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$\mathbf{\text{1)}}$ ${\displaystyle \frac{5}{x}+\frac{x^2}{7}}$

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The answer is ${\displaystyle \frac{x^3+35}{7x}}$

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${\displaystyle \frac{5}{x}+\frac{x^2}{7}}$


${\displaystyle \frac{7}{7}\cdot \frac{5}{x}+\frac{x^2}{7}\cdot \frac{x}{x}}$


${\displaystyle \frac{35}{7x}+\frac{x^3}{7x}}$


${\displaystyle \frac{x^3+35}{7x}}$

$\mathbf{\text{2)}}$ ${\displaystyle \frac{x-5}{x+3}+\frac{x+2}{x^2+5x+6}}$

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The answer is ${\displaystyle \frac{x-4}{x+3}}$

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


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


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


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


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

${\displaystyle \frac{x^2+6x+8}{(x+2)(x+3)}}$


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


${\displaystyle \frac{x-4}{x+3}}$

$\mathbf{\text{3)}}$ ${\displaystyle \frac{5}{x+3}+\frac{x}{x^2-9}}$

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The answer is ${\displaystyle \frac{3(2x-5)}{(x-3)(x+3)}}$

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${\displaystyle \frac{5}{x+3}+\frac{x}{x^2-9}}$


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


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


${\displaystyle \frac{5x-15}{(x-3)(x+3)}+\frac{x}{(x-3)(x+3)}}$


${\displaystyle \frac{6x-15}{(x-3)(x+3)}}$


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

$\mathbf{\text{4)}}$ ${\displaystyle \frac{5}{x-2}-\frac{8}{2-x}}$

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The answer is ${\displaystyle \frac{13}{x-2}}$

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${\displaystyle \frac{5}{x-2}-\frac{8}{2-x}}$


${\displaystyle \frac{5}{x-2}-\frac{8}{-(x-2)}}$


${\displaystyle \frac{5}{x-2}+\frac{8}{(x-2)}}$


${\displaystyle \frac{13}{x-2}}$

$\mathbf{\text{5)}}$ ${\displaystyle \frac{1}{2x}+\frac{x}{3x}+\frac{3}{4x}}$

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The answer is ${\displaystyle \frac{4x+15}{12x}}$

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${\displaystyle \frac{1}{2x}+\frac{x}{3x}+\frac{3}{4x}}$


${\displaystyle \frac{1}{2x}\cdot \frac{6}{6}+\frac{x}{3x}\cdot \frac{4}{4}+\frac{3}{4x}\cdot \frac{3}{3}}$


${\displaystyle \frac{6}{12x}+\frac{4x}{12x}+\frac{9}{12x}}$


${\displaystyle \frac{6+4x+9}{12x}}$


${\displaystyle \frac{4x+15}{12x}}$

$\mathbf{\text{6)}}$ ${\displaystyle \frac{5}{x+3}+4}$

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The answer is ${\displaystyle \frac{4x+17}{x+3}}$

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${\displaystyle \frac{5}{x+3}+4}$


${\displaystyle \frac{5}{x+3}+\frac{4}{1}\cdot \frac{x+3}{x+3}}$


${\displaystyle \frac{5}{x+3}+\frac{4x+12}{x+3}}$


${\displaystyle \frac{4x+17}{x+3}}$

$\mathbf{\text{7)}}$ ${\displaystyle \frac{x}{x+5}+\frac{3x+1}{x+2}}$

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The answer is ${\displaystyle \frac{4x^2+18x+5}{(x+2)(x+5)}}$

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${\displaystyle \frac{x}{x+5}+\frac{3x+1}{x+2}}$


${\displaystyle \frac{x+2}{x+2}\cdot \frac{x}{x+5}+\frac{3x+1}{x+2}\cdot \frac{x+5}{x+5}}$


${\displaystyle \frac{x^2+2x}{(x+2)(x+5)}+\frac{(3x+1)(x+5)}{(x+2)(x+5)}}$


${\displaystyle \frac{x^2+2x}{(x+2)(x+5)}+\frac{3x^2+16x+5}{(x+2)(x+5)}}$


${\displaystyle \frac{4x^2+18x+5}{(x+2)(x+5)}}$

$\mathbf{\text{8)}}$ ${\displaystyle \frac{2x+5}{x^2+3x+2}+\frac{x-1}{x^2+7x+10}}$

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The answer is ${\displaystyle \frac{3x^2+15x+24}{(x+5)(x+1)(x+2)}}$

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${\displaystyle \frac{2x+5}{x^2+3x+2}+\frac{x-1}{x^2+7x+10}}$


