What is (a-b)/(a^2+ab)+(a+b)/(ab)-a/(ab+b^2) ?

The solution to (a-b)/(a^2+ab)+(a+b)/(ab)-a/(ab+b^2) is 3/(b+a)

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simplify (a-b)/(a^2+ab)+(a+b)/(ab)-a/(ab+b^2)

simplify

\frac{a - b}{a ^{2} + ab} + \frac{a + b}{ab} - \frac{a}{ab + b ^{2}}

\frac{3}{b + a}

Solution steps

\frac{a - b}{a ^{2} + ab} + \frac{a + b}{ab} - \frac{a}{ab + b ^{2}}

Least Common Multiplier of

a^{2} + ab, ab, ab + b^{2} : ab (a^{2} + ab) (b^{2} + ab)

Adjust Fractions based on the LCM

= \frac{( a - b ) ab ( b ^{2} + ab )}{ab ( a ^{2} + ab ) ( b ^{2} + ab )} + \frac{( a + b ) ( a ^{2} + ab ) ( b ^{2} + ab )}{ab ( a ^{2} + ab ) ( b ^{2} + ab )} - \frac{a ^{2} b ( a ^{2} + ab )}{ab ( a ^{2} + ab ) ( b ^{2} + ab )}

Apply the fraction rule:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{( a - b ) ab ( b ^{2} + ab ) + ( a + b ) ( a ^{2} + ab ) ( b ^{2} + ab ) - a ^{2} b ( a ^{2} + ab )}{ab ( a ^{2} + ab ) ( b ^{2} + ab )}

Simplify

(a - b) ab (b^{2} + ab) + (a + b) (a^{2} + ab) (b^{2} + ab) - a^{2} b (a^{2} + ab) : 3 a^{3} b^{2} + 3 a^{2} b^{3}

= \frac{3 a ^{3} b ^{2} + 3 a ^{2} b ^{3}}{ab ( a ^{2} + ab ) ( b ^{2} + ab )}

Cancel

\frac{3 a ^{3} b ^{2} + 3 a ^{2} b ^{3}}{ab ( a ^{2} + ab ) ( b ^{2} + ab )} : \frac{3}{b + a}

= \frac{3}{b + a}

3 - \frac{5 x - 1}{10} = \frac{x - 1}{5} - \frac{x - 3}{2}

- x^{3} + 6 x^{2} - 12 x + 9 < 0

3 t^{2} + 4 t + 8 = - 6 t

- 42 \geq 40 + 41 c

\frac{1}{e ^{- 2}}

What is (a-b)/(a^2+ab)+(a+b)/(ab)-a/(ab+b^2) ?
The solution to (a-b)/(a^2+ab)+(a+b)/(ab)-a/(ab+b^2) is 3/(b+a)