Question

adding and subtracting rational expressions
rational expressions are similar when they share a common denominator. addition and subtraction of such expressions can be performed by combining the numerators with the given operation/s while keeping the common denominator.
\\(\frac{a - 3}{a^2 - 1}+\frac{2}{a^2 - 1}\\) \\(\frac{x}{2x + 6}+\frac{3x}{2x + 6}\\) \\(\frac{2n - 2}{n^2 + n - 6}-\frac{n - 5}{n^2 + n - 6}\\) \\(\frac{2y^2 + y}{y^2 - 1}-\frac{y^2 + 2y}{y^2 - 1}\\)

addition and subtraction of rational expressions with different denominators involve converting each expression into equivalent rational expressions with the least common denominator.
\\(\frac{3}{x - 5}+\frac{x}{x + 2}\\) \\(\frac{m}{m + 3}-\frac{2}{m - 1}\\) \\(\frac{4 - 2p}{p^2 - p - 2}+\frac{p + 2}{p + 1}\\)
\\(\frac{4 - a}{a - 3}-\frac{a - 3}{a^2 - 6a + 9}\\) \\(\frac{3}{u^2 + 7u + 10}-\frac{2}{u^2 + 8u + 15}\\) \\(\frac{1}{n^2 - 1}+\frac{3}{n^2 - 4n + 3}\\)
\\(\frac{8}{y + 9}-\frac{9}{3y + 27}\\) \\(\frac{1}{b + 1}+\frac{2}{b^2 - 1}\\) \\(\frac{10}{x - 5}-\frac{2x^2 + 4x}{x^2 - 3x - 10}\\)

Explanation:

Section 1: Common Denominators

Step1: Combine numerators, keep denominator

$\frac{n-3 + 2}{n^2 - 1}$

Step2: Simplify numerator

$\frac{n - 1}{n^2 - 1}$

Step3: Factor denominator, cancel terms

$\frac{n - 1}{(n-1)(n+1)} = \frac{1}{n+1}$

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Step1: Combine numerators, keep denominator

$\frac{x + 3x}{2x + 6}$

Step2: Simplify numerator, factor denominator

$\frac{4x}{2(x+3)}$

Step3: Reduce fraction

$\frac{2x}{x+3}$

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Step1: Combine numerators, keep denominator

$\frac{(2n-2) - (n-5)}{n^2 + n - 6}$

Step2: Simplify numerator

$\frac{2n - 2 - n + 5}{n^2 + n - 6} = \frac{n + 3}{n^2 + n - 6}$

Step3: Factor denominator, cancel terms

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

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Step1: Combine numerators, keep denominator

$\frac{(2y^2 + y) - (y^2 + 2y)}{y^2 - 1}$

Step2: Simplify numerator

$\frac{2y^2 + y - y^2 - 2y}{y^2 - 1} = \frac{y^2 - y}{y^2 - 1}$

Step3: Factor, cancel common terms

$\frac{y(y-1)}{(y-1)(y+1)} = \frac{y}{y+1}$

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Section 2: Different Denominators

Step1: Find LCD: $(x-5)(x+2)$

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

Step2: Expand numerator

$\frac{3x + 6 + x^2 - 5x}{(x-5)(x+2)}$

Step3: Simplify numerator

$\frac{x^2 - 2x + 6}{(x-5)(x+2)}$

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Step1: Find LCD: $(m+3)(m-1)$

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

Step2: Expand numerator

$\frac{m^2 - m - 2m - 6}{(m+3)(m-1)}$

Step3: Simplify numerator

$\frac{m^2 - 3m - 6}{(m+3)(m-1)}$

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Step1: Factor denominator, find LCD: $(p-2)(p+1)$

$\frac{4-2p + (p+2)(p-2)}{(p-2)(p+1)}$

Step2: Expand and simplify numerator

$\frac{4-2p + p^2 - 4}{(p-2)(p+1)} = \frac{p^2 - 2p}{(p-2)(p+1)}$

Step3: Factor numerator, cancel terms

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

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Step1: Factor denominator, find LCD: $(a-3)^2$

$\frac{(4-a)(a-3) - (a-3)}{(a-3)^2}$

Step2: Factor out $(a-3)$ in numerator

$\frac{(a-3)(4-a - 1)}{(a-3)^2}$

Step3: Simplify, cancel terms

$\frac{3 - a}{a-3} = -1$

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Step1: Factor denominators, find LCD: $(u+2)(u+3)(u+5)$

$\frac{3(u+3) - 2(u+2)}{(u+2)(u+3)(u+5)}$

Step2: Expand and simplify numerator

$\frac{3u + 9 - 2u - 4}{(u+2)(u+3)(u+5)} = \frac{u + 5}{(u+2)(u+3)(u+5)}$

Step3: Cancel terms

$\frac{1}{(u+2)(u+3)}$

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Step1: Factor denominators, find LCD: $(n-1)(n+1)(n-3)$

$\frac{1(n-3) + 3(n+1)}{(n-1)(n+1)(n-3)}$

Step2: Expand and simplify numerator

$\frac{n - 3 + 3n + 3}{(n-1)(n+1)(n-3)} = \frac{4n}{(n-1)(n+1)(n-3)}$

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Step1: Factor denominator, find LCD: $3(y+9)$

$\frac{8(3) - 9}{3(y+9)}$

Step2: Simplify numerator

$\frac{24 - 9}{3(y+9)} = \frac{15}{3(y+9)}$

Step3: Reduce fraction

$\frac{5}{y+9}$

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Step1: Factor denominator, find LCD: $(b-1)(b+1)$

$\frac{1(b-1) + 2}{(b-1)(b+1)}$

Step2: Simplify numerator

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

Step3: Cancel terms

$\frac{1}{b-1}$

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Step1: Factor denominator, find LCD: $(x-5)(x+2)$

$\frac{10(x+2) - (2x^2 + 4x)}{(x-5)(x+2)}$

Step2: Expand and simplify numerator

$\frac{10x + 20 - 2x^2 - 4x}{(x-5)(x+2)} = \frac{-2x^2 + 6x + 20}{(x-5)(x+2)}$

Step3: Factor numerator, cancel terms

$\frac{-2(x^2 - 3x - 10)}{(x-5)(x+2)} = \frac{-2(x-5)(x+2)}{(x-5)(x+2)} = -2$

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Answer:

s:

1. $\frac{1}{n+1}$
2. $\frac{2x}{x+3}$
3. $\frac{1}{n-2}$
4. $\frac{y}{y+1}$
5. $\frac{x^2 - 2x + 6}{(x-5)(x+2)}$
6. $\frac{m^2 - 3m - 6}{(m+3)(m-1)}$
7. $\frac{p}{p+1}$
8. $-1$
9. $\frac{1}{(u+2)(u+3)}$
10. $\frac{4n}{(n-1)(n+1)(n-3)}$
11. $\frac{5}{y+9}$
12. $\frac{1}{b-1}$
13. $-2$