denominators: $y^2 - 1$ becomes $(y-1)(y+1)$.
* Find LCD: The LCD is $(y-1)(y+1)$.
* Adjust Fractions:
* First term needs $(y-1)$: $\frac{4(y-1)}{LCD} = \frac{4y-4}{LCD}$
* Middle term is already good: $\frac{y+2}{LCD}$
* Last term needs $(y+1)$: $\frac{3(y+1)}{LCD} = \frac{3y+3}{LCD}$
* Combine Numerators:
$$ (4y - 4) + (y + 2) + (3y + 3) $$
$$ 8y + 1 $$
* Result:
$$ \frac{8y + 1}{(y-1)(y+1)} \quad \text{or} \quad \frac{8y+1}{y^2-1} $$
5. $\frac{p+6}{5p+10} - \frac{p-2}{4p+8}$
* Factor Denominators:
* $5p + 10 = 5(p+2)$
* $4p + 8 = 4(p+2)$
* Find LCD: The LCD is $20(p+2)$.
* Adjust Fractions:
* First term needs 4: $\frac{4(p+6)}{20(p+2)} = \frac{4p+24}{20(p+2)}$
* Second term needs 5: $\frac{5(p-2)}{20(p+2)} = \frac{5p-10}{20(p+2)}$
* Combine Numerators:
$$ (4p + 24) - (5p - 10) $$
$$ 4p + 24 - 5p + 10 $$
$$ -p + 34 $$
* Result:
$$ \frac{34 - p}{20(p+2)} $$
6. $2 + \frac{x-3}{x+1}$
* Find LCD: Treat 2 as $\frac{2}{1}$. The LCD is $(x+1)$.
* Adjust Fractions:
$$ \frac{2(x+1)}{x+1} + \frac{x-3}{x+1} $$
$$ \frac{2x + 2 + x - 3}{x+1} $$
* Simplify:
$$ \frac{3x - 1}{x+1} $$
7. $\frac{5ab}{a^2-b^2} - \frac{a-b}{a+b}$
* Factor Denominators: $a^2 - b^2 = (a-b)(a+b)$.
* Find LCD: The LCD is $(a-b)(a+b)$.
* Adjust Fractions:
* First term is already good.
* Second term needs $(a-b)$: $\frac{(a-b)(a-b)}{(a+b)(a-b)} = \frac{a^2 - 2ab + b^2}{LCD}$
* Combine Numerators:
$$ 5ab - (a^2 - 2ab + b^2) $$
$$ 5ab - a^2 + 2ab - b^2 $$
$$ -a^2 + 7ab - b^2 $$
* Result:
$$ \frac{-a^2 + 7ab - b^2}{(a-b)(a+b)} $$
8. $\frac{7x+4}{x^2+3x+2} + \frac{3x-2}{x^2+3x+2}$
* Check Denominators: They are already the same!
* Combine Numerators:
$$ (7x + 4) + (3x - 2) = 10x + 2 $$
* Current Fraction:
$$ \frac{10x + 2}{x^2 + 3x + 2} $$
* Simplify: Factor both top and bottom to see if anything cancels.
* Top: $2(5x + 1)$
* Bottom: $(x+2)(x+1)$
* Nothing cancels, so we leave it factored or expanded. Usually, factored form is preferred for the denominator.
$$ \frac{2(5x+1)}{(x+2)(x+1)} $$
Final Answer:
1. $\frac{-z^2 + 5z}{(z-1)(z+1)}$
2. $\frac{2}{x(x-1)}$
3. $\frac{4t^2 - 2t - 14}{(t-2)(t+2)}$
4. $\frac{8y + 1}{(y-1)(y+1)}$
5. $\frac{34 - p}{20(p+2)}$
6. $\frac{3x - 1}{x+1}$
7. $\frac{-a^2 + 7ab - b^2}{(a-b)(a+b)}$
8. $\frac{2(5x+1)}{(x+2)(x+1)}$
Parent Tip: Review the logic above to help your child master the concept of operations with rational expressions worksheet.
