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-\frac{1}{3x}
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-\frac{1}{3x}
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\frac{3x+1}{2x\left(x-3\right)}-\frac{x+2}{3\left(x-3\right)}+\frac{2x-1}{6x}
Factor 2x^{2}-6x. Factor 3x-9.
\frac{3\left(3x+1\right)}{6x\left(x-3\right)}-\frac{\left(x+2\right)\times 2x}{6x\left(x-3\right)}+\frac{2x-1}{6x}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x\left(x-3\right) and 3\left(x-3\right) is 6x\left(x-3\right). Multiply \frac{3x+1}{2x\left(x-3\right)} times \frac{3}{3}. Multiply \frac{x+2}{3\left(x-3\right)} times \frac{2x}{2x}.
\frac{3\left(3x+1\right)-\left(x+2\right)\times 2x}{6x\left(x-3\right)}+\frac{2x-1}{6x}
Since \frac{3\left(3x+1\right)}{6x\left(x-3\right)} and \frac{\left(x+2\right)\times 2x}{6x\left(x-3\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{9x+3-2x^{2}-4x}{6x\left(x-3\right)}+\frac{2x-1}{6x}
Do the multiplications in 3\left(3x+1\right)-\left(x+2\right)\times 2x.
\frac{5x+3-2x^{2}}{6x\left(x-3\right)}+\frac{2x-1}{6x}
Combine like terms in 9x+3-2x^{2}-4x.
\frac{\left(-x+3\right)\left(2x+1\right)}{6x\left(x-3\right)}+\frac{2x-1}{6x}
Factor the expressions that are not already factored in \frac{5x+3-2x^{2}}{6x\left(x-3\right)}.
\frac{-\left(x-3\right)\left(2x+1\right)}{6x\left(x-3\right)}+\frac{2x-1}{6x}
Extract the negative sign in 3-x.
\frac{-\left(2x+1\right)}{6x}+\frac{2x-1}{6x}
Cancel out x-3 in both numerator and denominator.
\frac{-\left(2x+1\right)+2x-1}{6x}
Since \frac{-\left(2x+1\right)}{6x} and \frac{2x-1}{6x} have the same denominator, add them by adding their numerators.
\frac{-2x-1+2x-1}{6x}
Do the multiplications in -\left(2x+1\right)+2x-1.
\frac{-2}{6x}
Combine like terms in -2x-1+2x-1.
\frac{3x+1}{2x\left(x-3\right)}-\frac{x+2}{3\left(x-3\right)}+\frac{2x-1}{6x}
Factor 2x^{2}-6x. Factor 3x-9.
\frac{3\left(3x+1\right)}{6x\left(x-3\right)}-\frac{\left(x+2\right)\times 2x}{6x\left(x-3\right)}+\frac{2x-1}{6x}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x\left(x-3\right) and 3\left(x-3\right) is 6x\left(x-3\right). Multiply \frac{3x+1}{2x\left(x-3\right)} times \frac{3}{3}. Multiply \frac{x+2}{3\left(x-3\right)} times \frac{2x}{2x}.
\frac{3\left(3x+1\right)-\left(x+2\right)\times 2x}{6x\left(x-3\right)}+\frac{2x-1}{6x}
Since \frac{3\left(3x+1\right)}{6x\left(x-3\right)} and \frac{\left(x+2\right)\times 2x}{6x\left(x-3\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{9x+3-2x^{2}-4x}{6x\left(x-3\right)}+\frac{2x-1}{6x}
Do the multiplications in 3\left(3x+1\right)-\left(x+2\right)\times 2x.
\frac{5x+3-2x^{2}}{6x\left(x-3\right)}+\frac{2x-1}{6x}
Combine like terms in 9x+3-2x^{2}-4x.
\frac{\left(-x+3\right)\left(2x+1\right)}{6x\left(x-3\right)}+\frac{2x-1}{6x}
Factor the expressions that are not already factored in \frac{5x+3-2x^{2}}{6x\left(x-3\right)}.
\frac{-\left(x-3\right)\left(2x+1\right)}{6x\left(x-3\right)}+\frac{2x-1}{6x}
Extract the negative sign in 3-x.
\frac{-\left(2x+1\right)}{6x}+\frac{2x-1}{6x}
Cancel out x-3 in both numerator and denominator.
\frac{-\left(2x+1\right)+2x-1}{6x}
Since \frac{-\left(2x+1\right)}{6x} and \frac{2x-1}{6x} have the same denominator, add them by adding their numerators.
\frac{-2x-1+2x-1}{6x}
Do the multiplications in -\left(2x+1\right)+2x-1.
\frac{-2}{6x}
Combine like terms in -2x-1+2x-1.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}