${\displaystyle \frac{2x+5}{(x+1)(x+2)}+\frac{x-1}{(x+5)(x+2)}}$


${\displaystyle \frac{x+5}{x+5}\cdot \frac{2x+5}{(x+1)(x+2)}+\frac{x-1}{(x+5)(x+2)}\cdot \frac{x+1}{x+1}}$


${\displaystyle \frac{(x+5)(2x+5)}{(x+5)(x+1)(x+2)}+\frac{(x-1)(x+1)}{(x+5)(x+2)(x+1)}}$


${\displaystyle \frac{2x^2+15x+25}{(x+5)(x+1)(x+2)}+\frac{x^2-1}{(x+5)(x+2)(x+1)}}$


${\displaystyle \frac{3x^2+15x+24}{(x+5)(x+1)(x+2)}}$


${\displaystyle \frac{3x^2+15x+24}{(x+5)(x+1)(x+2)}}$

$\mathbf{\text{9)}}$ ${\displaystyle \frac{x+7}{x^2+4x+4}+\frac{x+1}{x^2-4}}$

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The answer is ${\displaystyle \frac{2(x-8)}{(x-2)(x+2{)}^2}}$

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${\displaystyle \frac{x+7}{x^2+4x+4}+\frac{x+1}{x^2-4}}$


${\displaystyle \frac{x+7}{(x+2)(x+2)}+\frac{x+1}{(x+2)(x-2)}}$


${\displaystyle \frac{x-2}{x-2}\cdot \frac{x+7}{(x+2)(x+2)}+\frac{x+1}{(x+2)(x-2)}\cdot \frac{x+2}{x+2}}$


${\displaystyle \frac{x^2+5x-14}{(x-2){(x+2)}^2}+\frac{x^2+3x+2}{(x-2){(x+2)}^2}}$


${\displaystyle \frac{2x^2+8x-12}{(x-2){(x+2)}^2}+\frac{x^2+3x+2}{(x-2){(x+2)}^2}}$

$\mathbf{\text{10)}}$ ${\displaystyle \frac{2x}{x+2}+\frac{4}{x+2}}$

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The answer is $2$

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${\displaystyle \frac{2x}{x+2}+\frac{4}{x+2}}$


${\displaystyle \frac{2x+4}{x+2}}$


${\displaystyle \frac{2(x+2)}{x+2}}$


$2$

$\mathbf{\text{11)}}$ ${\displaystyle \frac{3x+1}{{(x+2)}^2}+\frac{3}{x+2}}$

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The answer is ${\displaystyle \frac{6x+7}{(x+2{)}^2}}$

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${\displaystyle \frac{3x+1}{(x+2{)}^2}+\frac{3}{x+2}}$


${\displaystyle \frac{3x+1}{(x+2{)}^2}+\frac{3}{x+2}\cdot \frac{x+2}{x+2}}$


${\displaystyle \frac{3x+1}{{(x+2)}^2}+\frac{3x+6}{{(x+2)}^2}}$


${\displaystyle \frac{6x+7}{(x+2{)}^2}}$

$\mathbf{\text{12)}}$ ${\displaystyle \frac{4}{x}+\frac{3}{x+1}}$

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The answer is ${\displaystyle \frac{7x+4}{x(x+1)}}$

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${\displaystyle \frac{4}{x}+\frac{3}{x+1}}$


${\displaystyle \frac{x+1}{x+1}\cdot \frac{4}{x}+\frac{3}{x+1}\cdot \frac{x}{x}}$


${\displaystyle \frac{4x+4}{x(x+1)}+\frac{3x}{x(x+1)}}$


${\displaystyle \frac{7x+4}{x(x+1)}}$

See Related Pages$$

$\bullet \text{ Rational Expression Calculator }$
$\text{(Symbolab.com)}$

$\bullet \text{ Ratios and Proportions}$
${\displaystyle \frac{4}{3}=\frac{d-4}{12}\dots }$

$\bullet \text{ Rational Expressions- Multiplying and Dividing}$
${\displaystyle \frac{x^2+3x-4}{(x+4)(x+5)}\cdot {\displaystyle \frac{x+5}{x-1}\dots }}$

$\bullet \text{ Rational Expressions- Adding and Subtracting}$
${\displaystyle \frac{x-5}{x+3}+\frac{x+2}{x^2+5x+6}\dots }$

$\bullet \text{ Direct, Inverse, and Joint Variation}$
$$ $\dots$

$\bullet \text{ Complex Fractions}$
${\displaystyle \frac{\frac{x}{5}+\frac{1}{3}}{\frac{1}{5}-\frac{1}{6}}\dots }$

$\bullet \text{ Partial Fraction Decomposition}$
${\displaystyle \frac{8x+10}{x^2+2x}={\displaystyle \frac{5}{x}+\frac{3}{x+2}\dots }}$

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