Solve (m)/((m)^(2)-5m+6)+(1)/(2-m)-(2m)/((3-m)(1-m)) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{ m }{ { m }^{ 2 } -5m+6 } + \frac{ 1 }{ 2-m } - \frac{ 2m }{ (3-m)(1-m) }$$

Evaluate

$-\frac{2m-1}{\left(1-m\right)\left(2-m\right)}$

Short Solution Steps

Factor $m^{2}-5m+6$.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(m-3\right)\left(m-2\right)$ and $2-m$ is $\left(m-3\right)\left(m-2\right)$. Multiply $\frac{1}{2-m}$ times $\frac{-\left(m-3\right)}{-\left(m-3\right)}$.

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

Since $\frac{m}{\left(m-3\right)\left(m-2\right)}$ and $\frac{-\left(m-3\right)}{\left(m-3\right)\left(m-2\right)}$ have the same denominator, add them by adding their numerators.

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

Do the multiplications in $m-\left(m-3\right)$.

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

Combine like terms in $m-m+3$.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(m-3\right)\left(m-2\right)$ and $\left(3-m\right)\left(1-m\right)$ is $\left(m-3\right)\left(m-2\right)\left(m-1\right)$. Multiply $\frac{3}{\left(m-3\right)\left(m-2\right)}$ times $\frac{m-1}{m-1}$. Multiply $\frac{2m}{\left(3-m\right)\left(1-m\right)}$ times $\frac{m-2}{m-2}$.

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

Since $\frac{3\left(m-1\right)}{\left(m-3\right)\left(m-2\right)\left(m-1\right)}$ and $\frac{2m\left(m-2\right)}{\left(m-3\right)\left(m-2\right)\left(m-1\right)}$ have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in $3\left(m-1\right)-2m\left(m-2\right)$.

$$\frac{3m-3-2m^{2}+4m}{\left(m-3\right)\left(m-2\right)\left(m-1\right)}$$

Combine like terms in $3m-3-2m^{2}+4m$.

$$\frac{7m-3-2m^{2}}{\left(m-3\right)\left(m-2\right)\left(m-1\right)}$$

Factor the expressions that are not already factored in $\frac{7m-3-2m^{2}}{\left(m-3\right)\left(m-2\right)\left(m-1\right)}$.

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

Extract the negative sign in $3-m$.

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

Cancel out $m-3$ in both numerator and denominator.

$$\frac{-\left(2m-1\right)}{\left(m-2\right)\left(m-1\right)}$$

Expand $\left(m-2\right)\left(m-1\right)$.

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

To find the opposite of $2m-1$, find the opposite of each term.

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

Expand

$-\frac{1-2m}{\left(1-m\right)\left(m-2\right)}$

Short Solution Steps

Factor $m^{2}-5m+6$.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(m-3\right)\left(m-2\right)$ and $2-m$ is $\left(m-3\right)\left(m-2\right)$. Multiply $\frac{1}{2-m}$ times $\frac{-\left(m-3\right)}{-\left(m-3\right)}$.

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

Since $\frac{m}{\left(m-3\right)\left(m-2\right)}$ and $\frac{-\left(m-3\right)}{\left(m-3\right)\left(m-2\right)}$ have the same denominator, add them by adding their numerators.

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

Do the multiplications in $m-\left(m-3\right)$.

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

Combine like terms in $m-m+3$.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(m-3\right)\left(m-2\right)$ and $\left(3-m\right)\left(1-m\right)$ is $\left(m-3\right)\left(m-2\right)\left(m-1\right)$. Multiply $\frac{3}{\left(m-3\right)\left(m-2\right)}$ times $\frac{m-1}{m-1}$. Multiply $\frac{2m}{\left(3-m\right)\left(1-m\right)}$ times $\frac{m-2}{m-2}$.

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

Since $\frac{3\left(m-1\right)}{\left(m-3\right)\left(m-2\right)\left(m-1\right)}$ and $\frac{2m\left(m-2\right)}{\left(m-3\right)\left(m-2\right)\left(m-1\right)}$ have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in $3\left(m-1\right)-2m\left(m-2\right)$.

$$\frac{3m-3-2m^{2}+4m}{\left(m-3\right)\left(m-2\right)\left(m-1\right)}$$

Combine like terms in $3m-3-2m^{2}+4m$.

$$\frac{7m-3-2m^{2}}{\left(m-3\right)\left(m-2\right)\left(m-1\right)}$$

Factor the expressions that are not already factored in $\frac{7m-3-2m^{2}}{\left(m-3\right)\left(m-2\right)\left(m-1\right)}$.

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

Extract the negative sign in $3-m$.

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

Cancel out $m-3$ in both numerator and denominator.

$$\frac{-\left(2m-1\right)}{\left(m-2\right)\left(m-1\right)}$$

Expand $\left(m-2\right)\left(m-1\right)$.

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

To find the opposite of $2m-1$, find the opposite of each term.

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